W, 110 Y, 224 L, 374 P even at high temperatures.
CoVID-19 pandemic is attracting unprecedented attention [1] [2] [3] [4] [5] in investigating the corona virus and its constituents. Corona virus involves a number of proteins, RNA and a huge list of crowded inter-and intra-cellular constituents in its assembly and replication. In an initial investigation even with a coarse-grained computer simulation model it is not feasible to consider all constituents that are involved in its assembly and replication. We examine the structural dynamics of a nucleocapsid (COVN) protein [6] consisting of 422 residues which plays a critical role in packaging the viral genome RNA into ribonucleocapsid and virion assembly [7] [8] [9] . For the sake of simplicity and to develop a clear understanding of the basic nature of the conformational evolution, it would be interesting to examine the structural response of a free COVN as a function of temperature before systematically including different types of proteins, solute, solvent etc. of the underlying host space.
'Protein folding' [10, 11] remains an open problem despite enormous efforts for over half a century. Because of the enormity of challenges (e.g. time scale for huge degrees of freedom with all-atom approaches), coarse-graining [12] [13] [14] [15] [16] [17] remains a viable choice to gain insight into the fundamental mechanism of conformational dynamics. Using a simplified yet efficient and effective coarse-grained model [18, 19] , a large-scale Monte Carlo simulation is performed to study the thermal response of COVN. Our coarse-grained model has already been used to investigate structural dynamics of such proteins as histones critical in assembly of chromatin [20] , lysozyme [21] and alpha-synuclein [22] key in amyloid, protein (VP40) in ebola virus [23], membrane proteins [18, 19] for selective transports, etc. COVN is represented by a chain of 422 residues in a specific sequence in a cubic lattice [18, 19] . Each residue interacts with surrounding residues within a range (rc) with a generalized Lennard-Jones potential, , rij < rc (1) where rij is the distance between the residues at site i and j; rc=8 and  = 1 in units of lattice constant. A knowledge-based [12] [13] [14] [15] [16] [17] residue-residue contact matrix (based on a large ensemble of protein structures in PDB) is used as input for the potential strength ij [14] in phenomenological (1) . With the implementation of excluded volume and limits on the covalent bond length constraints, each residue performs its stochastic movement with the Metropolis algorithm, i.e. with the Boltzmann probability exp(-E/T) where E is the change in energy between new and old position. Attempts to move each residue once defines unit Monte Carlo time step. All quantities are measured in arbitrary unit (i.e. spatial length in unit of lattice constant) including the temperature T which is in reduced units of the Boltzmann constant.
Simulations are performed on a 550 3 lattice for a sufficiently long time (10 7 ) steps with a number of independent samples (1001000) over a wide range of temperatures. Different sample sizes are also used to verify the reliability of the qualitative trends from our data presented here. A number of local and global physical quantities such as radius of gyration, root mean square displacement of the center of mass, structure factor, contact map, etc. are examined as a function of temperature. The conformation of the protein exhibits a monotonous response from a randomcoil of folded (globular) segments in native phase to tenuous fibrous conformations on raising the temperature; it exhibits a non-monotonic response with a re-entrant conformation involving enhanced globularity before reaching a steady-state conformation on further heating. While most segmental folds disappear in denatured phase while some persist even at a very high temperature (see below). Figure 1 shows the variation of the average radius of gyration (Rg) with the temperature. At low temperatures (T=0.010 -0.015), the radius of gyration remains almost constant with its lowest magnitude (Rg  22.5) in its native phase. Unlike many proteins (globular in native phase), COVN appears to be expanded into a random coil (see below) signature of an intrinsically disordered [7] Representative snapshots (figure 1, see also figure S1) of the protein at selected temperatures shows the variations in nature of the self-organizing structures over the range of temperature. For example, in native phase (T=0.010, 0.014) we see local segmental folding with a chain of folded blobs in a random-coil-like conformation (see below) in contrast to a global folding one generally expects. Local folds begin to disappear at high temperatures but still persist in smaller sizes. Segmental folds appear to be distributed along the entire protein backbone at both maxima and at high temperatures in denature phase while the segmental folds at the minimum and in native phase are localized. Size of the self-organized segmental assembly represents the degree of globularization. In snapshots, gold spheres represent residues in contact, the large black sphere is the first residue 1 M and large grey sphere is the last 422 A (see figure S1 ).
How to quantify the distribution of residues over length scales? To assess the mass (distribution), we have analyzed the structure factor S(q) defined as,
where rj is the position of each residue and |q| = 2/ is the wave vector of wavelength . Using a power-law scaling S(q)  q -1/ , one may be able to evaluate the power-law exponent  and estimate the spread of residues over the length scale . Overall size of the protein chain is described by its radius of gyration (Rg). Therefore, the structure factor over the length scale comparable to protein size ( In the native phase (T = 0.0150) where the radius of gyration is minimum (Rg  22.5), the effective dimension D  2.053 of the protein shows that the overall spread is not globular. It is rather random-coil, a chain of segmental globules (see figure 1 ). In unfolding-transition regime (T=0.020), the effective dimension D  1.726 decreases while retaining its partial folding towards C-terminal (see below). Continuous increasing the temperature leads to maximum unfolding (T = 0.0230) where the protein chain stretches to its maximum gyration radius (Rg  55) with lowest effective dimension D  1.579 with a couple of unfolded segments (see below). Further heating leads to contraction with a lower radius of gyration (Rg  22.5, T = 0.0246) with a higher effective dimension D  2.389, which indicates more compact conformation than that in its native phase, a thermal-induced folding. The effective dimension begin to reduce with increasing the temperature further as the protein conformation approaches a tenuous structure, i.e. D  1.579 at T=0.036. Let us look closer into the local structures by examining the contact map in depth as presented in figure 3 (see also figure S2 ). First, we notice that the number of residues (Nr) within the range of interaction of each residues along the backbone, is higher at lower temperatures. However, the distribution of Nr is highly heterogenous and concentrated towards specific segments ( 65 L, 110 Y, 224 L, 257 K, 370 K, 374 K). The degree of folds appears to be significant at these globularization centers (in particular segment 367 T-380 A) even at higher temperatures although it is highest in native to denature transition region (see also figure 1 ). Thermal response of contact profiles of each center of folding appears similar except 65 L which exhibits a non-linear (somewhat oscillatory) response (see the right section of figure 3 ). Thus, the thermal response of COVN protein is non-linear with a random coil of folded blobs in native phase to a systematic unfolding, refolding, and unfolding as the protein denatures on increasing the temperature. The radius of gyration increases on raising the temperature, first monotonically from a minimum in its native state to a maximum value. Further heating leads to a sharp decline (the protein contracts) in a narrow temperature range followed by increase (protein expands) again to a second maximum with a local minimum in between. The radius of gyration at the local minimum is larger than that in its native state but the segmental globularization is localized towards the second half (C-terminal) while the first half (N-terminal) of the protein acquire a fibrous configuration. Continued heating causes COVN to approach a steady-state value with a small contraction rate.
Scaling analysis of the structure factor is critical in quantifying the overall spread of COVN Details of contact map presented in figures S2 and S3 (for entire protein) shows how the degree of folding in different segments in native phase (T=0.0150) decreases with unfolding on raising the temperature. For example, the degree of folding in segments 118 P-164 Q and 276 G-344 G reduces dramatically by increasing the temperature T = 0.0150 -0.0230 as the protein chain denatures to its maxiumum extension (T=0.0230). Further heating (T=0.0230 -0.0246) leads to eradicating a large fraction of these folds while the remaining folds in specific segments e.g. 62 K -70 G, 81 P-114 L, 226 T-239 A, 248 T-270 N, 367 T-380 A appear to induce contraction in spread of the protein. The degree of folding reduced on continued heating but the persistence of some folded segments e.g. 367 T-380 A (along with other segments with comparably low folds) lead to futher expansion of the protein (T=0.0268) before it reaches to a stable conformation.