Since the first protein structures were solved in the 1950s, the protein data bank has grown to include over one hundred thousand macromolecular structures ranging in size from small peptides to large viral capsids. These experiments have shown that proteins exhibit a diverse range of structure and function and that these two aspects are closely related. In fact, it is often possible to predict a protein's function from its structure alone. Much of the focus to date has been on the more static regions of proteins for theoretical and practical reasons. However, it is important to note that even well folded proteins experience everlasting fluctuations due to the constant influence from outside forces, which drive motions that are relevant to function such as sidechain fluctuations and conformational shifts. The possible movements that can arise from these fluctuations are determined by a protein's structure. This means flexibility, or the ability to deform from the current conformation under external forces, is an intrinsic property of all proteins, and is closely tied to function. In order to better study protein function in ordered or disordered proteins, we require accurate, efficient, multiscale tools for evaluating flexibility.This work puts forward a multiscale, multiphysics and multidomain model, the flexibility-rigidity index (FRI), to estimate the flexibility and conformational motions of macromolecular structures. The basic assumption of the present FRI theory is that the geometry or structure of a given protein, together with its specific environment, completely determines the biological function and properties including flexibility and charge. To this end, we utilize monotonically decreasing functions to measure the geometric compactness of a protein and quantify the topological connectivity of atoms or residues in the proteins and nucleic acids. We define the total rigidity of a molecule by a summation of atomic rigidities. A practical validation of the proposed FRI for flexibility analysis is provided by the prediction of B-factors, or temperature factors of proteins, measured by X-ray crystallography. We employ a test set of 263 structurally distinct proteins to examine the validity and robustness of the proposed FRI method for B-factor estimation or flexibility prediction. The basic FRI algorithm outperforms GNM on this test set by about 20\%. After validation of the basic FRI method we introduce a multikernel-based multiscale FRI (mFRI) strategy to analyze macromolecular flexibility. The essential idea is to employ two or three kernels each parameterized with a different scale to capture the multiple characteristic interaction scales of complex biomolecules. Based on an expanded test set containing 364 proteins, we show that the mFRI method is about 22\% more accurate than the GNM method in B-factor prediction. Most importantly, we demonstrate that the present mFRI gives rise to excellent flexibility analysis for many proteins that are difficult cases for GNM and the previously introduced single-scale FRI methods. Finally, for a protein of $N$ residues, we illustrate that the computational complexity of the proposed mFRI is of linear scaling ${\cal O}(N)$, in contrast to the order of ${\cal O}(N^3)$ for GNM.
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