Biology subjects

Thirumalai, D.

Publications and source records attributed to Thirumalai, D..

At least 37 records · Page 2Linked to original sources

bioRxiv2024

Rotational migration in human pancreatic ductal organoids depends on actin and myosin activity

Rotational migration is one specific form of collective cell migration when epithelial cells are confined in a spherical geometry, such as in the epithelial acini. This tissue-level rotation motion is crucial for the morphogenesis of multiple epithelial systems. Here, we introduce a new primary human model for the study of rotational migration, pancreatic ductal organoids. Live imaging revealed the persistent rotation of the organoids over time. By tracking the nuclei, the three-dimensional trajectory of the cellular movement was reconstructed and the velocity of the rotation was quantified. The presence of focal adhesion clusters and prominent actin stress fibers were observed at the basal side of the organoids, suggesting the interactions between the cells and the surrounding extracellular matrix. Finally, our inhibition study showed the dependence of pancreatic ductal organoid rotational migration on myosin activity, actin polymerization, and actin branching. We hope that this model will enable future studies with human primary cells, which are more faithful to normal epithelial cells.

cell biology↗

bioRxiv2024

Emergence of cellular nematic order is a conserved feature of gastrulation in animal embryos

Cells undergo dramatic morphological changes during embryogenesis, yet how these changes affect the formation of ordered tissues remains elusive. Here, we show that a phase transition leading to the formation of a nematic liquid crystal state during gastrulation in the development of embryos of fish, frogs, and fruit flies occurs by a common mechanism despite substantial differences between these evolutionarily distant animals. Importantly, nematic order forms early before any discernible changes in the shapes of cells. All three species exhibit similar propagation of the nematic phase, reminiscent of nucleation and growth mechanisms. The spatial correlations in the nematic phase in the notochord region are long-ranged and follow a similar power-law decay (y~x-) with less than unity, indicating a common underlying physical mechanism. To explain the common physical mechanism, we created a theoretical model that not only explains the experimental observations but also predicts that the nematic phase should be disrupted upon loss of planar cell polarity (frog), cell adhesion (frog), and notochord boundary formation (zebrafish). Gene knockdown or mutational studies confirm the theoretical predictions. The combination of experiments and theory provides a unified framework for understanding the potentially universal features of metazoan embryogenesis, in the process shedding light on the advent of ordered structures during animal development.

developmental biology↗

bioRxiv2024

Sequence Complexity and Monomer Rigidity Control the Morphologies and Aging Dynamics of Protein Aggregates

Understanding the biophysical basis of protein aggregation is important in biology because of the potential link to several misfolding diseases. Although experiments have shown that protein aggregates adopt a variety of morphologies, the dynamics of their formation are less well characterized. Here, we introduce a minimal model to explore the dependence of the aggregation dynamics on the structural and sequence features of the monomers. Using simulations we demonstrate that sequence complexity (codified in terms of word entropy) and monomer rigidity profoundly influence the dynamics and morphology of the aggregates. Flexible monomers with low sequence complexity (corresponding to repeat sequences) form liquid-like droplets that exhibit ergodic behavior. Strikingly, these aggregates abruptly transition to more ordered structures, reminiscent of amyloid fibrils, when the monomer rigidity is increased. In contrast, aggregates resulting from monomers with high sequence complexity are amorphous and display non-ergodic glassy dynamics. The heterogeneous dynamics of the low and high-complexity sequences follow stretched exponential kinetics, which is one of the characteristics of glassy dynamics. Importantly, at non-zero values of the bending rigidities, the aggregates age with the relaxation times that increase with the waiting time. Informed by these findings, we provide insights into aging dynamics in protein condensates and contrast the behavior with the dynamics expected in RNA repeat sequences. Our findings underscore the influence of the monomer characteristics in shaping the morphology and dynamics of protein aggregates, thus providing a foundation for deciphering the general rules governing the behavior of protein condensates. Significance StatementProtein aggregates exhibit diverse morphology, exemplified by amyloid fibrils, gel-like structures, and liquid-like condensates. Differences in the morphologies in identical proteins play important functional roles in several diseases. Simulations using a minimal model show that such structures are encoded in the sequence complexity and bending rigidity of the monomers. The low-complexity flexible sequences form liquid droplets, whose relaxation dynamics are ergodic. In contrast, rigid low and high-complexity sequences, which form ordered nematic fibril-like structures and amorphous aggregates, exhibit heterogenous, non-ergodic dynamics. The relaxation times under these conditions increase as the waiting time increases, which is a signature of aging. The implications of our findings for aging in intrinsically dis-ordered proteins and repeat RNA sequences are outlined.

biophysics↗

bioRxiv2024

The Folding of Germ Granule mRNAs Controls Intermolecular Base Pairing in Germ Granules and Maintains Normal Fly Development

Drosophila germ granules enrich mRNAs critical for fly development. Within germ granules, mRNAs form multi-transcript clusters marked by increased mRNA concentration, creating an elevated potential for intermolecular base pairing. However, the type and abundance of intermolecular base pairing in mRNA clusters is poorly characterized. Using single-molecule super-resolution microscopy, chemical probing for base accessibility, phase separation assays, and simulations, we demonstrated that mRNAs remain well-folded upon localization to germ granules. While most base pairing is intramolecular, mRNAs still display the ability for intermolecular base pairing, facilitating clustering without high sequence complementarity or significant melting of secondary structure. This base pairing among mRNAs is driven by scattered and discontinuous stretches of bases appearing on the surface of folded RNAs, providing multivalency to clustering but exhibits low probability for sustained interactions. Notably, engineered germ granule mRNAs with exposed GC-rich complementary sequences (CSs) presented within stable stem loops induce sustained base pairing in vitro and enhanced intermolecular interactions in vivo. However, the presence of these stem loops alone disrupts fly development, and the addition of GC-rich CSs exacerbates this phenotype. Although germ granule mRNAs contain numerous GC-rich CSs capable of stable intermolecular base pairing, they are primarily embedded by RNA folding. This study emphasizes the role of RNA folding in controlling the type and abundance of intermolecular base pairing, thereby preserving the functional integrity of mRNAs within the germ granules.

molecular biology↗

bioRxiv2024

Proliferation-driven mechanical feedback regulates cell dynamics in growing tissues

Local stresses in a tissue, a collective property, regulate cell division and apoptosis. In turn, cell growth and division induce active stresses in the tissue. As a consequence, there is a feed-back between cell growth and local stresses. However, how the cell dynamics depend on local stress-dependent cell division and the feedback strength is not fully understood. Here, we probe the consequences of stress-mediated growth and cell division on cell dynamics using agent-based simulations of a two-dimensional growing tissue. We discover a rich dynamical behavior of individual cells, ranging from jamming (mean square displacement, {Delta}(t) [~] t with less than unity), to hyperdiffusion ( > 2) depending on cell division rate and the strength of the mechanical feedback. Strikingly, {Delta}(t) is determined by the tissue growth law, which quantifies cell proliferation (number of cells N (t) as a function of time). The growth law (N (t) [~] t{lambda} at long times) is regulated by the critical pressure that controls the strength of the mechanical feedback and the ratio between cell division-apoptosis rates. We show that{lambda} [~] , which implies that higher growth rate leads to a greater degree of cell migration. The variations in cell motility are linked to the emergence of highly persistent forces extending over several cell cycle times. Our predictions are testable using cell-tracking imaging techniques.

biophysics↗

bioRxiv2023

On the role of mechanical feedback in synchronous to asynchronous transition during embryogenesis

The initial stage of Zebrafish morphogenesis is characterized by a synchronous to asynchronous transition (SAT) in cell divisions. The cells divide in unison in the synchronous phase, unlike in the asynchronous phase. Despite the widespread observation of SAT in multiple organisms, there is no theoretical framework to predict the critical number of cell cycles n* that marks the beginning of asynchronous division. Here, by probabilistically modeling cell cycle progression under the assumption that the distribution of cell division times is broadened, we predict n* and the time at which the SAT occurs. The theory, supplemented by agent-based simulations, supports the hypothesis that the SAT emerges as a consequence of biomechanical feedback on cell division. Our results are in excellent agreement with the experiments while also explaining the cell cycle lengthening that arises as a result of biomechanical feedback. The emergence of the asynchronous phase is due to increasing fluctuations in the cell cycle times with each round of cell division. We also make several testable predictions that further sheds light on the role of biomechanical feedback in growing multicellular systems, such as during tissue and tumor growth.

biophysics↗

bioRxiv2023

Salt-dependent self-association of trinucleotide repeat RNA sequences

Low complexity repeat RNA sequences self-associate by homotypic interactions to form condensates. Using simulations of a coarse grained Single-Interaction Site model for (CAG)n (n = 30 and 31), we show that the salt-dependent free energy gap, {triangleup}GS, between the ground (perfect hairpin) and the excited state (slipped hairpin (SH) with one CAG overhang) of monomer (n even) is the primary factor that determines the rates and yield of self-assembly. For odd n, the SH ground state free energy (GS) is used to predict self-association kinetics. As the monovalent salt concentration, CS, increases {triangleup}GS and GS increases, which in turn decreases the self-association rates. In contrast, {triangleup}GS for scrambled sequences, with the same length and sequence composition as (CAG)31 but with higher complexity, is larger which greatly suppresses the propensities to aggregate. Although demonstrated explicitly for (CAG)30 and (CAG)31 polymers, the finding that there is an inverse correlation between CS-dependent {triangleup}GS or GS and RNA aggregation is general. Our predictions are amenable to experimental tests.

biophysics↗

bioRxiv2023

Competition between stacking and divalent cation mediated electrostatic interactions determines the conformations of short DNA sequences

Interactions between divalent cations (Mg2+ and Ca2+) and highly charged single stranded DNA (ssDNA) and double stranded DNA (dsDNA), as well as stacking interactions, are important in a variety of problems, including nucleosome stability and phase separation in nucleic acids. Quantitative techniques accounting for ion-DNA interactions are needed to obtain insights into these and related problems. Towards this end, we created a computational model that explicitly takes into account monovalent and divalent ions, within the framework of the sequence-dependent coarse-grained Three Interaction Site (TIS) model for DNA. Molecular simulations of the rigid 24 base-pair (bp) dsDNA and flexible ssDNA sequences, dT30 and dA30, in a buffer containing Na+ and Cl-, with varying amounts of the divalent cations, are used to show that the calculated excess number of ions around the dsDNA and ssDNA agree quantitatively with ion-counting experiments. Using an ensemble of all-atom structures generated from coarse-grained simulations, we calculated the Small Angle X-ray Scattering (SAXS) profiles, which are also in excellent agreement with experiments. Strikingly, recapitulation of all the experimental findings was achieved without adjusting any of the parameters in the energy function to fit the data. At a molecular level, we find that Mg2+ and Ca2+ sense the differences between the major and minor grooves in dsDNA even though they are masked in ion-counting and SAXS experiments. The smaller Mg2+ binds predominantly to the minor grooves and phosphate groups whereas Ca2+ binds specifically only to the minor groove. The dA30 conformations are dominated by stacking interactions, resulting in structures with considerable helical order. In contrast, the near cancellation of the favorable stacking and unfavorable electrostatic interactions leads to dT30 populating an ensemble of heterogeneous conformations.

biophysics↗

bioRxiv2023

Brewing COFFEE: a sequence-specific coarse-grained energy function for simulations of DNA-protein complexes

DNA-protein interactions are pervasive in a number of biophysical processes ranging from transcription, gene expression, to chromosome folding. To describe the structural and dynamic properties underlying these processes accurately, it is important to create transferable computational models. Toward this end, we introduce Coarse grained force field for energy estimation, COFFEE, a robust framework for simulating DNA-protein complexes. To brew COFFEE, we integrated the energy function in the Self-Organized Polymer model with Side Chains for proteins and the Three Interaction Site model for DNA in a modular fashion, without re-calibrating any of the parameters in the original force-fields. A unique feature of COFFEE is that it describes sequence-specific DNA-protein interactions using a statistical potential (SP) derived from a dataset of high-resolution crystal structures. The only parameter in COFFEE is the strength ({lambda}DNAPRO) of the DNA-protein contact potential. For an optimal choice of{lambda} DNAPRO, the crystallographic B-factors for DNA-protein complexes, with varying sizes and topologies, are quantitatively reproduced. Without any further readjustments to the force-field parameters, COFFEE predicts the scattering profiles that are in quantitative agreement with SAXS experiments as well as chemical shifts that are consistent with NMR. We also show that COFFEE accurately describes the salt-induced unraveling of nucleosomes. Strikingly, our nucleosome simulations explain the destabilization effect of ARG to LYS mutations, which does not alter the balance of electrostatic interactions, but affects chemical interactions in subtle ways. The range of applications attests to the transferability of COFFEE, and we anticipate that it would be a promising framework for simulating DNA-protein complexes at the molecular length-scale. Graphical TOC Entry O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=81 SRC="FIGDIR/small/544064v2_ufig1.gif" ALT="Figure 1"> View larger version (22K): org.highwire.dtl.DTLVardef@1190c18org.highwire.dtl.DTLVardef@169098eorg.highwire.dtl.DTLVardef@f24e75org.highwire.dtl.DTLVardef@1fd0bd1_HPS_FORMAT_FIGEXP M_FIG C_FIG

biophysics↗

bioRxiv2023

Sizes, Conformational fluctuations, and SAXS profiles for Intrinsically Disordered Proteins

The preponderance of Intrinsically Disordered Proteins (IDPs) in the eukaryotic proteome, and their ability to interact with each other, folded proteins, RNA, and DNA for functional purposes, have made it important to quantitatively characterize their biophysical properties. Toward this end, we developed the transferable Self-Organized Polymer (SOP-IDP) model to calculate the properties of several IDPs. The values of the radius of gyration (Rg) obtained from SOP-IDP simulations are in excellent agreement (correlation coefficient of 0.96) with those estimated from SAXS experiments. For AP180 and Epsin, the predicted values of the hydrodynamic radii (Rhs) are in quantitative agreement with those from Fluorescence Correlation Spectroscopy (FCS) experiments. Strikingly, the calculated SAXS spectra for thirty-six IDPs are also nearly superimposable on the experimental profiles. The dependence of Rg and the mean end-to-end distance (Ree) on chain length, N, follows Florys scaling law, R {approx} aN 0.588 ( = g, and e), suggesting that globally IDPs behave as synthetic polymers in a good solvent. The values of ag, and ae are 0.20 nm and 0.48 nm respectively. Surprisingly, finite size corrections to scaling, expected on theoretical grounds, are negligible for Rg and Ree. In contrast, only by accounting for the finite sizes of the IDPs, the dependence of experimentally measurable Rh on N can be quantitatively explained using{nu} = 0.588. Although Flory scaling law captures the estimates for Rg, Ree, and Rh accurately, the spread of the simulated data around the theoretical curve is suggestive of of sequence-specific features that emerge through a fine-grained analysis of the conformational ensembles using hierarchical clustering. Typically, the ensemble of conformations partitiones into three distinct clusters, having different equilibrium populations and structural properties. Without any further readjustments to the parameters of the SOP-IDP model, we also obtained excellent agreement with paramagnetic relaxation enhancement (PRE) measurements for -synuclein. The transferable SOP-IDP model sets the stage for several applications, including the study of phase separation in IDPs and interactions with nucleic acids.

biophysics↗

bioRxiv2023

A method for extracting effective interactions from Hi-C data with applications to interphase chromosomes and inverted nuclei

Contact probabilities between loci, separated by arbitrary genomic distance, for a number of cell types have been reported using genome-wide chromosome conformation capture (Hi-C) experiments. How to extract the effective interaction energies between active euchromatin (A) and inactive het-erochromatin (B) directly from the experimental data, without an underlying polymer model, is unsolved. Here, we first calculate the pairwise effective interaction energies (A-A, B-B, or A-B) for interphase chromosomes based on Hi-C data by using the concept of Statistical Potential (SP), which assumes that the interaction energy between two loci is proportional to the logarithm of the frequency with which they interact. Polymer simulations, using the extracted interaction energy values without any parameters, reproduce the segregation between A and B type loci (compartments), and the emergence of topologically associating domains (TADs), features that are prominent in the Hi-C data for interphase chromosomes. Remarkably, the values of the SP automatically satisfy the Flory-Huggins phase separation criterion for all the chromosomes, which explains the mechanism of compartment formation in interphase chromosomes. Strikingly, simulations using the SP that accounts for pericentromeric constitutive heterochromatin (C-type), show hierarchical structuring with the high density of C-type loci in the nuclear center, followed by localization of the B type loci, with euchromatin being confined to the nuclear periphery, which differs from the expected nuclear organization of interphase chromosomes, but is in accord with imaging data. Such an unusual organization of chromosomes is found in inverted nuclei of photoreceptor rods in nocturnal mammals. The proposed method without free parameters and its applications show that compartment formation in conventional and inverted nuclei is best explained by the inequality between the effective interaction energies, with heterochromatin attraction being the dominant driving force.

biophysics↗

bioRxiv2023

Watching ion-driven folding kinetics of a ribozyme one molecule at a time

Folding of ribozymes into well-defined tertiary structures usually requires divalent cations. How Mg2+ ions direct the folding kinetics has been a long-standing unsolved problem because experiments cannot detect the positions and dynamics of ions. To address this problem, we used molecular simulations to dissect the folding kinetics of the Azoarcus ribozyme by monitoring the path each molecule takes to reach the folded state. We quantitatively establish that Mg2+ binding to specific sites, coupled with counter-ion release of monovalent cations, stimulate the formation of secondary and tertiary structures, leading to diverse pathways that include direct rapid folding and trapping in misfolded structures. In some molecules, key tertiary structural elements form when Mg2+ ions bind to specific RNA sites at the earliest stages of the folding, leading to specific collapse and rapid folding. In others, the formation of non-native base pairs, whose rearrangement is needed to reach the folded state, is the rate-limiting step. Escape from energetic traps, driven by thermal fluctuations, occurs readily. In contrast, the transition to the native state from long-lived topologically trapped native-like metastable states is extremely slow. Specific collapse and formation of energetically or topologically frustrated states occur early in the assembly process. Graphical abstract O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=108 SRC="FIGDIR/small/527349v3_ufig1.gif" ALT="Figure 1"> View larger version (27K): org.highwire.dtl.DTLVardef@1a69418org.highwire.dtl.DTLVardef@144e1a3org.highwire.dtl.DTLVardef@139b8e2org.highwire.dtl.DTLVardef@993ca2_HPS_FORMAT_FIGEXP M_FIG C_FIG

biophysics↗

bioRxiv2023

Odd-even disparity in the population of slipped hairpins in RNA repeat sequences with implications for phase separation

Low complexity nucleotide repeat sequences, which are implicated in several neurological disorders, undergo liquid-liquid phase separation (LLPS) provided the number of repeat units, n, exceeds a critical value. Here, we establish a link between the folding landscapes of the monomers of trinucleotide repeats and their propensity to self-associate. Simulations using a coarse-grained Self-Organized Polymer (SOP) model for (CAG)n repeats in monovalent salt solutions reproduce experimentally measured melting temperatures, which are available only for small n. By extending the simulations to large n, we show that the free energy gap, {Delta}GS, between the ground state (GS) and slipped hairpin (SH) states is a predictor of aggregation propensity. The GS for even n is a perfect hairpin (PH) whereas it is a SH when n is odd. The value of {Delta}GS (zero for odd n) is larger for even n than for odd n. As a result, the rate of dimer formation is slower in (CAG)30 relative to (CAG)31, thus linking {Delta}GS to RNA-RNA association. The yield of the dimer decreases dramatically, compared to the wild type, in mutant sequences in which the population of the SH is decreases substantially. Association between RNA chains is preceded by a transition to the SH even if the GS is a PH. The finding that the excitation spectra, which depends on the exact sequence, n, and ionic conditions, is a predictor of self-association, should also hold for other RNAs (mRNA for example) that undergo LLPS.

biophysics↗

bioRxiv2022

On the origin of viscosity saturation at high densities during zebrafish morphogenesis

A recent experiment on zebrafish blastoderm morphogenesis showed that the viscosity ({eta}) of a non-confluent embryonic tissue grows sharply until a critical cell packing fraction ({phi}S). The increase in{eta} up to{phi} S is similar to the behavior observed in several glass forming materials, which suggests that the cell dynamics is sluggish or glass-like. Surprisingly,{eta} is a constant above{phi} S. To determine the mechanism of this unusual dependence of{eta} on{phi} , we performed extensive simulations using an agent-based model of a dense non-confluent two-dimensional tissue. We show that polydispersity in the cell size, and the propensity of the cells to deform, results in the saturation of the available free area per cell beyond a critical packing fraction. Saturation in the free space not only explains the viscosity plateau above{phi} S but also provides a relationship between equilibrium geometrical packing to the dramatic increase in the relaxation dynamics.

biophysics↗

bioRxiv2022

Topological transitions, turbulent-like motion and long-time-tails driven by cell division in biological tissues

The complex spatiotemporal flow patterns in living tissues, driven by active forces, have many of the characteristics associated with inertial turbulence even though the Reynolds number is extremely low. Analyses of experimental data from two-dimensional epithelial monolayers in combination with agent-based simulations show that cell division and apoptosis lead to directed cell motion for hours, resulting in rapid topological transitions in neighboring cells. These transitions in turn generate both long ranged and long lived clockwise and anticlockwise vortices, which gives rise to turbulent-like flows. Both experiments and simulations show that at long wavelengths the wave vector (k) dependent energy spectrum E(k) {approx} k-5/3, coinciding with the Kolmogorov scaling in fully developed inertial turbulence. Using theoretical arguments and simulations, we show that long-lived vortices lead to long-time tails in the velocity auto-correlation function, Cv(t) [~] t-1/2, which has the same structure as in classical 2D fluids but with a different scaling exponent.

biophysics↗

bioRxiv2022

On the origin of the integrity of a subset of Topologically Associating Domains upon Cohesin loss in Interphase Chromosomes

Compartment formation in interphase chromosomes is a result of spatial segregation between eu- and heterochromatin on a few mega base pairs (Mbp) scale. On the sub-Mbp scales, Topologically Associating Domains (TADs) appear as interacting domains along the diagonal in the ensemble averaged Hi-C contact map. Hi-C experiments showed that most of the TADs vanish upon deleting cohesin, while the compartment structure is maintained, and perhaps even enhanced. However, closer inspection of the data reveals that a non-negligible fraction of TADs is preserved (P-TADs) after cohesin loss. Imaging experiments show that, at the single-cell level, TAD-like structures are present even without cohesin. To provide a structural basis for these findings, we first used polymer simulations to show that certain TADs with epigenetic switches across their boundaries survive after depletion of loops. More importantly, the three-dimensional structures show that many of the P-TADs have sharp physical boundaries. Informed by the simulations, we analyzed the Hi-C maps (with and without cohesin) in mouse liver and human colorectal carcinoma cell lines, which affirmed that epigenetic switches and physical boundaries (calculated using the predicted 3D structures using the data-driven HIPPS method that uses Hi-C as the input) explain the origin of the P-TADs. Single-cell structures display TAD-like features in the absence of cohesin that are remarkably similar to the findings in imaging experiments. Some P-TADs, with physical boundaries, are relevant to the retention of enhancer-promoter/promoter-promoter interactions. Overall, our study shows that preservation of a subset of TADs upon removing cohesin is a robust phenomenon that is valid across multiple cell lines.

biophysics↗

bioRxiv2022

Reduction of RBD Binding Affinity to Glycosylated ACE2 is Entropic in Origin

The spike protein in the virus SARS-CoV-2 (the causative agent of COVID-19) recognizes the host cell by binding to the peptidase domain (PD) of the extracellular enzyme Angiotensin-converting Enzyme 2 (ACE2). A variety of carbohydrates could be attached to the six asparagines in the PD, resulting in a heterogeneous population of ACE2 glycoforms. Experiments have shown that the binding affinity of glycosylated and deglycosylated ACE2 to the virus is virtually identical. In most cases, the reduction in glycan size correlates with stronger binding, which suggests that volume exclusion, and hence entropic forces, determine the binding affinity. Here, we quantitatively test the entropy-based hypothesis by developing a lattice model for the complex between ACE2 and the SARS-CoV-2 spike protein Receptor-binding Domain (RBD). Glycans are treated as branched polymers with only volume exclusion, which we justify using all atom molecular dynamics simulations in explicit water. We show that the experimentally measured changes in the ACE2-RBD dissociation constants for a variety of engineered ACE2 glycoforms are well accounted for by our theory, thus affirming that ACE2 glycans have only a weak, entropic effect on RBD binding.

biophysics↗

bioRxiv2022

Residue-dependent transition temperatures and denaturant midpoints in the folding of a multi-domain protein

As a consequence of the finite size of globular proteins, it is expected that there should be dispersions in the global melting temperature (Tm) and the denaturation midpoint (Cm). Thermodynamic considerations dictate that the dispersions, {Delta}Tm in Tm and {Delta}Cm in Cm, should decrease with N, the number of residues in the protein. We performed coarse-grained simulations of the Self-Organized Polymer (SOP) model of the multi-domain protein, Adenylate Kinase (ADK) with N = 214, in order to calculate thermal and denaturation unfolding titration curves. The results show that [Formula] and [Formula] are non-zero and follow the previously established (Phys. Rev. Lett. 93 268107 (2004)) thermodynamic [Formula] scaling for proteins accurately. For ADK, the dispersions are small ({approx} 0.004), which implies that the melting temperature is more or less unique, which is unlike in BBL (N =40) where [Formula].

biophysics↗