DEVELOPING A GENERAL DESCRIPTION OF THE UNFOLDED STATE OF A PROTEIN
By
Yujie Chen

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Physics
2011

ABSTRACT
DEVELOPING A GENERAL DESCRIPTION OF THE UNFOLDED STATE OF A PROTEIN
By
Yujie Chen
This thesis focuses on both the dynamical and structural aspects of the unfolded states of a
protein and develops a general description of these unfolded states over all denaturing conditions.
Investigating the unfolded states under physiological conditions can provide insights into the
entire folding pathway as they define the most likely conformations before the folding transition.
Also, since the initial motion of a protein in solvent early in the folding process is driven by
diffusion, the intramolecular diffusion rate may set an upper speed limit for protein folding.
In this thesis, the experimental measurement of the intramolecular diffusion is achieved by a
pump-probe spectroscopy using the technique of Trp/Cys contact quenching. Three typical
proteins in different folding categories have been studied: the intrinsically disordered
apocytochrome C, the aggregation-prone HypF-N and the well-folded ACBP (Acyl-coenzyme
A-binding Protein). For ACBP, I also used a microfluidic mixer coupled with the pump-probe
spectroscopy to study the intramolecular loop contact formation in its refolding process.
In order to extract the effective diffusion coefficient from the observed contact rates, I
employed a polymer theory by Szabo, Schulten and Schulten (SSS) which requires a probability
distribution of equilibrium Trp/Cys distances. In an attempt to capture the transient residual
structures and intramolecular interactions of unfolded polypeptides under folding conditions, I
integrated sequence specificities into a traditional non-overlapping worm-like chain model. The

new model can statistically re-weight the distribution to favor those conformations with more
hydrophobic contacts and it yields quite convergent results over many tested sequences. In the
thesis, I will also show that this model quantitatively predicted paramagnetic resonance
enhancement (PRE) measurements of ACBP and DrkN with no adjustable parameters.
After applying the SSS theory with the Trp/Cys distance distributions produced by the energy
re-weighted worm-like chain model to calculating the intramolecular diffusion coefficients of
those experimentally studied proteins, I found an intrinsic relationship between the
intramolecular diffusion rates and the aggregation propensities and I propose a dynamic range of
conformational reorganization within which partially or fully unfolded states are prone to
aggregation.

ACKNOWLEDGEMENTS

First, I would like to thank my advisor, Dr. Lisa Lapidus, for her guidance and support
through my graduate years. She has been a mentor, a friend, and a great inspiration to my
researches. She always does her best to provide me opportunities to explore the field of
experimental biophysics. Because of her trust and unselfishness, I was very thankful and honored
having been able to work on several promising projects and to work with so many talented
colleagues and collaborators. She has also given me enough freedom to be adventurous and
creative, which led me to another level as an independent researcher. On a personal level, I’m
especially thankful for her kindness and genuine friendship.

It has truly been a blessing

working with her for the past five years.
I would like to thank Dr. William Wedemeyer for his assistance in my computational projects.
His intelligence and broad knowledge in molecular biology were great inspirations to me.
I would also like to thank Terry Ball, who made almost all the protein samples for my
projects. Without him, none of these works would be possible.
Thanks to all the group members: Vijay Singh, Steven Waldauer, Stephen Decamp, Michael
King, Eric Buscarino, Li Zhu, and Basir Ahmad. It’s been a wonderful experience working with
you all. Thank you for your friendship and great memories.
Finally, I would like to thank my parents, who have always been there supporting me with
every decision I made. Special thanks to my lovely wife, who always give me honest yet
constructive opinions. Her love and support was the driving force behind my accomplishments.

iv

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................................ vii
LIST OF FIGURES ..................................................................................................................... viii
1

Introduction to Protein Folding, Misfolding and Aggregation ............................................... 1
1.1 Amino Acids, Peptides and Proteins .................................................................................. 1
1.2 Protein Folding: Mechanics, Energy and Time Scale ........................................................ 2
1.3 Introduction to Experimental Approaches ......................................................................... 9
1.4 The Importance of Studying the Unfolded States .............................................................11
1.5 Polymer Models and Computational Methods ................................................................ 12

2

Spectroscopic Study of the Dynamics of Unfolded Proteins................................................ 16
2.1 Introduction...................................................................................................................... 16
2.2 Monitoring Tryptophan Triplet Quenching Using Pump-probe Spectroscopy................ 18
2.3 Reaction-limited and Diffusion-limited Rates ................................................................. 24
2.4 Data Analysis Using SSS theory and a Wormlike Chain Model ..................................... 26
2.5 Tryptophan Triplet Quenching Experiments in a Microfluidic Mixer............................. 28

3

Experiments on Several Different Proteins Using Trp/Cys Quenching Spectroscopy ......... 33
3.1 Conformational properties of unfolded HypF.................................................................. 33
3.2 Intramolecular Loop Formation of Intrinsically Unfolded Proteins ................................ 45
3.3 Diffusion of Unfolded Acyl-coenzyme A-binding Protein over a Complete Range of
Denaturant.............................................................................................................................. 53

4

Sequence Specific Wormlike Chain Model .......................................................................... 58
4.1 Introduction to the Heteropolymer Model ....................................................................... 58
4.2 Boltzmann Energy Reweighting ...................................................................................... 61
4.3 Secondary Structure Propensity (SSP)............................................................................. 66
4.4 Convergence of σ ............................................................................................................. 68
4.5 Intrachain Contacts and Hydrophobic Energy................................................................. 71
4.6 Comparison between Two Protein L Mutants.................................................................. 76
4.7 PRE Prediction................................................................................................................. 79
4.8 Charge Effects.................................................................................................................. 82
4.9 Conclusion ....................................................................................................................... 86

5

Conclusions........................................................................................................................... 89

v

BIBLIOGRAPHY......................................................................................................................... 98

vi

LIST OF TABLES

Table 3.1 Observed rates of tryptophan triplet decay of W81F at various times after dilution in
3% TFE ......................................................................................................................................... 45
Table 3.2 Wormlike chain parameters and effective diffusion coefficients for both mutants
studied ........................................................................................................................................... 45
Table 3.3 Wormlike chain parameters and effective diffusion coefficients for apocytochrome C 47
Table 4.1 Energy-reweighted wormlike chain parameters and effective diffusion coefficients for
drkN .............................................................................................................................................. 68
Table 4.2 Energy reweighted worm-like chain parameters and effective diffusion coefficients for
AavLEA without charge ............................................................................................................... 83
Table 4.3 reweighting with both σ and γ for AavLEA T58C ........................................................ 83
Table 4.4 reweighting with only γ for AavLEA S2C .................................................................... 85
Table 5.1 Several protein sequences studied in our lab ................................................................ 90

vii

LIST OF FIGURES

Figure 1.1 Two amino acids form a peptide bond between the carboxyl group and the amino
group. (http://upload.wikimedia.org/wikipedia/commons/6/6d/Peptidformationball.svg) ............ 2
Figure 1.2 Twenty standard amino acid classified by their polarity (hydrophobicity) ................... 3
Figure 1.3 cartoon of the HypF-N protein showing (1) α-helix (red), (2) β-strand (blue), (3)
tryptophan 27 and 81 (purple) and (4) cysteine 65 (yellow) .......................................................... 5
Figure 1.4 Time span of protein folding events .............................................................................. 9
Figure 1.5 an energy landscape for protein folding created by HP model.................................... 14
Figure 2.1 The fraction of unfolded molecules (circles) of the T57C mutant of protein L as
measured by various spectroscopic probes: fluorescence intensity, wavelength of peak
fluorescence, relative amplitude of intramolecular contact formation (fraction of fast rate), and
the magnitude of the unquenched tryptophan triplet lifetime (slow rate). The last two probes will
be described in detail in the following sections. The lines are fits to two-state models............... 17
Figure 2.2 Schematic electronic energy levels of tryptophan. 289nm UV laser is used to excite
the tryptophan to its singlet states. The intersystem cross from S1 to T1 is achieved through the
reversal of the spin of an excited electron. Triplet states T1 absorb blue light (442nm) and have a
long lifetime .................................................................................................................................. 19
Figure 2.3 Cartoon of loop formation and quenching. A tryptophan residue (W) is first excited to
the triplet states and then along with a cysteine residue (C) at the other end diffuses towards and
away from each other at rates kD+ and kD- respectively. q is the quenching rate ........................ 20
Figure 2.4 Schematic of the tryptophan triplet quenching experiment......................................... 23
Figure 2.5 Geometry of the chaotic advection mixing microfluidic-chip. (A) Buffer inlet. (B)
Protein solution inlet. (C) Filter section (one of three total per chip). Each filter post is a 10 by 10
μm square and they are spaced 10 μm apart. (D). The serpentine mixing region is composed of
five turns and the channel is 30 μm wide. (E) The narrow observation region is 120 μm wide and
1.4 mm long. (F) The wide observation region is 1000 μm wide and 10.4 mm long. (G) Outlet

viii

port ................................................................................................................................................ 31
Figure 2.6 Schematic of the tryptophan triplet quenching experiment coupled with a microfluidic
mixer ............................................................................................................................................. 32
Figure 3.1 Kinetics of tryptophan triplet lifetime for the HypF mutant W81F(long loop) at
5 -1

various concentrations of GdnHCl. The fast decays (~10 s ) are due to intramolecular contact
4 -1

quenching by C65 and the slow decays (~10 s ) are due to the natural lifetime of the triplet in
an aqueous environment ............................................................................................................... 36
Figure 3.2 a) Observed exponential rates of the tryptophan triplet decay versus [GdnHCl] for the
two mutants, W81F (white) and W27F (black). The triangles are the slow rate and the circles
are the fast rate as illustrated in Figure 3.1. b) The relative amplitude of the fast rate vs.
[GdnHCl] for W27F (black circles) and W81F (white triangles). The grey line is a sigmoidal fit
to all data for both mutants ........................................................................................................... 37
Figure 3.3 The observed fast rates vs. viscosity for W27F (a,c,e) and W81F (b,d,f) in 6 M (a,b),
2.3 M (c,d) and 1.25 M (e,f) GdnHCl. Each temperature is plotted as a separate color. The
lines are fits of all the data in each plot to Eqs. 2.5 and 2.6.......................................................... 38
Figure 3.4 Reaction-limited (black circles) and diffusion-limited (white triangles) rates vs.
[GdnHCl] for a) W27F (short loop) and b) W81F (long loop) at 20 C and 1 cP. The points at 0
M GdnHCl are those measured in 3% TFE and determined from the slope and intercept of Figure
3.5b................................................................................................................................................ 40
Figure 3.5 a) Tryptophan triplet decay of W81F in 3% TFE, T=20 C at various times after
dilution. b) The observed fast rates vs. viscosity for W81F in 3% TFE for T= 0, 10, 20, 30, and
40 C ~6 minutes after dilution ...................................................................................................... 42
Figure 3.6 Reaction-limited (red triangles) and diffusion-limited (blue triangles) rates vs.
[GdnHCl] for apocytochrome C ................................................................................................... 47
Figure 3.7 Structural cartoon of Fexch and Uexch of DrkN........................................................... 48
Figure 3.8 Kinetics of tryptophan triplet lifetime for the drkN mutant C60 at various
concentrations of GdnHCl ............................................................................................................ 49
Figure 3.9 The observed fast rates vs. viscosity for C2 (c, d) and C60 (a, b) in 6 M (a, c), 2 M (b,

ix

d) GdnHCl. Each temperature is plotted as a separate color. The lines in (a) and (c) are fits of
all the data to Eqs. 2.5 and 3.1. ..................................................................................................... 50
Figure 3.10 Reaction-limited (black) and diffusion-limited (red) rates vs. [GdnHCl] for C2
(triangle) and C60 (circle) at 20 C and 1 cP ................................................................................. 51
Figure 3.11 Reaction-limited (filled) and diffusion-limited (open) rates vs. [GdnHCl] for S2C
(black circle) and T58C (red triangle) at 20 C and 1 cP. Both loops are equally 28-residue long.53
Figure 3.12 Trp/Cys contact-quenching studies of ACBP unfolded states. (a) Observed quenching
rates vs. time after mixing for I86C. Inset: observed quenching rate vs. viscosity. (b) The
effective diffusion coefficients at various GdnHCl concentrations. (c) Reaction-limited (filled)
and diffusion-limited (open) rates for I86C (d) Reaction-limited (filled) and diffusion-limited
(open) rates for T17C. Black triangles are measured values from molecular dynamics simulations.
....................................................................................................................................................... 55
Figure 3.13 Observed quenching rates vs. time after mixing for T17C. Dashed line represents the
natural lifetime of tryptophan triplet state. ................................................................................... 57
Figure 4.1 Hydrophobicity scale used in the energy reweighted WLC model. The values are
taken from the Miyazawa-Jernigan scale and normalized to span the interval 0 to 1. ................. 61
Figure 4.2 Rank correlations between radius of gyration (Rg) and Trp47/Cys23 (r23-47) (a) and
between Rg and Trp47/Cys57 (r47-57) (b) .................................................................................... 65

Figure 4.3 Second structure propensity (SSP) of Uexch (red) and Ugdn (blue) states of drkN..... 66

Figure 4.4 Average total energy <ETOT>r of wormlike chains with (black) and without (red)
secondary structure constraints ..................................................................................................... 67
Figure 4.5 a) Probability distribution of Trp-Cys distances in protein L K23C before (black) and
after (red) reweighting for σ = 2.5. b) Contour plot of probability vs r and E for b) 　σ= 0.2
and c) 　σ= 2.5 ............................................................................................................................. 70
Figure 4.6 The values of which best reproduce the measured reaction-limited rates using
equation 4.4 for various sequences and [GdnHCl] ....................................................................... 71

x

Figure 4.7 Values of σ that best fit measured reaction-limited rates for various forms of ETOT
2

1/r potential:

ETOT = −

∑

ei , j

2
i − j >1 ( ri − r j )

ETOT = −

Square-well potential:

⎡⎛
4
Lennard-Jones potential: ETOT = ∑ ei , j ⎢⎜
⎢⎜ ri − r j
i − j >1
⎢⎝
⎣

12

⎞
⎟
⎟
⎠

⎛ 4
−⎜
⎜ ri − r j
⎝

⎞
⎟
⎟
⎠

∑

i − j >1

ei, j

6⎤

⎥ for dα ≤ r − r ≤ 6.5 Å
i
j
⎥
⎥
⎦

1/r, no cutoff: equation 4.1 in the text for dα ≤ ri − r j < ∞....................................................... 72
Figure 4.8 Values of that best fit measured reaction-limited rates for various definitions of
hydrophobicity .............................................................................................................................. 73
Figure 4.9 a) Values of that best fit measured reaction-limited rates for hydrophobic-hydrophobic
interactions only. b) Comparisons of unweighted (P(r)) and weighted (Z(r)) probability
distributions for weighting schemes using hydrophilic interactions (green and red lines) and only
hydrophobic interactions (blue line). ............................................................................................ 74
Figure 4.10 Average number of favorable contacts of each residue in protein L K23C (black) and
F22A K23C (grey). The vertical dashed line shows the position of the mutation F22A........... 76
Figure 4.11 Contact maps of protein L K23C F22A (a, c) and K23C (b, c) at σ =0 (c) and σ =2.5
(a,b). Contact map (d) is the subtraction of (a) by (b). The lower-right triangle region shows the
average distance between residues i and j and the upper-left triangle region gives the standard
deviation of that average distance................................................................................................. 78
Figure 4.12 Measured (bars) and calculated (lines) PRE per residue for drkN SH3 domain (a and
b) and ACBP (c and d). In each graph the green lines correspond to a completely random set of
wormlike chains and the red lines correspond to an ensemble of wormlike chains with some
residual secondary structure given by measured NMR chemical shifts. The blue lines
correspond to the ensemble with secondary structure using a modified version of equation 4.1 in
which only interactions more than 4 residues apart in sequence are included (σ = 6). The
paramagnetic labels are located at residue 2 (a), the N-terminus (b), residue 36 (c) and residue 65
(d). For (a) τC=0.5 ns, a scaling factor in equation 4.7, is adjusted to best match the measured
data. Pearson Correlation coefficient between bars and red lines equals to 0.89 (a), 0.41 (b), 0.35
(c) and 0.59(d)............................................................................................................................... 81
Figure 4.13 Probability distribution of Trp-Cys distances in AavLEA T58C before (black) and

xi

after reweighting for σ = 2.5 (red), γ= 31 (green) and σ = 2.5, γ= 31 (blue) ................................ 85
a

Figure 4.14 The cumulative probability ( C (a) = Ν ∫ P( x)dx , N is a normalization constant) of
0

buried surface area for hydrophobic and hydrophilic residues of protein L at σ =0 and σ = 2.5.
Buried surface area is defined as the cross-section of overlap between two links on the wormlike
chain assuming each link has spherical volume of diameter 6.5 Å. Since no two links can be
2

closer than dα = 4 Å, the maximum buried surface area is 20.6 Å per residue........................... 87
Figure 4.15 The values of D which best reproduce the measured diffusion-limited rates using
equation 4.5 with values of σ shown in Figure 4.6....................................................................... 88
Figure 5.1 Experimental reaction-limited (black) and diffusion-limited (white) rates measured at
various concentrations of denaturant for (a) ACBP T17C (b) ACBP I86C (c) HypF-N W81F, (d)
apocytochrome c, (e) Protein L K23C and (f) Protein G E19C. The errors in these rates are
typically 10%. The triangular points in (a) and (b) are measured values from molecular
dynamics simulations. The triangular points in (e) are the destabilizing mutation K23C F22A in
which the unfolded state is observable in as little as 0.25 M GdnHCl. The reaction-limited rate
in (e) (black circle) at 0 M GdnHCl was calculated from a Trp-Cys distribution derived from a
37

molecular dynamics simulation

using equation 2.9.

All other points were derived from

measured tryptophan triplet decay rates at various temperatures and viscosities using equations
2.5 and 2.6..................................................................................................................................... 91
Figure 5.2 Intramolecular diffusion coefficients of several different proteins. Long polyQ is an
aggregation-prone protein like HypF-N. One single mutation F22A destabilizes well-behaved
widetype protein L tremendously and gives it a tendency to aggregate. ...................................... 94
Figure 5.3 Formation of aggregated species, Agg, according to the model outlined in the text. For
these simulations, kbi = kagg = 1, K = 0.1 and k-1 varies as indicated on the plot ....................... 96

xii

Chapter 1

Introduction to Protein Folding, Misfolding and Aggregation

1.1 Amino Acids, Peptides and Proteins
Proteins (polypeptides), like nucleic acids and carbohydrates, are macromolecules. They are
constructed in a linear combination of a set of 20 amino acids. Each type of amino acid has a
homogenous backbone including a carboxyl group and an amino group (Figure 1.1) and a
distinctive side chain (R-group) bonded to the Cα atom. Amino acids are usually classified by the
chemical properties of their side chains (Figure 1.2). Amino acids in a protein, sometimes also
called residues, are linked by peptide bonds between the carboxyl group and the amino group
(Figure 1.1). Thus, each peptide contains only one free non-bonded carboxyl group known as the
C-terminus and only one free amino group known as the N-terminus. By convention, the amino
acid sequence in a protein is numbered from the N terminus to the C-terminus. The linked series
of carbon, nitrogen and oxygen atoms excluding the R-group are usually referred to as the main
chain or backbone.
To perform physiological functions in cells, proteins need to fold into specific 3-dimentional
structures determined by the sequence of amino acids. For a protein with a single peptide chain,
its structure can be divided into three different levels: primary structure, secondary structure and
tertiary structure (Figure 1.3). Primary structure is just the unique amino acid sequence of this
protein. Secondary structure includes α-helix, β-strand and hairpin bend, all of which result from
interactions between local amino acids. The arrangement of these secondary structure elements
1

make up the tertiary structure, which reflects interactions between distant residues.

Figure 1.1 Two amino acids form a peptide bond between the carboxyl group and the amino
group. (http://upload.wikimedia.org/wikipedia/commons/6/6d/Peptidformationball.svg)
For interpretation of the references to color in this and all other figures, the reader is referred to
the electronic version of this dissertation.

1.2 Protein Folding: Mechanics, Energy and Time Scale
Proteins are synthesized in the ribosome through a series of processes including the
transcription of DNA to RNA, RNA splicing and then the translation of RNA to proteins. Thus,
the 1-D primary structure (sequence) and the final folded structure of a protein can be
determined by the genetic codes. Some proteins fold into their native state spontaneously with
2

little assistance provided by other intracellular mechanisms (chaperones) and only when folded
into the correct structure can they perform their intended physiological functions, such as
providing cellular structure in the cytoskeleton of cells, acting as the contractile machinery in
1

muscles, and working as enzymes to catalyze reactions . Protein misfolding can cause human
diseases, such as prion disease, Huntington’s chorea and Alzheimer’s disease. The famous
2, 3

Parkinson disease is associated with the aggregation of α-synuclein

.

Nonpolar, hydrophobic
Phenylalanine
(Phe)

Methionine
(Met)

Tryptophan
(Trp)

Isoleucine
(Ile)

Leucine
(Leu)

Cysteine
(Cys)

Valine
(Val)

Figure 1.2 Twenty standard amino acid classified by their polarity (hydrophobicity)

3

Polar, charged
Aspartic acid
(Asp)

Glutamic acid
(Glu)

Arginine
(Arg)

Lysine
(Lys)

Polar, uncharged
Alanine
(Ala)

Asparagine
(Asn)

Glutamine
(Gln)

Histidine
(His)

Glycine
(Gly)

Threonine
(Thr)

Proline
(Pro)

Serine
(Ser)

Figure 1.2 (cont’d)

4

Tyrosine
(Tyr)

Figure 1.3 cartoon of the HypF-N protein showing (1) α-helix (red), (2) β-strand (blue), (3)
tryptophan 27 and 81 (purple) and (4) cysteine 65 (yellow).

5

Therefore, in order to understand the connection between protein structures and their functions,
and between the folding pathways and the misfolding mechanism, it is essential to understand
how proteins fold into their native conformations from unfolded states, which is the so called
protein folding problem.
If we consider a protein in isolation, the folding process is essentially a competition between
enthalpy and entropy. First, there are several interactive forces and solvent effects that can drive
proteins towards folding. The hydrophobic effect is a major solvent effect that causes protein
collapse, in which non-polar residues cluster into a hydrophobic core surrounded by
4

solvent-exposed polar residues . Other factors include Van der Waals forces, disulfide bonds and
electrostatic Coulomb interactions. Hydrogen-bonding is due to Coulomb interactions. It
facilitates the formation of α-helices and β-sheets and also stabilizes a protein structure. On the
other hand, the entropic effect tends to unfold a protein as self-organization upon formation of
secondary and tertiary structures and volume compaction induced by the hydrophobic effect,
decreases the configurational entropy. Thus, the Gibbs free energy is widely used in protein
folding studies to combine these two opposing effects, which essentially results in the marginal
stability of the native state and the complexity of folding pathways.
Does the native state correspond to the free energy minimum? Christian B. Anfinsen, winner
of the 1972 Nobel Prize in Chemistry, proposed a positive answer to this question based on his
experiments on Ribonuclease A which demonstrated that folding and unfolding is reversible by
5

adding or removing Urea (a chemical denaturant used to unfold proteins) . He stated that the
protein would spontaneously fold into the most stable state and hence all the information needed
6

5

to determine the protein native conformation was in the sequence itself .
A small protein (<100 residues) usually folds on a millisecond timescale, however it would
take longer than the age of the universe for a 100-residue polypeptide to sample all the possible
6

conformations (Levinthal’s Paradox) . Therefore, it is much more likely that the folding pathway
towards the native state is biased to those with low activation energies in order for the protein to
fold in a biologically relevant timescale. Most scientists today take both thermodynamic and
kinetic theory into consideration in the analysis of protein folding

7, 8

.

Proteins fold on a wide range of timescales, from nanoseconds to hundreds of seconds.
Figure 1.4 shows a comparison of different protein folding events based on their associated
timescales. According to the following Arrhenius equation which was later developed into the
transition state theory (TST),

k = A exp(− Ea / RT )

(1.1)

the rate constant k of chemical reactions is directly connected to the activation energy Ea.
However, we generally have little knowledge of the pre-exponential factor A. Therefore, rather
than calculating the absolute reaction rates, the transition state theory has been used to derive the
9-14

free energies, entropies and enthalpies from experimentally measured reaction rates

. Also the

transition state theory was initially developed to describe reactions in the gas phase, so it might
be too simplified to model the dynamics of protein folding, in which the interactions between the
proteins and solvent play an important role. In this regard, an alternative, the Kramers’
15-19

theory

, might be more suited because it models the folding reaction as a diffusional crossing

of an energy barrier. The barriers described in the Kramers’ theory are explored by Brownian
7

motion and can be crossed repeatedly. In a limit of strong frictional dumping from the solvent,
the reaction rate can be simplified as

k = (ωaωb / 2πγ ) exp(− Ea / k BT )

(1.2)

where, ωa , ωb are the curvature of the unfolded well and the barrier top, Ea is the barrier
height and γ is the frictional drag coefficient, which is proportional to the solvent viscosity η.
This suggests an inverse linear relationship between the folding rate k and the solvent viscosity
18

η . This diffusion-based theory also inspires researchers to experimentally estimate the upper
speed limit of protein folding by measuring the intramolecular loop contact rates of unfolded
proteins

20-24

, because a protein cannot fold any faster than the formation of those elementary

folding events, such as loops, turns, α-helices and β-structures.

8

Timeline

ps

ns

μs

ms

Sidechain rotation Loop formation
Secondary structure
Bond stretching
formation

s
Protein folding

Figure 1.4 Time span of protein folding events

1.3 Introduction to Experimental Approaches
Experiments on protein folding have been performed both in vivo and in vitro. From the
thermodynamic point of view, one should observe similar dynamics in either environment as
long as the physiological conditions (pH, salt concentration and so on) are the same. This
provides scientists the chance to investigate the folding process of proteins in quartz cuvettes
instead of human vessels and on optical tables instead of living organisms. In vivo experiments
are still irreplaceable in some cases when other intracellular molecules that assist folding are
involved. These assistant molecules are called chaperone proteins which to some extent disprove
the thermodynamic hypothesis proposed by Anfinsen which states that the native structure is
determined only by the protein’s amino acid sequence. While molecular chaperones raise very
interesting biomolecular questions, this section will focus only on in vitro experiments.
In vitro protein folding experiments usually involve a perturbant to unfold or partially unfold

9

a protein. The perturbant can be a chemical denaturant such as urea and guanidine hydrochloride
(GdnHCl), heat or even some sort of mechanical force. Guanidine hydrochloride is commonly
used in the denaturation of proteins by decreasing solvent polarity and consequently disrupting
non-covalent interactions. An example of heat denaturant is the high energy laser pulse used in
25, 26

T-jump experiment where the protein is unfolded by a sudden change of temperature

. The

advantage of using mechanical force as denaturant is that instead of global denaturation, forces
can be exerted on designed sites exclusively. Such experiments usually employ techniques such
as AFM and optical tweezers and are extremely hard to perform because they require
27, 28

manipulation of proteins on the single molecule level

.

Another key element for protein folding experiments is the probe. While new techniques are
always being developed, several major probes widely employed by researchers include NMR,
X-ray crystallography, circular dichroism (CD), UV fluorescence and some other optical
approaches. The first two are commonly used in determining molecule structures. CD and UV
fluorescence, on the other hand, are very sensitive to rapid conformational changes, such as CD
to secondary structures and tryptophan fluorescence to its local environment, and hence are
suitable for timed folding experiments.
Advanced experimental techniques enable scientists to monitor the course of the entire or
partial folding process in real time by removing the perturbant at early times (ns to ms). For
mechanical denaturant, this can be simply accomplished by stopping pulling. For chemical
denaturant, a mixer device is usually employed to dilute proteins in high denaturant
concentrations into aquatic buffer. But no matter how small the ratio you mix the sample into
10

solution, there will always exist a small amount of denaturant that may shift the protein from its
native state. The time resolution of folding kinetics experiments is greatly influenced by the class
of probes. However, a continuous-flow microfluidic mixer device can be utilized to mix on the
29, 30

μs timescale

. In such a mixer, solutions flow through the observation channel at a constant

flow rate so that observations taken at different positions correspond to different times after
mixing. This mixer not only shortens the measurement dead time due to turbulence but also
allows for longer integration time for each measurement.

1.4 The Importance of Studying the Unfolded States
It is evident that insights into the protein folding pathways and energetics can be gained from
studying the unfolded states of proteins according to the energy landscape theory. If it is
impossible for proteins to sample all possible conformations in physiological relevant time as
6

proposed by Levinthal’s paradox , there must be some specific folding pathways associated with
specific starting unfolded structures that are biased towards the native configuration. And no
matter what the folding mechanism is, there will be some unfolded conformations that are more
favorable than others. The topology of these favorable unfolded states in folding conditions may
favor certain side chain interactions that can trigger or catalyze protein folding. Recent
experiments on several proteins using NMR spectroscopy support this model by showing the
31-35

presence of residual structures in unfolded states

. People used to consider unfolded states to

be completely random polymers switching configurations rapidly through diffusion, suggesting a
very smooth surface on energy landscape. However, several high resolution experiments and
11

computer simulations have shown that the diffusion of unfolded states of well behaved proteins
36-38

is indeed relatively slow, suggesting a rugged energy surface

. To add these complexities to

the current polymer model for describing the distribution of unfolded conformations, an energy
reweighted wormlike chain model will be brought into discussion in chapter 4.
Experimentally, monitoring the dynamics of unfolded states is no easy task because of the
nanosecond to microsecond relaxation time scale. Traditional equilibrium experiments performed
in different denaturing conditions can only be used to calculate thermodynamic parameters,
which give no insights into kinetics. The stopped-flow mixer has a mixing time longer than
milliseconds so it focuses mostly on the later stages of the folding process. However, a technique
with ns time resolution, probing the intramolecular contact frequency between Trp and Cys
residues can be utilized to capture the loop formation of unfolded polypeptides, which provides
23, 39-41

direct insight into the intramolecular diffusional dynamics

. Such experiments can be

38

performed in either equilibrium conditions or in a fast mixer . Chapters 2 and 3 will
demonstrate a series of results covering proteins with various aggregation propensities using this
method.

1.5 Polymer Models and Computational Methods
Unfolded proteins are one special type of polymer and can be elucidated well using
simplified polymer models. If one considers a completely denatured peptide with N residues as a
random walk in 3 dimensions with step size lk (Kuhn length), the end-to-end distance r of this
polymer will have a Gaussian distribution.
12

3/ 2

⎛ 3 ⎞
P(r ) = ⎜
⎟
⎜ 2π Nl 2 ⎟
k ⎠
⎝

⎛ 3r 2 ⎞
exp ⎜ −
⎟
⎜ 2 Nl 2 ⎟
k ⎠
⎝

(1.3)

This approximation tends to fail for short peptides because unfolded proteins are not real
freely-joined chains and they have a certain stiffness due to chemical bonds which can only be
bended to limited angles. One may also want to take into account the excluded volume effect of
biomolecules to create self-avoiding chains. A wormlike chain model has been used to include
these constraints and will be discussed in chapter 2. Another factor worth considering is the
heterogeneity of peptides in low denaturant concentrations where intramolecular interactions
start to take place.

13

42

Figure 1.5 an energy landscape for protein folding created by HP model

More sophisticated models are required to describe the complexity of protein folding
dynamics. The energy landscape theory has served as a central theory for many computer-based
43-48

protein folding simulations and experiments

. It takes a statistical approach to the energetics

of protein conformations and demonstrated that proteins can fold into their native states through
any arbitrary pathway on a funneled energy landscape (Figure 1.5), rather than through a single
2-state mechanism. Molecular Dynamics (MD) is a powerful computational tool that allows

14

atoms and molecules to interact in real time based on a physics-based force field and statistical
mechanics. MD simulation, however, is limited to the folding of small proteins on the
microsecond timescale due to the high computational cost at all-atom level. Enhanced sampling
methods applied to help proteins cross energy barriers are commonly used to fold large,
49

slow-folding proteins . Also, hardware improvement, such as the development of GPU
50

computing

44, 51

and distributed computing

, has made it possible to gather more folding

trajectories in less computational time.

15

Chapter 2

Spectroscopic Study of the Dynamics of Unfolded Proteins

2.1 Introduction
Spectroscopy, microscopy and other optical methods have been widely used in the study of
protein folding. Light sources cover the wavelengths from X-ray to IR. In the presence of several
aromatic amino acids or with other dyes attached (fluorophore or luminophore), the process of
protein folding can be monitored at various time scales with different experimental systems.
Through different approaches reflecting different aspects of folding kinetics, a combination of
these results should reveal a whole picture of protein folding pathways-the energy landscape.
Conventional in vitro equilibrium experiments are not hard to accomplish and they can
provide very meaningful results. Fluorescence spectroscopy is a typical probe. For example,
Trp47 in B1 domain of protein L under continuous wave excitation at 280 nm or 295 nm yields
an emission spectrum with a peak that is monotonically dependent on the local polarity
52,

(hydrophobicity). A blue-shift occurs when the tryptophan residues are more buried from water
53

. This provides an excellent tool to study the structural change of proteins under various

denaturing conditions. The fraction of unfolded proteins under different denaturing conditions
can be determined by plotting the wavelength or intensity of spectrum peaks against denaturant
concentrations (Figure 2.1). These fluorescence data fit well with a two-state model.
However a recent study of the early folding process (before 100 μs) of protein L using a rapid
continuous-flow mixer revealed the complexity of the folding pathway of this well recognized
16

two-state folder, interpreted as the ruggedness in folding

1.0
0.8

f

0.6
0.4
0.2

fraction of fast rate
fluorescence intensity

0.0
1.0
0.8

f

0.6
0.4
0.2

Slow rate
fluorescence wavelength

0.0
0

1

2

3

4

5

6

[GdnHCl]
Figure 2.1 The fraction of unfolded molecules (circles) of the T57C mutant of protein L as
measured by various spectroscopic probes: fluorescence intensity, wavelength of peak
fluorescence, relative amplitude of intramolecular contact formation (fraction of fast rate), and
the magnitude of the unquenched tryptophan triplet lifetime (slow rate). The last two probes will
be described in detail in the following sections. The lines are fits to two-state models

17

36

landscape . So understanding the early events in protein folding not only helps us find the true
folding speed limit but also refines the energy landscape. In this chapter, a pump-probe
spectroscopy for measuring intramolecular loop formation will be introduced and followed by
the data analysis method. Finally, a microfluidic mixer device is coupled with the pump-probe
spectroscopy to discovery the intramolecular dynamics of unfolded proteins before they fold.

2.2 Monitoring Tryptophan Triplet Quenching Using Pump-probe Spectroscopy
Nanosecond intramolecular loop contacts can be detected experimentally using a pump-probe
laser spectroscopy. The mechanism is the quenching of tryptophan triplet energy state (probe) by
cysteine

23

(both tryptophan and cysteine are natural amino acids). Protein samples are

engineered and mutated with minimum influence on their original structures and stability to
contain only one tryptophan (Trp) and one cysteine (Cys) so that the measured quenching rate
reflects only one intrachain contact. Normally, for each protein studied, multiple mutations are
made to cover different loops over the full protein length.
Tryptophan can be excited to long-lived triplet energy states upon the absorption of 289 nm
UV light (Figure 2.2). It has been shown previously that the tryptophan triplet lifetime is about
53

40 μs in water and longer in the hydrophobic core of a folded protein . The lifetime of the
triplet state will be shortened when cysteine is nearby. Cysteine is a very efficient quencher with
a quenching rate 400-fold faster than other amino acids at physiological pH. The mechanism of
the quenching is electron transfer so the quenching rate has an exponential dependence on the

18

40, 54

intramolecular Trp/Cys distance

:

q (r ) = q0 exp [ − β (r − a0 ) ]

(2.1)
-1

where r is the intramolecular distance, a0 is the distance of closest approach (4 Å), q0 =4.2ns
-1

and β = 40nm .

S2
T2

S1
289 nm

5 ns

442 nm
T1
> 40 μs

S0
Figure 2.2 Schematic electronic energy levels of tryptophan. 289nm UV laser is used to excite
the tryptophan to its singlet states. The intersystem cross from S1 to T1 is achieved through the
reversal of the spin of an excited electron. Triplet states T1 absorb blue light (442nm) and have a
long lifetime.

The triplet state of tryptophan absorbs visible light at 442nm wavelength, which can be used
as a probe to monitor the population of the triplet state. The observed decay rate is a combination
55

of several kinetic rates :

19

kobs = k0 + ∑ kiuni + ∑ kibi [i ]
i

i

(2.2)

where k0 is the decay rate without quencher i, kiuni and kibi [i ] are the unimolecular and
bimolecular quenching rates respectively. To make the unimolecular rate more significant than
others, we first need to keep the sample concentration low (~30 μM) so that the contribution of
8

-1 -1

23

the bimolecular rate (2 x 10 M s ) is negligible . We also need to make sure that tryptophan
and cysteine are engineered close enough in sequence so that the unimolecular contact rate is
4 -1

faster than k0 (2 x 10 s ) and hence distinguishable. With these experimental controls, the
kinetics happening on the microsecond time scale can be simplified to a 2-step kinetic model
(Figure 2.3).
Λ

Ω

C

kD+

hv
W

Σ

*

Φ

q

*

kD-

Figure 2.3 Cartoon of loop formation and quenching. A tryptophan residue (W) is first excited to
the triplet states and then along with a cysteine residue (C) at the other end diffuses towards and
away from each other at rates kD+ and kD- respectively. q is the quenching rate.

1. The excited tryptophan and cysteine (quencher) diffuse towards each other at a rate of kD+.
2. Cysteine either quenches the tryptophan at quenching rate q or diffuses away at rate kD- .

20

rd

By solving the chemical reaction equations with the assumption of no accumulation at the 3
stage in Figure 2.3,

d [Σ]
= 0 = k D + [Ω] − ( k D − + q)[Σ]
dt
⎛ k
⎞
d [Φ ]
≡ kobs [Ω] = q[Σ] = q ⎜ D + ⎟ [Ω]
dt
⎝ q + kD − ⎠
the observed rate is given as

kobs =

kD + q
q + kD −

(2.3)

If q >> kD-, then kobs = kD+, but if that condition does not hold then equation 2.3 can be rewritten
as

1
kobs

=

1
1
1
1
+
=
+
qK k D + k R (T ) k D + (T ,η )

(2.4)

where K ≡ kD+/kD- is the equilibrium constant for forming the Trp/Cys encounter complex and
kR ≡ qK, which is the reaction-limited rate. kD+ is called the diffusion-limited rate.
Figure 2.4 demonstrates the instrumentation for measuring the transient absorption of the
tryptophan triplet states using a transmission pump-probe spectroscopy. Two different laser
beams are involved in this collinear geometry setup. The Nd:YAG pulse laser (Continuum
Surelite II-10, pulse duration: 8 ns, interval time: 10 ms) is employed to pump the tryptophan
residues into their triplet states. The wavelength of the pulse beam is shifted from 266 nm to 289
56

nm using a Raman cell (filled with methane at 250 psi) to minimize the photodamage .
Between each pulse, a continuous wave (CW) 441 nm laser beam (He-Cd) is used to monitor the
tryptophan triplet states. It is split into two parts: a reference beam and a sample probe beam. The
21

transmitted probe beam is aligned collimated to the pulse beam so that it passes through those
excited molecules. The intensities of the probe beam and the reference beam are measured using
two silicon photodiodes (New-Focus), recorded with digital oscilloscopes (Tektronix TDS),
subtracted, amplified and then stored on a computer using GPIB interface. The absorption decay
trace reflects the lifetime of the tryptophan triplet states which can range from nanosecond to
submillisecond. So two oscilloscopes are used to cover the wide temporal range: one with a 10μs
window and one with a 10 ms window. Two additional amplifiers (LeCroy DA1886A differential
amplifier and SR445A, SRS Inc four channel preamplifier with 5x gain at each stage) have been
recently incorporated into the instrument to allow for lower sample concentration and lower
pulse beam power.
To eliminate any pulse leakage and high frequency cable noise, a background with the pump
but without the probe beam is recorded each time at the beginning of each experiment and
thereafter subtracted from all other recorded decay traces. The UV pulse also tends to generate
hydrated electrons and neutral radicals that absorb light near 450 nm and decay within 3 μs

23, 57

.

Furthermore, O2 is a good quencher of the tryptophan triplet state. Therefore, before each
experiment, those free electrons are scavenged and oxygen molecules are removed by degassing
the sample solution with N2O for at least one hour to eliminate from the signal trace the
amplitude cause by these photo-effects. Another undesired photo-effect is thermal lensing, in
which the heat induced by the pulse laser changes the local density and consequently the
refraction of index of the solution, causing the probe beam to slightly reflect and move across the
detector. Because the sensitivity is not uniform over the entire detector, thermal lensing may
22

result in a pseudo-decay trace on the millisecond timescale. Ways to diminish the effect of
thermal lensing include decreasing the power of pulse laser, realigning the optics of the pulse and
probe beams and adjusting the position of the detector.
The sample cuvette is placed in a Peltier temperature controlled sample holder (Quantum
Northwest). Benzophenone has a decay time of about 200 ns at 50 μM and is regularly used for
the initial optical alignment of the instrument. N-acetyl-L-tryptophanamide (NATA) is used to
optimize the alignment because it is the uncharged analogue of tryptophan and has a 40 μs decay
53

266 nm

Sample
Holder

Raman
Cell

cw 442 nm

UV
filter

Nd:YAG

HeCd

time in the presence of deionized water in the absence of any additional quencher .

289 nm

Oscilloscopes
Silicon Diode

Figure 2.4 Schematic of the tryptophan triplet quenching experiment

23

2.3 Reaction-limited and Diffusion-limited Rates
In an ideal contact quenching model, quenching is infinitely rapid (diffusion-limited) and cut
off at a short distance. Excited probes are immediately quenched upon loop closure and hence
the measured quenching rate is diffusion limited. However, for the Trp/Cys system, polymer
dynamics is more or less coupled with reaction (quenching) kinetics. For instance, equation 2.1
shows the quenching of the tryptophan triplet by cysteine exhibits an exponential decay with
base quenching rate q0. Because q0 is comparable to kD- , there is a fairly good chance for
tryptophan to escape the quenching from a close encounter (Figure 2.3). Consequently the decay
profile observed in experiment may actually reflect 2 or more processes of loop formation and
deformation. In this case, the measured rate is a mixture of reaction-limited and diffusion-limited
rates and does not solely reflect diffusional dynamics.
Because of the complexity of the experimentally measured kinetics of loop formation shown
above, efforts have been made to extract a meaningful diffusion-limited rate from the measured
effective quenching rate. Equation 2.4 enables us to separate reaction-limited and
diffusion-limited rates by performing experiments at various temperatures and viscosity. The
reaction-limited rate, kR , depends only on temperature because temperature affects the chemical
reaction speed but viscosity does not. We assume the diffusion-limited rate, kD+ , depends on
both temperature and viscosity. Because we observed a linear relationship between the inverse of
the observed rate and the solvent viscosity, we decided to use the following two empirical
equations to fit our experimentally measured rates,

24

⎛ E (T − T0 ) ⎞
k R (T ) = k R 0 exp ⎜ 0
⎟
⎝ RTT0 ⎠

(2.5)

k
T
k D + (T ,η ) = D + 0 exp ( γ (T − T0 ) )
ηT0

(2.6)
-1

-1

where T is the absolute temperature, T0 = 273 K, R = 0.00198 kcal•K mol , η is the solution
viscosity and kR0, kD+0, E0, and γ are fitting parameters. Based on equations 2.4, 2.5 and 2.6, a
plot of 1/kobs against viscosity η at constant temperature T exhibits a linear relationship. The
intercept gives the inverse of reaction-limited rate, 1/kR , and the slope is proportional to 1/kD+.
In experiments, observed exponential decay rates were determined using TableCurve 2D v5.01
and fitted with equations 2.5 and 2.6 using Matlab to extract the reaction-limited rate and
diffusion-limited rate.
The linear 1/kobs vs η relationship may breakdown in some cases when a power-law
δ

dependence 1/kobs ∝ η needs to be applied instead. In systems of 20-30 amino acids long loops,
58

δ has been estimated to be very close to 1 . Therefore variations due to these two different
fittings are negligible.
A brief analysis of kR and kD+ is as follows. The reaction-limited rate, kR, is the effective
quenching rate that can be observed only when diffusion is sufficiently fast. It should depend on
the individual quenching rate, q, and the Trp/Cys distance distribution. The diffusion-limited rate,
kD+, is the true rate of bringing two ends of the loop together. It should depend on both the
Trp/Cys distance distribution and the diffusional dynamics of this particular chain.

25

2.4 Data Analysis Using SSS theory and a Wormlike Chain Model
Reaction-limited and diffusion-limited rates can be determined analytically using the
polymer theory of Szabo, Schulten and Schulten (SSS), which models intrachain diffusional
dynamics as motion on a 1-dimensional potential of mean force which is determined by the
59

probability distribution of intrachain distances . Using a Smoluchowski-like diffusion equation
with a distance dependent quenching rate under the potential defined as

U (r ) = −k BT ln P (r )

(2.7)

where k B is the Boltzmann constant and P(r ) is the equilibrium Trp/Cys distance distribution,
the observed rate of bringing two ends of chain in close contact can be expressed as

1
kobs

=

1
1
+ 2
kR kR D

∫

a0

lc

dr ⎧
⎪
⎨
P(r ) ⎪
⎩

∫

lc

r

⎫
( q ( x ) − kR ) P( x)dx ⎪
⎬
⎪
⎭

54

2

(2.8)

where a0 is the closest contact distance, lc is the contour length of the loop, D is the effective
intramolecular diffusion constant and q(r) is the distance dependent quenching rate. The
reaction-limited rate is given by,

∞

kR =

∫ q(r ) P(r )dr

(2.9)

a0

and the second term of equation 2.8 yields the expression for the diffusion-limited rate,

1
kD +

=

1
2
kR D

∫

a0

lc

dr ⎧
⎪
⎨
P(r ) ⎪
⎩

∫
r

lc

⎫
⎪
q ( x ) − k R ) P( x)dx ⎬
(
⎪
⎭

2

(2.10)

If we take the reaction-limited and diffusion-limited rates from experiment and want to

26

numerically solve for diffusion constant D, the only unknown information in equations 2.9 and
2.10 is the Trp/Cys distance distribution P(r). One simple way to generate such a distribution is
using freely-joined or Gaussian chains. According to the Gaussian probability distribution
(equation 1.2) the reaction-limited rate for r>>dα is

kR =

(

4π qa03
2

3π r

2

)

3

2

⎛
2
⎜ − 3a0
exp
⎜
⎜ 2 r2
⎝

⎞
⎟
⎟
⎟
⎠

(2.11)

which tells us reaction-limited rate kR is inversely proportional to the average volume

r2

3

2

of

unfolded chains. The diffusion-limited rate of Gaussian chains is given by

kD + =

(

4π Ddα
2

3π

r

2

)

3

2

(2.12)

Therefore kD+ is directly proportional to the diffusion constant D while it is also inversely
proportional to the chain volume. Though a Gaussian chain is a very simplified model, these
qualitative relationships between rates and chain properties hold true in more sophisticated
studies of intramolecular dynamics.
41, 60

A wormlike chain model has been widely used in our lab for generating chains

. This is

a backbone bead model that excludes all the side-chain effects, but has been proven to be a good
approximation because of two important constraints added to an ideal Gaussian chain model.
First, we apply a persistence length lp to characterize the real chain stiffness due to chemical
bonds. Second, we account for the excluded volume effect by introducing a hard-sphere shell
with diameter dα to the end of each virtual peptide bond. The distance between two non-neighbor

27

spheres must exceed dα . Both persistence length and excluded radius prevent intrachain clashes.
Adjusting either of them can change the equilibrium distribution P(r).
61

Wormlike chains are computationally simulated using the method of Hagerman and Zimm
as described in refs.

41, 54

. We typically generate 2 million chains in order to make a normalized

histogram of Trp/Cys distances smooth enough for numerical calculation. Chains are grown from
the N-terminus by vectorially adding links (1/10 of the length of a peptide bond) to the chain with
a random azimuthal angle (φ) between 0 and 2π radians and a axial angle (θ) which is a Gaussian
distributed random number with a variance 4 lc /lp around zero. A Monte Carlo algorithm is
applied in the construction of each chain with one modification to speed up the computational
41

time : when a clash (less than dα between any two non-neighbor residues) is detected after the
addition of every 10 links, the chain is truncated 3 persistence lengths before the clash and the
chain is regenerated, otherwise additional links are added until the full length of the protein is
reached. This algorithm has been proven to be statistically equivalent to simply discarding chains
with clashes.

2.5 Tryptophan Triplet Quenching Experiments in a Microfluidic Mixer
In this section, I will briefly introduce an instrument first built by Steven Waldauer in our
lab to measure the intramolecular diffusion kinetically before the protein folds using a
microfluidic mixer coupled to the pump-probe spectroscopy. Figure 2.5 shows the geometry of
the serpentine mixing chip used in our experiments. It utilizes two different aspects of chaotic

28

62

advection , Dean and corner vortices and features a good mixing efficiency at a Reynolds
number higher than 60 and a mixing time of ~250 μs at a typical flow rate of ~1 m/s. Details on
the serpentine chip’s design and fabrication can be found in ref.

63

.

The instrument for the kinetics experiment is built next to the equilibrium Trp/Cys contact
quenching instrument (Figure 2.6), sharing the same pulse and CW lasers. Simply by adjusting
two flip mirrors, the pump and probe beams can be redirected into the new instrument, whose
optical design is very similar to the equilibrium set up. The microfluidic serpentine chip manifold
is mounted on an automated 3 axis translation stage (9066-COM-E, New Focus) which can be
controlled either by computer or by hand for motion in the plane normal to the beam but has to
be adjusted manually to move in the longitudinal direction. To accurately position the narrow
observation region in the beam path, we use a piezo-actuator mechanical screw drive (Picomotor
model 8301, New Focus, San Jose CA) capable of 10 nm step increments in both the horizontal
and vertical directions.
There is a possible synchronization problem that can be brought into the microfluidic mixing
experiment by a two-beam system. That is, the small sample volume excited by the pump beam
has to be illuminated by the probe beam long enough to trace the tryptophan triplet population.
While this requirement is self-fulfilled after the alignment in an equilibrium experiment,
additional optics are necessary for the microfluidic mixer because the volume excited by the
pump beam travels out the probe beam volume at a constant flow rate, typically 1 m/s. A UV
objective (Objectives for Research LMU-3X-266, Cantwell NJ) is therefore added to focus the
pump beam into an even smaller spot on the observation channel. For example, considering a
29

probe beam with a diameter of 100 μm, if the pump beam has a diameter of 1 μm on the focus
plane and has been aligned directly in the center of the probe beam, it will take the excited
molecules 50 μs to escape the volume illuminated by the probe beam. The small pump beam
diameter also helps increase the time resolution of the instrument. Following the previous
example, the time resolution should be considered as 1μs (1 μm / 1 m/s = 1 μs) even though the
probe beam time window is 100 μs long. In reality, the pump beam has to be focused onto a
plane in front of or behind the observation channel plane in order not to burn the microfluidic
chip and is about 10 μm in diameter.
Both amplifiers, the LeCroy DA1886A differential amplifier and the SR445A, SRS Inc four
channel preamplifier, must be used in order to compensate for the small signal caused by the
much shorter path length in a serpentine chip (about 30~40 μm) compared to a 1 cm cuvette. The
differential amplifier helps the baseline subtraction and also outputs a 10x amplitude of the
original signal. Typically, one or two stages on the preamplifier are used, providing a total gain
of 50-250x. Most elements in the instrument, including the pulse laser, the oscilloscope and the
syringe pumps which inject the protein solution and mixing buffer into the serpentine chip are all
controlled via LabVIEW VIs on a data collecting computer.

30

Figure 2.5 Geometry of the chaotic advection mixing microfluidic-chip. (A) Buffer inlet. (B)
Protein solution inlet. (C) Filter section (one of three total per chip). Each filter post is a 10 by 10
μm square and they are spaced 10 μm apart. (D). The serpentine mixing region is composed of
five turns and the channel is 30 μm wide. (E) The narrow observation region is 120 μm wide and
1.4 mm long. (F) The wide observation region is 1000 μm wide and 10.4 mm long. (G) Outlet
63

port.

31

Figure 2.6 Schematic of the tryptophan triplet quenching experiment coupled with a microfluidic
mixer

63

32

Chapter 3

Experiments on Several Different Proteins Using Trp/Cys Quenching Spectroscopy

3.1 Conformational properties of unfolded HypF

Amyloidgenic Protein HypF
Amyloid is one type of protein aggregation characterized by a cross-beta sheet quaternary
structure. Abnormal accumulation of these insoluble fibrous amyloids in organs can cause
64

serious amyloidosis and some other neurodegenerative diseases. HypF-N , the fragment of
65

maturation factor of prokaryotic hydrogenases , is one of the most amyloidgenic proteins and
66

has been shown to have one of the fastest aggregation rates in vitro . It is thought that the
67, 68

disordered oligomers produce the toxic species

, so it is quite relevant to study the

conformational properties of the unfolded HypF.
It is of medical importance to answer questions like why some proteins tend to aggregate
while some don’t and what determines the aggregation rates. Recent work has shown a strong
correlation between aggregation rates and hydrophobicity, hydrophobic patterning and charge of
66, 69-73

a sequence

; additionally helix and sheet propensity correlates significantly with
70, 74-77

mutational changes in aggregation rates within a sequence

. From these statistical

analyses a general model of the relevant physicochemical parameters has emerged

33

66, 70

, but it

does not fully define the physical basis for how aggregation happens. One possible mechanism is
that the aggregation precursor is fairly disordered and transiently exposes hydrophobic residues
to the solvent for sufficient time to form bimolecular interactions. Therefore there may be some
correlation between intramolecular diffusion rates of aggregation precursors and aggregation
propensity.
Campioni et al. have shown that the conformation populated at low pH can form prefibrillar
aggregates and can be described as a pre-molten globule, defined by Uversky as an intermediate
78

between a fully disordered chain and a molten globule . Furthermore, ANS fluorescence
measurements during folding have shown that a collapsed state formed within the first 1 ms of
folding has a significant number of hydrophobic residues exposed to solvent

79

.

These results

support the idea that a fairly disordered state, which may even be the unfolded state under
folding conditions, is the aggregation precursor for HypF-N, and therefore the rate of
intramolecular diffusion should determine the rate of bimolecular association.

Experimental Results for HypF
The mutants of HypF used in this study are a kind gift from Niccolò Taddei and Claudia
78

Parrini. Two mutants were made , W27F/C7S/C40S (referred to as W27F) and
W81F/C7S/C40S (W81F), to remove two cysteines and one of these tryptophans so that
intramolecular contact is observed between positions 65 and 81 (short loop) and positions 27 and
65 (long loop) independently. For both mutants at every concentration of denaturant, we observe
one or two exponential decays of the tryptophan triplet state (Figure 3.1). The fast decay rate
34

represents the Trp-Cys quenching kinetics of the population of unfolded proteins in the sample
and a slow rate corresponds the natural lifetime of tryptophan for folded proteins since the
positions of tryptophan and cysteine are designed to be distant in folded structure. Figure 3.1
shows the relative amplitude of the slow rates (fraction of folded states) increases with
decreasing concentration of denaturant. The observed contact rates for the short loop are
generally faster but similar at 6M and 1.25M GdnHCl compared to the long loop (Figure 3.2(a)).
4 -1 23

The slow rates for both mutants are similar to that observed for free Trp in water (~10 s )

,

suggesting that the tryptophans in both mutants are still solvated in their folded or intermediate
states, not buried in the hydrophobic core of the folded protein. Figure 3.2(b) shows the fraction
of the unfolded state for both W81F and W27F. Based on this measure of protein stability, the
midpoint of folding for HypF is at about 1.25M GdnHCl.

35

Normalized Absorbance

1.0

6M
2.3M
1.25M
1M

0.8
0.6
0.4
0.2
0.0

10-6

10-5

10-4

10-3

time (s)
Figure 3.1 Kinetics of tryptophan triplet lifetime for the HypF mutant W81F (long loop) at
5 -1

various concentrations of GdnHCl. The fast decays (~10 s ) are due to intramolecular contact
4 -1

quenching by C65 and the slow decays (~10 s ) are due to the natural lifetime of the triplet in
an aqueous environment.

The global fitting of measured decay time (1/kobs) against viscosity at multiple temperatures
guarantees the minimization of uncertainty raised by either a faulty single experiment or possible
change of experimental conditions, such as optical alignment, for each measurement. Fits at
different concentrations of GdnHCl are shown in Figure 3.3. Each temperature is plotted as a
different color and the viscosity is increased by the addition of sucrose. Sucrose has been shown to
stabilize folded proteins, but we assume that it won’t change the dynamics of the unfolded states at
the protein concentrations we use, especially in the presence of GdnHCl. For both mutants at 6 M

36

GdnHCl, each temperature lies on a different line, indicating that kR depends strongly on
temperature. However, at 1.25 M GdnHCl, W81F still shows separate lines for each temperature
while W27F has converged on a single line for all temperatures, suggesting both kR and kD+ have
lost any dependence on temperature. The loss of temperature dependence on kR might be due to
41

the fact that the equilibrium rate, K, had increased far above 1 at low denaturant . The loss of
temperature dependence on kD+ indicates that γ in equation 2.6 is infinitely small and the effect of
viscosity change on decay time has become more significant than temperature change (a steeper
slope of 1/kobs). Thus the 65-81 (short) loop appears to be highly collapsed (1/kR ~ 0) at low
denaturant and the intramolecular diffusion rate (kobs ~ kD+) is slow enough to be independent of
temperature.

105

104

1.0

a

Unfolded fraction

-1
kobs (s )

106

b

0.8
0.6
0.4
0.2
0.0

0

1

2

3

4

5

6

0

[GdnHCL](M)

1

2

3

4

5

6

[GdnHCL](M)

Figure 3.2 a) Observed exponential rates of the tryptophan triplet decay versus [GdnHCl] for the
two mutants, W81F (white) and W27F (black). The triangles are the slow rate and the circles
are the fast rate as illustrated in Figure 3.1. b) The relative amplitude of the fast rate vs.
[GdnHCl] for W27F (black circles) and W81F (white triangles). The grey line is a sigmoidal fit
to all data for both mutants.

37

Figure 3.3 The observed fast rates vs. viscosity for W27F (a,c,e) and W81F (b,d,f) in 6 M (a,b),
2.3 M (c,d) and 1.25 M (e,f) GdnHCl. Each temperature is plotted as a separate color. The lines
are fits of all the data in each plot to Eqs. 2.5 and 2.6.

38

35

35
0C
10C
20C
30C
40C

1/kobs(μs)

25
20
15
10
5
0

2

10

4

6

8

10

η(cp)

12

14

16

0

18

0C
10C
20C
30C
40C

b
0

2

4

6

8

10

η(cp)

12

14

16

18

25
20
15
10
5

25
20
15
10
5

d
0

2

4

6

8

0

10 12 14 16 18

η(cp)

35

d
0

2

4

6

8

10 12 14 16 18

η(cp)

35
0C
10C
20C
30C
40C

25
20
15
10
5

25
20
15
10
5

e
0

2

4

0C
10C
20C
30C
40C

30

1/kobs(μs)

30

0

0C
10C
20C
30C
40C

30

1/kobs(μs)

1/kobs(μs)

15

35

30

1/kobs(μs)

20

5

35

0

25

a

0

0C
10C
20C
30C
40C

30

1/kobs(μs)

30

6

8

10

η(cp)

12

14

16

0

18

f
0

Figure 3.3

39

2

4

6

8

10

η(cp)

12 14

16

18

107

a

-1

k (s )

-1

k (s )

107

106

105

0

1

2

3

4

5

6

b

106

105

[GdnHCL](M)

0

1

2

3

4

5

6

[GdnHCL](M)

Figure 3.4 Reaction-limited (black circles) and diffusion-limited (white triangles) rates vs.
[GdnHCl] for a) W27F (short loop) and b) W81F (long loop) at 20 C and 1 cP. The points at 0 M
GdnHCl are those measured in 3% TFE and determined from the slope and intercept of Figure
3.5b.

Figure 3.4 shows the reaction-limited (red) and diffusion-limited (blue) rates after the
decomposition using equations 2.5 and 2.6. Between 6 M and 2.3 M GdnHCl, kR and kD+ for
both mutants increase slightly, but below 2.3 M GdnHCl, the diffusion-limited rates start to
decrease, more significantly for W27F than for W81F. For the W81F (long loop) mutant, the
reaction-limited rate continues to increase, while the W27F (short loop) mutant has no measurable
7 -1

reaction-limited rate and is presumed to be greater than 1x10 s

(see the y-axis intercept in

Figure 3.39(e)). Measurements of kR and kD+ were not possible below 1.25 M GdnHCl because
the addition of sucrose stabilized the folded state of the protein, preventing the observation of
quenching in the unfolded state.

Loop Formation Kinetics in TFE
40

2,2,2-Trifluoroethanol (TFE) at low concentrations acts as a protein denaturant. However,
unlike GdnHCl, TFE unfolds proteins without providing solvent protection of hydrophobes so it
74, 78, 80

promotes bimolecular association and consequent aggregation

.

Equilibrium

measurement of Trp/Cys quenching in TFE enables us to estimate the polymer dynamics of the
unfolded state, which may also be the aggregation precursor, in aqueous solution. In these
experiments, the protein was dissolved and measurements began within a few minutes.
Distinguishable fast decay traces have only been observed for W81F, probably because the 65-81
loop remains same compact and viscous as in 1.25M GdnHCl. Two TFE concentrations (v/v) 6%
and 3% have been used for W81F and there was visible aggregation in the sample within 30
minutes and 2 hours, respectively. The observed kinetics is similar between 3% and 6% TFE
concentrations but the lower concentration allows sufficient time for temperature dependent
measurements. Figure 3.5 shows the tryptophan triplet decay at various times after dissolution in
3% TFE solution. The first trace shows that the majority of the population decays before 10 μs
while later measurements still have significant population at that time. Each of these traces can
be fit to 2 exponentials between 400 ns and 1 ms. The relative amplitude of unfolded state and
decay times of these fits are shown in Table 3.1.
We attribute the slow decay to solvent quenching of the tryptophan in either the folded or
78

aggregated state. Since W27F in W81F mutant is solvent exposed in the native state

with a

lifetime of 40 μs, I assume the increase in the slow lifetime over the measurement period is due to
a contribution from an aggregate in which the Trp is somewhat buried from solvent.

41

Tryptophan Triplet Population

1.0
a

0.8
0.6
6 minutes
35 minutes
75 minutes
110 minutes

0.4
0.2
0.0

10-7

10-6

Time(s)

10-5

10-4

20

1/kobs(μs)

16
12
8
4
0
0.0

b
0.4

0.8

1.2

1.6

2.0

η(cp)
Figure 3.5 a) Tryptophan triplet decay of W81F in 3% TFE, T=20 C at various times after dilution.
b) The observed fast rates vs. viscosity for W81F in 3% TFE for T= 0, 10, 20, 30, and 40 C ~6
minutes after dilution.

I assume that the fast decay in 3% TFE is a transiently populated unfolded state that is the
precursor to aggregation. Directly determining the reaction-limited and diffusion-limited rates of
the unfolded state was not possible because the addition of sucrose stabilized the folded state of the
protein. Therefore I measured the triplet lifetime at various temperatures and assumed that there
was a minimal temperature dependence for kR or kD+ and fit the rates at all temperatures to a

42

single line (Figure 3.5b). Assuming 3% TFE approximately represents the solvent conditions in
the absence of GdnHCl, I plot these rates at [GdnHCl] = 0 M on Figure 3.4b. If there is a
temperature dependence for kR and kD+ such that k vs. η can be fit to separate lines at different
temperatures, then our fit in Figure 3.5b of the intercept is underestimated and the slope is
overestimated. Therefore the points at 0 M GdnHCl in Figure 4 represent an upper limit on kR and
a lower limit on kD+. I conclude that the intramolecular diffusion rate for W81F decreases by no
more than a factor of 10 between 6 M and 0 M GdnHCl. This means the long loop remains fairly
diffusive in native conditions.

Diffusional Dynamics of Unfolded HypF
The underlying polymeric dynamics that produces the observed contact rates was determined
using SSS theory and WLC model (Chapter 2.4). Chains with full peptide length of 91 residues
have been constructed. Since the reaction-limited rate is relatively insensitive to the persistence
41

length , it is again kept fixed at lp = 4 Å and the excluded diameter (dα) is varied with denaturant
concentration. In detail, once two million chains were generated using the method described in
chapter 2.4 at a chosen dα, the corresponding distribution of Trp/Cys distance was plugged into
equation 2.9 for numerical integration to find kR. The next procedure is to compare this
theoretically calculated kR to the experimental value and if they match each other, we believe that
an accurate distribution has been found for that particular GdnHCl concentration. Otherwise, the
value of dα must be adjusted accordingly until agreement is reached. One needs to decrease dα to

43

match a larger kR since the kR is inversely proportional to chain volume. The parameters of the
appropriate distribution P(r) for each denaturant concentration and the correspondingly
determined effective diffusion coefficient, D, are listed in Table 3.2. One might notice that
excluded diameter dα needs to be adjusted to 0 to match the fastest kR. Since dα cannot be
negative, this means the limitation of the WLC model on compaction of the chain conformations
has been reached. In order to overcome it, additional attractive potentials that can further collapse
WLC conformations need to be introduced (see chapter 4).
From <r> and D in Table 3.2, we see that the degree of compaction and loss of diffusivity is
different in separate loops in the chain below 2.3 M GdnHCl. The C65-W81 loop compacts and
loses diffusivity significantly more than the W27-C65 loop. The W27-C65 loop encompasses
one of the putative amyloidogenic regions of the protein, based on mutational studies of the
homologous protein acylphosphatase, while the C65-W81 loop does not

81, 82

, so these results

suggest that the amyloidgenic region of the chain is more diffusive than surrounding regions.
The physical basis for the increased diffusivity of the long loop could be due to the fact that
there is significant net charge in the long loop while the short loop, like the whole chain, is almost
neutral. This would force the long loop to adopt more extended conformations. However, the
83

long loop still contains many hydrophobic residues. Using the Kyte-Doolittle scale , the mean
hydrophobicity (-0.41) and the standard deviation (3.36) of the long loop is almost the same as the
short loop or the entire chain. Also, the hydrophobic pattern of the entire chain oscillates rapidly,
with no large clusters of hydrophobic residues to direct collapse. Therefore the W27-C65 region
of the chain, forced into extended conformations by electrostatic repulsion that are not
44

counterbalanced by large hydrophobic clusters pulling the chain together, will have hydrophobes
that are more often exposed to solvent and to other, nearby chains, making bimolecular
hydrophobic interactions more likely.

Table 3.1 Observed rates of tryptophan triplet decay of W81F at various times after dilution in
3% TFE
-1

-1

fast rate (s )

slow rate (s )

94364
123948
113546
113459

time(min)
6
35
72
110

24102
13157
11647
10630

fraction fast rate
0.70
0.27
0.27
0.24

Table 3.2 Wormlike chain parameters and effective diffusion coefficients for both mutants
studied
-1

-1

6

2 -1

D (x10 cm s )

kR (s )

kD+(s )

lp(Å)

dα(Å)

<r> (Å)

1.25
2.3
6

1990089
347950
203920
144830

107408
896190
1401100
1049700

4
4
4
4

0
2.75
3.20
3.50

30.6
35.5
37.0
38.1

0.02
0.51
1.15
1.07

1.25
2.3
6

4.8E+14
448200
185960

433088
1913100
1167500

4
4
4

0
3.20
4.10

19.9
24.2
25.1

0.02
0.58
0.47

[GdnHCl]
W81F
a

0

W27F

a

The solution contained 3% TFE and was measured 6 minutes after dilution.

3.2 Intramolecular Loop Formation of Intrinsically Unfolded Proteins

45

Apocytochrome C
Intrinsically unstructured (natively unfolded) proteins provide the possibility to study the
dynamics of unfolded polypeptides under both denaturing and physiological conditions. After the
84

heme is removed from horse heart cytochrome C, this protein becomes unstructured . The
wildtype sequence of apocytochrome C has two cysteine residues at position 14 and 17, either of
which can make contact with the tryptophan residue at position 59. Since these two cysteine
residues are close, the formation of either loop will demonstrate very similar dynamics.
Apocytochrome C is unfolded at all concentrations of denaturant so measurement of tryptophan
triplet lifetime yields one exponential decay rate due to contact quenching.
As usual, experiments were carried out at four different GdnHCl concentrations (0, 1.5 M,
2.3 M and 6 M) and the decay trace of the tryptophan triplet absorbance at 441nm was recorded
at various temperatures (0 C, 10 C, 20 C, 30 C and 40 C) and viscosities (v/v: 0%, 10%, 20%,
30% and 40% of sucrose) for each GdnHCl concentration. From Figure 3.6, we can see kR
increases as GdnHCl concentration decreases, indicating that unfolded apocytochrome C
molecules become more compact in the absence of denaturant. On the other hand, the
diffusion-limited rate decreases as GdnHCl concentration decreases, indicating a less diffusive
natively unfolded state.
When generating wormlike chains for apocytochrome C, I set the contour length of the
Typ-Cys loop to be 42 residues and each terminal tail to be 5 residues since it was proven that the
85

reaction-limited rate is relatively insensitive to the length of the tail . As a result, each chain
was only 52 residues long after the truncation and computational time decreased about a half.
46

WLC parameters and corresponding D for different GdnHCl concentrations are listed in Table
3.3. We can see that D decreases by a factor of 5 from 6 M to 0 M GdnHCl. However compared
41

to that of protein L in the same nondenaturing condition , this effective diffusion coefficient is
still 2 orders greater, indicating that intrinsically unstructured protein apocytochrome C is much
more diffusive than well-folded protein L in aqueous solution.

107

rates(s-1)

kD+
kR
106

105

0

1

2

3

4

5

6

[GdnHCl](M)
Figure 3.6 Reaction-limited (red triangles) and diffusion-limited (blue triangles) rates vs.
[GdnHCl] for apocytochrome C

Table 3.3 Wormlike chain parameters and effective diffusion coefficients for apocytochrome C

-1

[GdnHCl]
0
1.5
2.3
6

kR (s )

-1

kD+(s )

lp(Å)

dα(Å)

626989
504390
287532 1074200
267253 1043900
157440 1051700

4
4
4
4

1.98
2.86
2.90
3.32

47

6

<r> (Å)
35.2545
37.7664
37.9181
39.4323

2 -1

D (x10 cm s )
0.2100
0.7700
0.7600
1.0900

Uexch

Fexch

Figure 3.7 Structural cartoon of Fexch and Uexch of DrkN

DrkN
Drosophila drk N-terminal (drkN) SH3 domain is a metastable protein, which exists in
86, 87

approximately 1:1 equilibrium between folded (Fexch) and unfolded (Uexch) states in water

(Figure 3.7). Thus, I expected to see some fast kinetics due to contact quenching under
nondenaturing conditions. However, I didn’t. Both mutants of drkN (C2 and C60) diffuse so
4 -1

slowly that I can still see a slow decay (~10 s ) even in 6 M GdnHCl (Figure 3.8). As a result, I
was only able to conduct temperature and viscosity dependent measurements of these two
mutants in 2 M and 6 M GdnHCl. In both conditions, the fraction of the amplitude of fast rate to
slow rate is about 1:1 (Figure 3.8). Because it is almost impossible to determine whether the slow
rate is due to contact quenching by slowly approaching cysteine residues or just a reflection of
the tryptophan natural lifetime, I decided to extrapolate the diffusion-limited and reaction-limited

48

rates only from the fast observed rates. Another interesting finding is the nonlinear dependence
of kobs

Absorbance

1.0
0.8
0.6
0.4
0.2
0.0

6M
2M
1M
0M

10-6

10-5

10-4

times(s)
Figure 3.8 Kinetics of tryptophan triplet lifetime for the drkN mutant C60 at various
concentrations of GdnHCl

on 1/η for both mutants in 6 M GdnHCl as shown in Figure 3.9, consequently equation 2.6 was
85

modified to account for the observed curvature .

k D + (T ) =

k D + 0T
exp ( γ (T − T0 ) )
2
(η − Aη )T0
49

(3.1)

In 2 M GdnHCl, both proteins seem to converge on a single straight line for all temperatures,
similar to what we saw for the W27F mutant of HypF. This suggests a complete loss of
temperature dependence of kR and kD+.

C60 in 6M GdnHCl

14

a

1/kobs(μs)

12
10

0C
10C
20C
30C
40C

8
6
4
2

0

2

6

8

10 12 14 16 18

C2 in 6M GdnHCL

8

1/kobs(μs)

4

18
16
14
12
10
8
6
4
2
0

C60 in 2M GdnHCl
b

0

4

6

8

10

12

14

C2 in 2M GdnHCl

10

c

6

2

d

4

5

2
0

0

2

4

6

8

10 12 14 16 18

η(cp)

0

0

2

4

6

8

η(cp)

10

12

14

Figure 3.9 The observed fast rates vs. viscosity for C2 (c, d) and C60 (a, b) in 6 M (a, c),
2 M (b, d) GdnHCl. Each temperature is plotted as a separate color. The lines in (a) and
(c) are fits of all the data to Eqs. 2.5 and 3.1.

The extracted reaction-limited and diffusion-limited rates are shown in Figure 3.10. We see
few changes of kR and kD+ for both C2 and C60 between 2 M Gdn and 6 M GdnHCl. This tells us
50

both portions of the drkN protein have the same compaction and diffusivity as the GdnHCl
concentration decreases. Thus, due to the slow and invariable kinetics, this protein seems not as
interesting as it is expected to be for the intramolecular loop formation study. But some additional
information available for this protein, such as the secondary structure propensity determined by
experimentally measured chemical shifts, does make it a good calibration object for an energy
reweighted WLC model, which will be described in detail in chapter 4.

107
C60 kR
C60 kD+
C2 kD+

-1

rates(s )

C2 kR

106

105

0

1

2

3

4

5

6

[GdnHCl](M)
Figure 3.10 Reaction-limited (black) and diffusion-limited (red) rates vs. [GdnHCl] for C2
(triangle) and C60 (circle) at 20 C and 1 cP

AavLEA
Late embyrogenesis abundant (LEA) is a special group of intrinsically disordered proteins.
They are natively unstructured in water but do begin to aggregate when undergoing dessication

51

88

or when they are placed in a non-polar solution . LEA proteins were first found in maturing
88, 89

plant seeds and have since been found in invertebrates and bacteria

. Most organisms are

dependent on the presence of water in a cell, however LEA can not only survive desiccation but
return to their pre-desiccation states of function after rehydration. This process of desiccation
tolerance is known as anhydrobiosis and LEA proteins take on a significant role in anhydrobiosis.
More importantly, they can also prevent aggregation in numerous other proteins during
88

desiccation .
The particular LEA protein used in our experiments was AavLEA1 which is found in the
88

nematode Aphelenchus avenae . This intrinsically disordered protein has a very high content of
charged residues that can prevent intrachain collapse. The measurements were not taken directly
by me but my lab colleagues Eric Buscarino and Basir Ahmad. Two mutants S2C and T58C were
made. Tryptophan was located at position 30 so the chain lengths between the tryptophan and
cysteine were 28 for both mutants. Due to their intrinsically disordered nature, AavLEA1 was not
at risk to aggregate quickly but to maintain consistency in our measured rates we used each
sample within one - three hours of thawing and dilution. The experiments were performed at pH
= 7.5 in order to neutralized histidine residues. Figure 3.11 shows the reaction-limited and
diffusion-limited rates of the two mutants of LEA. The kinetics of these two loops is very similar
and this protein is still highly diffusive and less compact even in 0 M GdnHCl. We believe the
50% charged residues in the protein sequence account for its extended and unstructured nature.
To better model this protein theoretically, I will introduce a method to include the charge effect
in the WLC model in the next chapter.
52

AavLEA
(28 residue loop)
kR S2C
kR T58C
kD+ S2C
kD+ T58C

rates (1/s)

106

105
0

1

2 3 4 5
[GdnHCl](M)

6

Figure 3.11 Reaction-limited (filled) and diffusion-limited (open) rates vs. [GdnHCl] for S2C
(black circle) and T58C (red triangle) at 20 C and 1 cP. Both loops are equally 28-residue long.

3.3 Diffusion of Unfolded Acyl-coenzyme A-binding Protein over A Complete Range of
Denaturant
In order to have a complete knowledge of the intramolecular diffusion of proteins with
different aggregation propensities, well-behaved acyl-coenzyme A-binding protein (ACBP) has
been studied over a wide range of denaturant in both equilibrium and using the microfluidic mixer
described in chapter 2. This 86-residue protein contains four helices and folds on a 10ms
90

90, 91

timescale . Even though ACBP has long been established as a two-state folder

, Teilum et al.

revealed the presence of a ~80 μs phase, suggesting a three-state model with a partially-folded
92, 93

intermediate

. Recent theoretical study using the Markov state models built from atomistic

53

simulations of ACBP, however, revealed a highly diffusive network of metastable unfolded states
instead of a single well-defined intermediate. Therefore, measuring intramolecular loop formation
using Trp/Cys contact quenching technique may provide an excellent tool to reveal the underlying
dynamics.
Two mutants of ACBP (T17C/W55A and W55A/I86C) were used in this study. The only
tryptophan after mutation sits at position 58 and can make contact with cysteine at either position
17 or position 86. Equilibrium Trp/Cys quenching experiments were performed at 6 M, 2.3 M and
1.5 M GdnHCl for both T17C and I86C. Because I86C has a denaturation midpoint slightly lower
35

than the other mutant , I was also able to collect equilibrium data also at 1.25 M GdnHCl for this
mutant. Reaction-limited and diffusion-limited rates were extracted from the observed quenching
rate using equations 2.5 and 2.6 and plotted in Figure 3.12 (c and d). As the denaturant
concentration decreases from 6 M to 1.25 M, these two mutants yield very similar trends, an
increasing reaction-limited rate and a decreasing diffusion-limited rate, which is also observed for
most other proteins studied using Trp-Cys quenching.

54

a

b

c

d

Figure 3.12 Trp/Cys contact-quenching studies of ACBP unfolded states. (a) Observed quenching
rates vs. time after mixing for I86C. Inset: observed quenching rate vs. viscosity. (b) The effective
diffusion coefficients at various GdnHCl concentrations. (c) Reaction-limited (filled) and
diffusion-limited (open) rates for I86C (d) Reaction-limited (filled) and diffusion-limited (open)
rates for T17C. Black triangles are measured values from molecular dynamics simulations.

An ultrarapid mixer as previously described in chapter 2.5 has also been employed to capture
the dynamics of ACBP before folding. Unfolded 1 mM ACBP protein in 4 M GdnHCl was pushed
through the mixer and combined with buffer without denaturant at a ratio of 1:20. After passing

55

through the serpentine mixing region, the protein enters the wide observation channel with a final
GdnHCl concentration of ~0.2 M. The protein itself is also diluted to 50 μM after mixing. The total
flow rate is 210 μL/min ((200+10) μL/min). Most observed rates measured along the exit channel
can be well fit with single exponential decay. Figures 3.12 (a) and 3.13 show the observed rate of
ACBP I86C and T17C, respectively, as a function of time after mixing based on the position in the
channel. The rates of T17C are overall slower and comparable to the natural lifetime of tryptophan
(gray dashed line), suggesting that the 17-58 loop might be too long for this kinetics study. But the
5 -1

58-86 loop (I86C) yields rates > 10 s . The decrease of the rate over time after mixing is
probably due to the collapse of protein during mixing and was fit into a single exponential decay.
The inset in Figure 3.12 (a) shows the observed rate of I86C 1432 μs after mixing as a function of
viscosity. The rates in various sucrose concentrations (v/v: 0%, 5%, 10%, 15% and 20%) can be fit
to a straight line with a zero intercept within error, indicating that the observed rate is
-1

diffusion-limited. So the observed rate at η =1cp directly gives kD+ = 125000 s . The
reaction-limited rate at 0.2 M GdnHCl in Figure 3.7 (c) is obtained from molecular dynamics
simulations performed at 300K and is used to determine the effective diffusion coefficient using
SSS theory, which gives D = 5 x 10

-9

2 -1

cm s . The effective diffusion coefficients at other

denaturant concentrations are exhibited in Figure 3.12 (b). So the experimental results exhibits
similarly slow diffusion as seen in molecular dynamics simulation, which gives D = 8.579+/-1.95
-9

x 10

2 -1

cm s

37

obtained by measuring the mean-squared displacement over time . Given the

average radius of gyration of the unfolded states of ACBP (2~3 nm), the reconfiguration time
through intramolecular diffusion can be estimated to be ~ 10 μs. These results validate the
56

suggestion of recent simulational study that the diffusion of the unfolded states of ACBP is too
slow to form an intermediate in the early microsecond timescale. This protein, however, is still
38

shown to be more diffusive than protein L in physiological conditions , suggesting a strong
sequence dependence of intramolecular diffusion.

4e+5

no sucrose
20 % sucrose

-1
kobs(s )

3e+5

2e+5

1e+5

0
200

400

600

800

1000 1200 1400 1600 1800 2000

time after mixing(us)
Figure 3.13 Observed quenching rates vs. time after mixing for T17C. Dashed line represents the
natural lifetime of tryptophan triplet state.

57

Chapter 4

Sequence Specific Wormlike Chain Model

4.1 Introduction to the Heteropolymer Model
There is considerable evidence that highly denatured states act globally like random,
non-interacting chains restricted by excluded volume of the chain and that this chain reconfigures
on the nanosecond-microsecond timescale

44

. Thus, a wormlike chain model with SSS theory is

sufficient to describe these states and has been shown to yield a corresponding Trp/Cys distance
distribution shift as one gradually decreases the excluded diameter (dα ) with denaturant
concentration. It is not clear if such a model can be extended to physiologically relevant conditions.
The model has been shown to lead to extremely small values of excluded volume for solvent
conditions that favor folding and we have found that for some sequences even dα = 0 does not
make a chain compact enough to match experimental kR. A growing number of unfolded proteins
have been shown by NMR methods to have some residual structure under folding conditions

31-35

,

but these methods do not fully characterize the distribution of conformations and cannot measure
the reconfiguration time within the distribution. The experimental work in this thesis shows that
unfolded proteins under folding conditions are much more compact than proteins in high
denaturant. All these observations in low denaturants suggest more complicated unfolded states
with transient intrachain interactions and secondary structure formation. Therefore a homogenous

58

wormlike chain model is obviously too simple to describe an unfolded protein under folding
conditions.
This chapter shows that a sequence-specific wormlike chain model that is statistically
reweighted to favor highly collapsed conformations of the chain can reproduce experimental
contact quenching rates measured under folding conditions. The motivation for this work is to find
a realistic description of the probability distribution of unfolded states that did not require
enormous amounts of computation. Similar efforts have been successful in comparing NMR data
31, 94

to randomly generated chains or molecular dynamics simulations

. But because I intended to

calibrate my computational model with an experimental method that is only sensitive to the tail of
the distribution of random conformations, we required many more than the ~10000 conformations
generated by previous methods in order to have statistical accuracy. Furthermore, I did not want
to bias our distribution with native-like structure that may not be representative of the full range of
accessible conformations. Therefore, I chose a simple wormlike chain model with excluded
volume and a bias for short-range contacts between residues of similar hydrophobicity. This model
reproduces experimental rates from different proteins under a variety of denaturant concentrations
that vary over more than an order of magnitude. After determining the magnitude of reweighting
from contact quenching, I then show that this model successfully predicts measured values of
paramagnetic resonance enhancement for two different proteins.
I introduce this model by first placing it in context of previous models and methods. Almost
20 years ago, Dill and coworkers sought to describe protein folding in terms of the balance

59

between preferential de-solvation of different amino acids and chain conformational entropy

95

.

In most subsequent work, the effect of denaturant is then modeled with a change in the solvent
accessible surface area (ASA) of various amino acids sidechains and/or backbone
of papers by Thirumalai and coworkers elucidates these ideas very well

96-99

. A series

100-102

. They first

simulate the dynamics of a coarse-grained model of a protein without denaturant and then use the
Tanford transfer model to produce statistical distributions of conformations at various denaturant
concentrations.

This model has been remarkably successful in reproducing average

intramolecular distances of denatured proteins measured by single-molecule FRET

102

.

For this work I have taken a complimentary approach. Because coarse-grained atomistic
models cannot produce sufficiently random conformations in high denaturant, I have chosen to
start with a wormlike chain model that has proven to be a realistic model of many highly denatured
proteins over a range of sampled conformations e.g. the average radius of gyration from SAXS, the
longest extensions from single-molecule stretching and the shortest distances from intramolecular
quenching

44, 85, 103

. Furthermore, because a wormlike chain model is inherently random, I

needed a method to choose those conformations which would be favored when denaturant is
decreased. Simply decreasing ASA does not explicitly do that because it does not contain
specific intramolecular interactions.

Therefore I assign an energy to each conformation based on

favorable interactions between residues of similar hydrophobicity.

60

4.2 Boltzmann Energy Reweighting
Wormlike chains are computationally generated in the same way as described in Chapter 2.
For each wormlike chain conformation of a particular sequence, I define a total energy, ETOT , that
α

is the sum of pairwise interactions between the C atoms,

ETOT = − ∑

i − j >1

ei , j
ri − rj

(4.1)

⎧0, hi − h j > 0.3
⎪
ei , j = ⎨
⎪σ , hi − h j ≤ 0.3
⎩
where, i and j are the indices of two non-adjacent residues,

ri − rj

is the distance between

these two residues which must be larger than dα and less than 6.5Å (a reasonable cutoff distance
104

for hydrophobic associations

), and hi and hj are the normalized hydrophobicities of the two

Normalized Hydrophobicity

residues as determined by the Miyazawa

1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Phe Met Ile Leu Cys Trp Val Tyr Ala His Thr Gly Arg Ser Gln Pro Asn Glu Asp Lys

Figure 4.1 Hydrophobicity scale used in the energy reweighted WLC model. The values are
taken from the Miyazawa-Jernigan scale and normalized to span the interval 0 to 1.

61

Jernigan scale (see Figure 4.1). ei,j is the strength of the hydrophobic potential between two
residues which equals zero if one is hydrophobic and one is hydrophilic and equals an adjustable
parameter, σ, if both are hydrophobic or hydrophilic. I used 0.3 as the cutoff value because it
corresponds to the spread of the normalized hydrophobicity distribution of all amino acids. σ (in
units of kT•Å) exhibits the strength of the hydrophobic interaction and is tunable according to
various solvent conditions.
Using ETOT and r as independent reaction coordinates, I can define a 2-D probability
distribution P(r , ETOT ) so that

1 = ∫ dr ∫ dETOT P(r , ETOT )

(4.2)

∫

and the 1-d Trp-Cys distance distribution is defined as P ( r ) = P ( r , ETOT ) dETOT .
Let Z (r ) represent the energy reweighted distribution of distance r between the Trp and Cys
α

C position. Z (r ) can be written as

Z (r ) = N ⋅ ∫ P(r , ETOT )exp(− ETOT / kT )dETOT
= N ⋅ ∫ P(r , ETOT )dETOT

∫
⋅

P(r , ETOT ) exp(− ETOT / kT )dETOT

= N ⋅ P(r ) ⋅ exp(− ETOT / kT )

∫ P( r , E

TOT

)dETOT
(4.3)

r

lc
1
= ∫ P(r ) ⋅ exp(− ETOT / kT ) r dr
N dα

N is a normalization constant. For any given set of conformations with the same r (within 0.1 Å),
the distribution of ETOT is typically Gaussian (central limit theorem) but there are often outlying
conformations that significantly skew Z(r) due to the exponential averaging. Therefore, the limits
62

of the integral in equation 4.3 are set at two standard deviations below and above the mean,
eliminating ~2% of conformations.
To compare Z(r) to experimental measures of intrachain distance I again need to use Szabo,
Schulten and Schulten (SSS) theory (Chapter 2.4). After substitution of P(r) by Z(r), equations
2.9 and 2.10 become
∞

kR =

∫ q(r )Z (r )dr

dα

1
1
= 2
kD+ kR D

lc

∫

dα

dr ⎧
⎪
⎨
Z (r ) ⎪
⎩

lc

∫
r

(4.4)

⎫
⎪
q ( x ) − k R ) Z ( x)dx ⎬
(
⎪
⎭

2

(4.5)

where r is the distance between the tryptophan and cysteine and Z(r) is given by equation 4.3. D
is the effective intramolecular diffusion constant and q(r) is the distance dependent quenching rate.
An important question one might ask is “Is the Trp/Cys distance r a good reaction
coordinate?”

The answer will be certainly yes if the distribution is Gaussian, because a

Gaussian chain should yield the same scaling for the radius of gyration, the end-to-end distance,
any intrachain distance or other length measurements. For sufficiently long wormlike chains (>
10 residues) with persistence length lp =4 Å, it has been shown that the distributions of the
54

end-to-end distance are well-described by a Gaussian distribution .

Since the radius of

gyration has been established as a good measure of the volume of polymers, especially proteins,
how well the Trp/Cys distance is correlated to radius of gyration should help us determine if this
measure is also a good reaction coordinate. Figure 4.2 shows the rank correlation between

63

radius of gyration (Rg) and Trp47/Cys23 (r23-47) (a) and between Rg and Trp47/Cys57 (r47-57)
(b) of protein L. These figures are created in the following way: the Trp/Cys distance and Rg of
chain i are each converted to a rank over 2 million chains, and then I use these two ranks to
generate a 2-D histogram. While r23-47 yields a very clear positive correlation with Rg, the
correlation between r47-57 and Rg is not so obvious. This is probably because the Trp47/Cys57
loop is so short that it might be heavily constrained by the stiffness of the chain. But as long as
these two ranks do not correlate in an opposite way, we are safe to use the Trp/Cys distance as
the reaction coordinate.

64

20 a

200

18

180

16

160

14

140

r23-47 12

120

rank 10

100

(x10 ) 8

5

80

6

60

4

40

2

20

0

0

2

4

6

8

10

12

14

16

18

20

0

20 b
18

200

16

160

14

140

r47-57 12

120

rank 10

100

(x10 ) 8

5

80

6

60

4

40

2

20

0

0
0

180

2

4

6

8

10 12 14
5
Rg rank (x10 )

16

18

20

Figure 4.2 Rank correlations between radius of gyration (Rg) and Trp47/Cys23 (r23-47) (a) and

between Rg and Trp47/Cys57 (r47-57) (b)

65

SSP

0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1 0
-0.2
-0.3
-0.4
-0.5

Uexch
Ugdn

10

20

30

40

50

60

70

Residue Index
Figure 4.3 Second structure propensity (SSP) of Uexch (red) and Ugdn (blue) states of drkN

4.3 Secondary Structure Propensity (SSP)

It is relatively straightforward to force certain links of the standard wormlike chain to have
axial and azimuthal angles of α-helix or β-strand secondary structure. Since secondary structure
is not stable in the unfolded state, each link in the chain can be assigned a secondary structure
propensity such as determined by relative NMR chemical shifts of an unfolded protein

31, 105

. If

a random number falls below this propensity, then all 10 links associated with one amino acid
adopt the canonical angles for helical (θ = 9.2o, φ = 176o) or strand structure (θ = 0 o and φ = 0 o).
Secondary structure was added to the WLC model of drkN. The SSPs of drkN were
α

calculated from the combined C

β

and C

31

secondary chemical shifts
66

(deviations of

experimental chemical shifts from their expected random-coil values). (see Figure 4.3) Figure
4.4 shows the comparison of the average hydrophobic energy <ETOT>r as a function of Trp/Cys
distance between chains with and without secondary structure constraints and we can clearly see
that adding secondary structure constraint lowers the total energy to some extent. However, if we
just compare the results in Table 4.1(a) (without SSP) and 4.1(b) (with SSP), such as the average
Trp/Cys distance, <r>, and the efficient diffusion constant, D, we can barely observe differences.
Therefore, for typical residual structure observed in unfolded proteins, this added feature has very
little effect on the average radius of gyration or other coarse measures of the unfolded ensemble,
but as will be shown below, it does improve prediction of more specific measures, such as
paramagnetic resonance enhancement.

-3.0
-3.5

r vs <Etot>(r) w/ SSP
r vs <Etot>(r) w/o SSP

<Etot>(r)

-4.0
-4.5
-5.0
-5.5
-6.0
-6.5
-7.0
0

20

40

60

80

r

Figure 4.4 Average total energy <ETOT>r of wormlike chains with (black) and without (red)

secondary structure constraints.

67

Table 4.1 Energy-reweighted wormlike chain parameters and effective diffusion coefficients for
drkN
(a) without SSP constraints
-1

-1

-6

2 -1

kD+(s )

lp(Å)

dα(Å)

<r> (Å)

σ(1/T)

D (x10 cm s )

2

338000 2250000

4

4

34.7435

1

1.4

6

680000 2380000

4

4

32.7247

1.6

1

2

580000

964000

4

4

29.6232

0.3

0.37

6

580000

870000

4

4

29.6232

0.3

0.34

kD+(s )

lp(Å)

dα(Å)

<r> (Å)

σ(1/T)

D (x10 cm s )

2

338000 2250000

4

4

34.2501

1.3

1.4

6

680000 2380000

4

4

32.1777

2

0.9

2

580000

964000

4

4

29.3944

0.4

0.38

6

580000

870000

4

4

29.3944

0.4

0.34

[GdnHCl]

kR (s )

C2

C60

(b) with SSP constraints
[GdnHCl]

-1

kR (s )

-1

-6

2 -1

C2

C60

4.4 Convergence of σ

This model has only one adjustable parameter, σ, which determines the strength of reweighting
due to finding two similar amino acids in close proximity within a wormlike chain. Figures 4.5(b)
and (c) shows the contour plots of the probability vs. ETOT and r for σ = 0.2 and 2.5 for protein L
K23C. Increasing σ lowers the energy of the smallest r relative to the largest r which then
differentially biases the probability at low r after reweighting, as shown in Figure 4.5(a). After
68

reweighting, the resulting probability function, Z(r), is used in equation 4.4 to determine the
reaction-limited rate of Trp-Cys contact quenching which can be compared to experimental values
under various solvent conditions. Figure 4.6(a) plots the values of σ that best fit experimental
values of kR for various proteins and mutants. The standard deviation in σ is about 30% at 2.3 M
GdnHCl and 20% at 0 M GdnHCl. At the highest values, kR is very sensitive to σ; for a 10%
change in σ, kR changes by about 25%, much larger than our uncertainty. There is good
convergence between sequences which suggests that this model is effective at capturing the
apparent collapse to varying degrees of all sequences in decreasing denaturant.

69

0.006

a

Z(r)

0.004

0.002

0.000
0

ETOT

b
-1
-2

c
-16

ETOT

-18

-20

-22
(Å)

-24
0

20

40

60

r
70

Figure 4.5 a) Probability
distribution of Trp/Cys
distances in protein L K23C
before (black) and after
(red) reweighting for σ =
2.5. b) Contour plot of
probability vs r and E for
b) σ= 0.2 and c) σ= 2.5.

6
apocytochrome C
HypF-N W81F
Protein L K23C
Protein L T57C
Protein G E19C
Protein G T51C
Protein L K23C F22A

5
σ (kT•Å)

4
3
2
1
0
0

1

2

3

4

5

6

[GdnHCl] (M)

Figure 4.6 The values of which best reproduce the measured reaction-limited rates using
equation 4.4 for various sequences and [GdnHCl].

4.5 Intrachain Contacts and Hydrophobic Energy

As shown in equations 4.1 and 4.3, I reweight the probability distribution based on close (1/r
with a cut off at 6.5 Å) interactions of residues of similar hydrophobicity. I find that this model
gives similar trends and good convergence for many different variations of the distance
2

dependence of ETOT. Figure 4.7 shows σ versus [GdnHCl] for 1/r distance dependence, a
Lennard-Jones distance dependence and no distance dependence (square well).
magnitude of σ varies from one model to another,

71

While the

2

Square Well Potential

0.8

20
15

0.4

10

σ

0.6
σ

1/r Potential

0.2
0.0

5

0

1

2

3

4

5

0

6

0

1

[GdnHCl] (M)

5

6

1.5

4
3

σ

σ

4

1/r Potential, no cutoff

2.0

5

2

1.0
0.5

1
0

3

[GdnHCl] (M)

Lennard-Jones Potential

6

2

0

1

2

3

4

5

[GdnHCl] (M)

6

0.0

0
1
apocytochrome c
HypF-N
Protein L K23C
Protein L T57C
Protein G E19C
Protein G T51C

2

3

4

5

6

[GdnHCl] (M)

Figure 4.7 Values of σ that best fit measured reaction-limited rates for various forms of ETOT.
2

1/r potential:

ETOT = −

∑

ei , j

2
i − j >1 ( ri − r j )

Square-well potential:

⎡⎛
4
Lennard-Jones potential: ETOT = ∑ ei , j ⎢⎜
⎢⎜ ri − r j
i − j >1
⎢⎝
⎣

12

⎞
⎟
⎟
⎠

⎛ 4
−⎜
⎜ ri − r j
⎝

1/r, no cutoff: equation 4.1 in the text for dα ≤ ri − r j < ∞

72

⎞
⎟
⎟
⎠

6⎤

ETOT = −

∑

i − j >1

ei, j

⎥ for d ≤ r − r ≤ 6.5 Å.
α i
j
⎥
⎥
⎦

Tyr excluded
6

σ (kT•Å)

5
4
3
2
1
0

0

1

2

3

4

5

6

[GdnHCl] (M)

Hydrophobic: VILFMYW
6
apocytochrome c
HypF-N
Protein L K23C
Protein L T57C
Protein G E19C
Protein G T51C

σ (kT•Å)

5
4
3
2
1
0

0

1

2

3

4

5

6

[GdnHCl] (M)

Hydrophobic: VILFMYWCA
6

σ (kT•Å)

5
4
3
2
1
0

0

1

2

3

4

5

6

[GdnHCl] (M)
73

Figure 4.8 Values of that best fit
measured reaction-limited rates for
various definitions of hydrophobicity.

Hydrophobic interactions only
10
8

σ (kT•Å)

apocytochrome c
HypF-N
protein L K23C
protein L T57C
protein G E19C
protein G T51C

a

6
4
2
0
0

1

2

3

4

5

6

[GdnHCL] (M)
0.006

b

P(r)
Z(r) 6M
Z(r) 0M
Z(r) 0M only Hydrophobics

0.005

P(r)

0.004
0.003
0.002
0.001
0.000

0

10

20

30

40

50

60

Trp-Cys Distance, r (Angstroms)

Figure 4.9 a) Values of that best fit measured reaction-limited rates for
hydrophobic-hydrophobic interactions only. b) Comparisons of unweighted (P(r)) and weighted
(Z(r)) probability distributions for weighting schemes using hydrophilic interactions (green and
red lines) and only hydrophobic interactions (blue line).

74

the trend of each sequence with denaturant concentration is similiar. However, if the energy
function does not cut off at 6.5 Å, then the convergence in σ between sequences degrades. This
suggests that the inclusion of longer range interactions introduces sequence-dependent complexity
to the reweighting function.
In my model I use a normalized Miyazawa-Jernigan scale to assign a numerical
hydrophobicity value to each residue in the sequence.

I assume that residues of similar

hydrophobicity will prefer to be near each other, regardless of whether the pair is hydrophobic or
hydrophilic. Therefore, only pairs with hydrophobicity values within 30% of each other are
included in the energy function in equation 4.1.

Figure 4.8 shows that this model is somewhat

sensitive to the method of defining hydrophobicity.

In the Miyazawa-Jernigan scale, amino

acids fall into two well separated regimes except for tyrosine. If interactions with tyrosine are
given no weight (ei,j = 0), there is almost no change in σ except for the T57C mutant of protein L,
which covers a very short loop (10 residues) that includes one tyrosine. Also, if the most
hydrophobic residues are given a hydrophobicity value of 1 and all other residues a value of 0
(based on categories used in hydrophobic cluster analysis), there is a greater dispersion in values of
σ for different sequences at the same [GdnHCl].

I have also considered the possibility that only

interactions between hydrophobic residues drive the collapse, as shown in Figure 4.9(a). The
convergence of σ is not as good as including all interactions (Figure 4.6) and increasing σ
primarily increases the probability at the lowest r (where the experimental method is most
sensitive) rather than significantly shifting the entire distribution as shown in Figure 4.9(b). From

75

these results I conclude that the primary driving force of collapse is the sequestration of residues of
similar hydrophobicity regardless of their overall polarity.

4.6 Comparison between Two Protein L Mutants

I also find this model is sensitive to a single mutation that changes the hydrophobicity of the
sequence. Figure 4.10 shows that with the mutation of Phe22 to Ala in protein L, the average
number of favorable contacts (ei,j ≠ 0) increased locally and decreased in several positions far from
position 22. Therefore mutant F22A K23C has fewer favorable conformations that act to collapse
the chain. Since the measured reaction-limited rates for K23C F22A are lower than for K23C at
the same denaturant concentrations and the chain compacts less, the values of σ required to match
the experimental data are approximately the same for both sequences.

average #contacts

5
4
3
2
1
0

10

20

30

40

50

60

Residue Index
Figure 4.10 Average number of favorable contacts of each residue in protein L K23C (black) and
F22A K23C (grey). The vertical dashed line shows the position of the mutation F22A.

76

I have also used residue contact maps generated by the WLC model before (Figure 4.11(c)) and
after (Figure 4.11(a,b,d)) reweighting to analyze the conformational changes of these two protein
L mutants. In Figure 4.11, the lower-right triangle region gives the average distance <ri,j >
between residue i and j and the upper-left triangle region gives the standard deviation (spread of
the distance distribution) of these two residues. This statistical model yields a trivial result at 6M
GdnHCl(Figure 4.11(c)), that is, distant residues have a longer average contact distance and a
wider distance distribution. Figures 4.11(a) and (b) show the contact maps for σ =2.5. The
average terminal distance decreased from 50 Å to about 30 Å and the curvature in these contour
lines also indicates an overall compaction of unfolded states. In order to measure the slight
difference between K23C F22A (Figure 4.11(a)) and K23C (Figure 4.11(b)), I subtract the
contact map of K23C from that of F22A K23C (Figure 4.11(d)) which shows the C-terminus of
F22A K23C mutant makes less contacts with the middle region of the protein than K23C mutant
but the N-terminus actually makes more contacts. One may also notice from Figures 4.11(a) and
(b) that those points with significant average distances also have smaller deviations, which may
be due to the overweighted contribution from those conformations with the lowest energies.

77

a

c

b

d

Figure 4.11 Contact maps of protein L K23C F22A (a, c) and K23C (b, c) at σ =0 (c) and σ =2.5
(a,b). Contact map (d) is the subtraction of (a) by (b). The lower-right triangle region shows the
average distance between residues i and j and the upper-left triangle region gives the standard
deviation of that average distance.

78

4.7 PRE Prediction

The best measure of a model is its ability to predict experiments de novo. With good
convergence of σ at low concentrations of denaturant for various sequences, I have used energy
reweighting to predict another measure of the unfolded ensemble in chains outside the initial data
set. Paramagnetic resonance enhancement (PRE) is a type of NMR measurement which reports
long-range pairwise interactions by using a spin label attached to the polypeptide chain. Typically
15

1

N- H HSQC experiments are recorded with the label in two states: oxidized (paramagnetic) and

reduced (diamagnetic). Either the transverse relaxation rate of the paramagnetic sample, R2P, or
the ratio between the intensity of a peak in the paramagnetic sample, Iox, and the diamagnetic
sample, Ired, reflects the distance from the spin-labeled site to every backbone H-N

R2 P = K (r i ) −6 (4τ C +

3τ C
)
2 2
1 + ω Hτ C

I ox R2 R exp(− R2 P t )
=
I red
R2 R + R2 P

(4.7)

(4.8)
−32

where R2R is the transverse relaxation rate in the diamagnetic sample, K is 1.23×10

6 −2

cm s

for the interaction between a single electron and a proton and ωH is the Larmor frequency of a
proton (750 MHz). τC is the effective correlation time, typically 2 ns. In each wormlike chain the
i

α

distance r between the C of the spin-labeled residue and that of the ith residue is calculated.
Individual R2P and Iox/ Ired can hence be calculated for each chain using equations 4.7 and 4.8
and the average (experimental) values can be calculated as

< R2 P >i = ∫ Z (r i ) R2 P (r i )dr i

79

(4.9)

< I ox I red >i = ∫ Z (r i ) I ox (r i ) I red (r i )dr i

(4.10)

i

where Z(r ) comes from equation 4.3. For a random coil, one would expect deviations from a ratio
of 1 only at residues nearby the spin label in sequence; deviations far in sequence indicate that the
spin labels are in contact with the unfolded chain, at least transiently.
Figure 4.12 shows the calculated and measured PRE values for multiple labels of drkN SH3
domain

31

and acyl-CoA binding protein (ACBP)

35

. I find remarkable agreement between

calculated and measured data. However the agreement between measured (black bars) and
calculated PRE is much better if residual secondary structure is included in the construction of the
wormlike chains (red lines) than without it (green lines). For ACBP, I find good agreement for
most of the chain but for certain regions, particularly residues 24-28, 29-35 and 50-60, there is
little non-local interaction, regardless of the position of the paramagnetic probe. Examination of
the sequence shows that these regions are almost exclusively hydrophobic (24-28) or hydrophilic
(29-35 and 50-60) so local interactions within these regions will contribute significantly to ETOT.
However if equation 4.1 is altered to include only interactions at least 4 residues away (|i-j|>4),
then agreement with the data is somewhat improved (blue lines) as non-local interactions are more
highly weighted (although σ increased because the total number of interactions decreased).
These results suggest there are still further refinements that can be made to this model with the
addition of different types of data.

80

60

1.0

a

50

0.8

Ired I ox
I ox /Ired

R2
R2PP

40
30
20

0.6
0.4
0.2

10
0

b

0.0

0

10

20

30

40

50

60

0

10

20

40

50

60

Residue

Residue
c

1.0

d

1.0
0.8
ox I
IIox/Ired
red

0.8

IIox Iredred
ox/ I

30

0.6
0.4

0.6
0.4

0.2

0.2

0.0

0.0
0

20

40

60

0

80

20

40

60

80

Residue

Residue

Figure 4.12 Measured (bars) and calculated (lines) PRE per residue for drkN SH3 domain (a and
b) and ACBP (c and d). In each graph the green lines correspond to a completely random set of
wormlike chains and the red lines correspond to an ensemble of wormlike chains with some
residual secondary structure given by measured NMR chemical shifts. The blue lines
correspond to the ensemble with secondary structure using a modified version of equation 4.1 in
which only interactions more than 4 residues apart in sequence are included (σ = 6). The
paramagnetic labels are located at residue 2 (a), the N-terminus (b), residue 36 (c) and residue 65

(d). For (a) τC=0.5 ns, a scaling factor in equation 4.7, is adjusted to best match the measured
data. Pearson Correlation coefficient between bars and red lines equals to 0.89 (a), 0.41 (b), 0.35
(c) and 0.59(d).
81

4.8 Charge Effects

Intrinsically disordered proteins generally have a high charge content and the net charge has
been shown to correlate with the radius of gyration. So in order to better model the polymer
dynamics of intrinsically disordered proteins, an inclusion of both hydrophobic and charge
effects may be necessary. After the addition of the electrostatic energy term, the total energy
(equation 4.1) of each chain becomes,

ETOT = Eh + Ee = Eh + γ

∑

i − j >1

qi q j
ri − rj

=

∑

i − j >1

γ qi q j − ei , j
ri − rj

(4.11)

⎧1, if i is Arg or Lys
⎪
qi = ⎨−1, if i is Asp or Glu
⎪0, otherwise
⎩

γ exhibits the strength of effective electrostatic interactions between two charged residues and qi
is the charge of residue i. The above expression of qi is only valid at pH = 7. At lower pH value,
one may need to consider the positive charges introduced by His. Also Arg and Lys will become
mostly uncharged in basic solution and Asp and Glu will become uncharged in acidic solution.
According to the Henderson-Hasselbalch equation, a charged residue possesses a certain
probability of gaining or losing an electron determined by its pKa and the solution’s pH. To take
into account this uncertainty in our simulation, for each chain generated, a random number (0 to
1) independently assigned to each residue was compared to the probability determined by
Henderson-Hasselbach equation. If this random number is smaller than the probability, this
residue shall carry charge for this particular chain.
AavLEA has a large fraction of charged amino acids (~50%) and has been shown in chapter
82

2 to be near equally extended and diffusive over all denaturing conditions (Figure 3.11). The
reaction-limited and diffusion-limited rates and corresponding WLC parameters are listed in
Table 4.2. We find the range of σ values is much lower than previously determined values for
other proteins (Figure 4.6), so there has to be some additional swelling forces opposing the
hydrophobic effects for this protein and the high volume of charges distributed over the entire
AavLEA peptide can be a good candidate.

Table 4.2 Energy reweighted worm-like chain parameters and effective diffusion coefficients for
AavLEA without charge
-1

-1

-6

2 -1

kD+(s )

lp(Å)

dα(Å)

<r> (Å)

σ(1/T)

D (x10 cm s )

0
1
6

256000 830000
260000 1024000
223000 696000

4
4
4

4
4
4

32.1233
32.1233
32.6823

0.6
0.6
0.45

0.55
0.65
0.5

0

267000

950000

4

4

32.0618

0.6

0.6

250000

643000

132000

811000

4
4

4
4

32.2638
33.9852

0.55
0.1

0.4
0.8

[GdnHCl]

kR (s )

S2C

T58C
1
6

Table 4.3 reweighting with both σ and γ for AavLEA T58C

[GdnHCl]

assumed σ(1/T)

γ(1/T)

0
1
6

2.5
1.8
0.5

31
19
5

83

One way to take into account the charge effects is to first assume an identical or different γ
value for each [GdnHCl] and then adjust σ to match measured reaction-limited rates. However,
because we have little knowledge of this newly introduced parameter, γ, we don’t know what to
choose for its value. So instead, I take a complimentary approach and assume a similar set of σ
values for AavLEA at different [GdnHCl] to those values shown in Figure 4.6. And then fit γ to
experimental data. The Trp-Cys quenching experiments of AavLEA were carried out in pH=7, so
I didn’t consider partial charges discussed above.
I find the addition of the charge effects to the WLC model yields very meaningful results for
the AavLEA T58C mutant. The fitted γ values to σ values are listed in Table 4.3. We can see γ is
consistently about 10 times larger than σ at all three GdnHCl concentrations, suggesting that the
strength of charge effects is homogenously greater than the hydrophobic effects at all [GdnHCl]
(note that qi in equation 4.11 only indicates the sign of the charge, so σ and γ have the same
dimension), which makes sense if one considers the charge effects as a representative of the
electrostatic forces and the hydrophobic effects as a result from the Van der Waals forces. The
consistent ratios of magnitude between 0M and 6M GdnHCl for both σ and γ also indicate that
these two parameters might represent well the solvent conditions. As the GdnHCl concentration
decreases, the ionic strength decreases accordingly causing a weaker charge shielding. Hence,
the increase of γ reflects a longer Debye length at low denaturant concentration. Similarly, the
increase of σ as [GdnHCl] decreases reflects a weaker polar shielding of the water molecules
from the guanidinium ion. Figure 4.13 shows the Trp/Cys distance distributions before and after
reweighting by σ and (or) γ. We can see an opposing effect of σ and γ on the distance distribution
84

P(r) and a combination of them results in a squeezed distribution with a decreasing in the
population of both compact and stretched conformations.
aavLEA
(W30-C58 loop)
0.006

no weighting
weighted with σ=2.5 γ=31
weighted with σ=2.5 γ=0
weighted with σ=0 γ=31

0.005

Z(r)

0.004
0.003
0.002
0.001
0.000

0

10

20

30 40 50
Trp/Cys distance

60

70

Figure 4.13 Probability distribution of Trp-Cys distances in AavLEA T58C before (black) and
after reweighting for σ = 2.5 (red), γ= 31 (green) and σ = 2.5, γ= 31 (blue)

Table 4.4 reweighting with only γ for AavLEA S2C

[GdnHCl]

-1

-1

-6

2 -1

kD+(s )

lp(Å)

dα(Å)

<r> (Å)

γ(1/T)

D (x10 cm s )

256000 830000
260000 1024000
223000 696000

4
4
4

4
4
4

31.4681
31.4681
32.0442

10
10
8

0.5
0.6
0.45

kR (s )

S2C
0
1
6

The charge effects on the S2C mutant seem to work on the opposite way: they further
compact the chains (Table 4.4). While it is good to see the heterogeneity between different loops

85

of the same protein, the homogenous effects of σ and γ make it impossible to fit both to the
experimental data. This suggests there are other effects than charge preventing the C2-W30 loop
from collapsing.

4.9 Conclusion

The main feature of this model is that a distribution of random conformations is reweighted to
favor those with close interactions between residues of similar hydrophobicity. This includes
residues that are relatively hydrophilic because I assume that in a random ensemble of collapsed
conformations, not all hydrophilic residues will be accessible to solvent and intramolecular
interactions between hydrophilics will also help collapse the chain. Figure 4.14 shows the
cumulative probability of buried surface area for hydrophobic (Phe to Tyr in Figure 4.1) and
hydrophilic residues of protein L. At σ = 0, they are trivially identical and remain fairly similar at
σ = 2.5 even though there are twice as many hydrophilic residues in the sequence than
hydrophobic residues. Therefore, the typical unfolded conformation does not have a hydrophobic
core, but in water-like solvent conditions hydrophobes are significantly more buried from water
than in highly denatured conditions.

Interestingly, I also found that binary HP models (Figure

4.8), in which all residues were grouped into only two categories of hydrophobicity, produced
more variance in σ than a more graduated model based on similarity of amino acids on the
Miyazawa-Jernigan scale (Figure 4.6) and that only including interactions of

86

cumulative probability

1.00
0.75
0.50
σ=0 hydrophobic
σ=2.5 hydrophobic
σ=0 hydrophillic
σ=2.5 hydrophillic

0.25
0.00

0

30

2

(Å
60 )

90

120

150

buried surface area
Figure 4.14 The cumulative probability ( C (a ) = Ν

a

∫0 P( x)dx , N is a normalization constant) of

buried surface area for hydrophobic and hydrophilic residues of protein L at σ =0 and σ = 2.5.
Buried surface area is defined as the cross-section of overlap between two links on the wormlike
chain assuming each link has spherical volume of diameter 6.5 Å. Since no two links can be
2

closer than dα = 4 Å, the maximum buried surface area is 20.6 Å per residue.
hydrophobic residues did not significantly collapse the chains at all (Figure 4.9). This suggests
that collapse of random chains in low denaturant conditions is driven by interactions of all amino
acids of similar polarity. I also found that convergence of σ required only very short range
r-dependence (less than 6.5 Å) but the form of the distance-dependence did not matter. This
suggests that only the closest intramolecular interactions drive the collapse of random chains.
As described in chapter 2, an intramolecular diffusion coefficient, D, which reflects the time
for unfolded conformations to reconfigure, can be determined from Trp-Cys quenching

87

measurements (see equation 4.5). Figure 4.15 shows D for various sequences and [GdnHCl]
using Z(r) from the model and measured kD+. There is some variance in D at high denaturant
which increases as denaturant decreases, with unstructured or unstable sequences (apocytochrome
c and protein L F22A) remaining fairly diffusive and well folded proteins (proteins L and G)
slowing dramatically. A recent measurement of Trp/Cys contact quenching of protein L after
5 -1

rapid dilution (250 μs) of denaturant in a microfluidic mixer determined kD+ ~10 s which gives
-9

D=2 x 10

2 -1 38

cm s

, more than two orders of magnitude slower than apocytochrome c.

Importantly, in this measurement diffusion was so slow that kobs ~ kD+ so kR could not be
determined directly. Therefore a predictive model of Z(r) is critical to determine D.

10-7

2 -1

D (cm s )

10-6

apocytochrome C
HypF-N W81F
Protein L K23C
Protein L T57C
Protein G E19C
Protein G T51C
Protein L F22A K23C

10-8

10-9

10-10

0

1

2

3

4

5

6

[GdnHCL]
Figure 4.15 The values of D which best reproduce the measured diffusion-limited rates using
equation 4.5 with values of σ shown in Figure 4.6.

88

Chapter 5

Conclusions

In the end, I would like to summarize the conformational and diffusional properties of several
typical proteins with very different folding profiles. Figure 5.1 shows the measured values of
reaction-limited and diffusion-limited rates for various sequences shown in Table 5.1. There is
little difference in rates at 6 M GdnHCl, but the reaction-limited rates at low concentrations of
GdnHCl (determined either from experiment or molecular dynamics simulation) vary by over an
order of magnitude. The slowest kR, which corresponds to a more expanded chain, are observed
in apocytochrome c, which is intrinsically disordered, and the amyloidogenic protein HypF-N.
Proteins L and ACBP are the fastest. Significantly, the mutation of a single phenylalanine to
alanine in protein L decreases kR significantly at all [GdnHCl]. Therefore sequence plays a
significant role in determining intramolecular contact formation under conditions that favor
folding.

89

Table 5.1 Several protein sequences studied in our lab

protein
apocytochrome c

sequence
length
probe position (mutation)
104
C17,W59

loop
length
42

HypF-N

91

W27, C65(C7S/C40S/W81F)

38

G

56

L

64

ACBP

86

C19, W43 (E19C)
W47, C51 (T51C)
C23, W47 (K23C)
C23, W47 (F22A/K23C)
W47, C57 (T57C)
C17, W58 (T17C/W55A)
W58, C86 (W55A/I86C)

23
8
23
23
10
28
28

90

Figure 5.1 Experimental reaction-limited (black) and diffusion-limited (white) rates measured at
various concentrations of denaturant for (a) ACBP T17C (b) ACBP I86C (c) HypF-N W81F, (d)
apocytochrome c, (e) Protein L K23C and (f) Protein G E19C. The errors in these rates are
typically 10%. The triangular points in (a) and (b) are measured values from molecular
dynamics simulations. The triangular points in (e) are the destabilizing mutation K23C F22A in
which the unfolded state is observable in as little as 0.25 M GdnHCl. The reaction-limited rate
in (e) (black circle) at 0 M GdnHCl was calculated from a Trp-Cys distribution derived from a
37

molecular dynamics simulation

using equation 2.9.

All other points were derived from

measured tryptophan triplet decay rates at various temperatures and viscosities using equations
2.5 and 2.6.

91

7

10

6

10

6

10

5

10

4

10

5

10

7

10

6

10

5

10

4

10

7

10

6

10

5

10

4

10

Figure 5.1

92

Theoretically, I developed a sequence specific worm-like chain model to compare to the
experimental results of those unfolded proteins, especially under folding conditions. I believe the
method of re-weighting worm-like chain distributions to account for intrachain interactions is
robust, realistic and predictive and may be a useful starting point for further models of protein
folding. For example, a recent study of molecular dynamics folding trajectories found that
randomly generated chains folded more slowly and less efficiently than chains generated with
native-like conformations determined by a knowledge-based potential

106

. This indicates that not

all conformations in the unfolded state are equally likely to proceed to the folded state and that
there may in fact be specific bottlenecks for leaving the unfolded basin. However, that study did
not examine how realistic or probabilistic the productive unfolded conformations are. That study
also shows the native-like chains are much more extended. If extension is required for folding
then finding the conformation that leads to folding may be extremely unlikely and knowing how
quickly a random unfolded conformation can access the productive conformation defines the
folding rate.
Figure 5.2 shows the intramolecular diffusion coefficients of several proteins in various
folding categories. We can see nonfolding peptides and intrinsically disordered proteins
(aavLEA and apocytochrome C) are highly diffusive in water while well-behaved proteins
(protein L and ACBP) diffuse very slowly. The diffusivities of aggregation-prone proteins (long
polyQ, L F22A and HypF-N) lie right in the middle between these two groups. Therefore I
propose that there is a dangerous dynamic range of diffusivity in which unfolded proteins or
polypeptides are more likely to aggregate.
93

Intramolecular Diffusion Coefficients in Water
10

Unstructured
IDPs
Aggregation prone
well behaved

D x 106 (cm2/s)

1

0.1

0.01

0.001

A
C
BP
pr
ot
ei
n
L
w
t

no
nf
ol
di
ng

pe
pt
id
es
aa
ap
vL
oc
EA
yt
oc
hr
om
ec
lo
ng
po
pr
ot
ly
Q
ei
n
L
F2
2A
H
yp
FN

0.0001

Figure 5.2 Intramolecular diffusion coefficients of several different proteins. Long polyQ is an
aggregation-prone protein like HypF-N. One single mutation F22A destabilizes well-behaved
widetype protein L tremendously and gives it a tendency to aggregate.

In this dynamic range, a freely diffusing chain reconfigures just fast enough to expose
hydrophobic residues to solvent long enough to form bimolecular associations. Polypeptides that
diffuse faster do not leave hydrophobes exposed for long, destabilizing bimolecular complexes,
and well behaved proteins diffuse slow enough that most hydrophobes are exposed to solvent
infrequently. This is illustrated by a simple kinetic model

94

A

k1

A*

k−1

kagg

A + A ⎯⎯ [ A A ] ⎯⎯→ Agg
→
*

*

kbi

*

*

↓ k −1
A + A*
*

where A is the unfolded ensemble of conformations and A is the conformational subset that can
* *

make bimolecular associations. [A A ] is the bimolecular encounter complex that can lead to
*

irreversible aggregation, Agg, but can also be returned to monomers by conversion of A into A.
The intramolecular reconfiguration rates, k1 and k-1, are related to the intramolecular diffusion
coefficient, D. kagg consists of a sequential steps of the aggregation process from the formation
of oligamers to fiberizations. Keeping kbi, kagg and K≡k1/k-1 constant, Figure 5.3 shows the
formation of Agg is substantially slower when k1 is much higher or much lower than kbi than when
they are the about the same. This is not a realistic model but the simplest model I can extract
from my experimental results. Although there is a big difference in the timescale between
intramolecular diffusion and aggregation, this model should still be considered intuitive as long
as kagg is irreversible. Proof of this hypothesis will certainly require more measurements of
aggregation-prone sequences, but a more complete understanding of hydrophobic encounter
complexes between proteins will also help to define the limits of this dynamical regime of
aggregation.

95

Aggregated fraction

1.0

k-1 1 = bi bi
k− = k k

0.8

kkbi = 1000 ∗ k −-1
bi = 1000*k 1

0.6
0.4

k −1-1= 100 ∗ kbi
k = 100*kbi
0.2
0.0

0

2000

4000

6000

8000

10000

time steps
Figure 5.3 Formation of aggregated species, Agg, according to the model outlined in the text.

For these simulations, kbi = kagg = 1, K = 0.1 and k-1 varies as indicated on the plot.

96

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97

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