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How long threadlike eukaryotic chromosomes fit tidily in the small volume of the nucleus without significant entanglement is just beginning to be understood, thanks to major advances in experimental techniques. Several polymer models, which reproduce contact maps that measure the probabilities that two loci are in spatial contact, have predicted the three-dimensional structures of interphase chromosomes. Data-driven approaches, using contact maps as input, predict that mitotic helical chromosomes are characterized by switch in handedness, referred to as "perversion". By using experimentally derived effective interactions between chromatin loci in simulations, structures of conventional and inverted nuclei have been accurately predicted. Polymer theory and simulations show that the dynamics of individual loci in chromatin exhibit subdiffusive behavior but the diffusion exponents are broadly distributed, which accords well with experiments. Although coarse-grained models are successful, many challenging problems remain, which require the creation of new experimental and computational tools to understand genome biology.

The Random First Order Transition Theory (RFOT) predicts that transport proceeds by cooperative movement of particles in domains whose sizes increase as a liquid is compressed above a characteristic volume fraction, $\phi_d$. The rounded dynamical transition around $\phi_d$, which signals a crossover to activated transport, is accompanied by a growing correlation length that is predicted to diverge at the thermodynamic glass transition density ($> \phi_d$). Simulations and imaging experiments probed the single particle dynamics of mobile particles in response to pinning all the particles in a semi-infinite space or randomly pinning (RP) a fraction of particles in a liquid at equilibrium. The extracted dynamic length increases non-monotonically with a peak around $\phi_d$. This finding is at variance with the results obtained using the small wave length limit of a four-point structure factor for unpinned systems. To obtain a consistent picture of the growth of the dynamic length, one that is impervious to the use of RP, we introduce a multi particle structure factor, $S^c_{mp}(q,t)$, that probes collective dynamics. The collective dynamic length, calculated from the small wave vector limit of $S^c_{mp}(q,t)$, increases monotonically as a function of the volume fraction in glass forming binary mixture of charged colloidal particles in both unpinned and pinned systems. This prediction, which also holds in the presence of added monovalent salt, may be validated using imaging experiments., Comment: 7 figures

Cells undergo dramatic changes in morphology during embryogenesis, yet how these changes affect the formation of ordered tissues remains elusive. Here we find that the emergence of a nematic liquid crystal phase occurs in cells during gastrulation in the development of embryos of fish, frogs, and fruit flies. Moreover, the spatial correlations in all three organisms are long-ranged and follow a similar power-law decay (y~$x^{-\alpha}$ ) with $\alpha$ less than unity for the nematic order parameter, suggesting a common underlying physical mechanism unifies events in these distantly related species. All three species exhibit similar propagation of the nematic phase, reminiscent of nucleation and growth phenomena. Finally, we use a theoretical model along with disruptions of cell adhesion and cell specification to characterize the minimal features required for formation of the nematic phase. Our results provide a framework for understanding a potentially universal features of metazoan embryogenesis and shed light on the advent of ordered structures during animal development., Comment: Main text: 29 pages, 6 figures. SI: 9 figures and 1 table

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 feedback 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, $\Delta (t) \sim t^{\alpha}$ with $\alpha$ less than unity), to hyperdiffusion ($\alpha > 2$) 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 ($N(t) \sim t^{\lambda}$ 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 $\lambda \sim \alpha$, 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., Comment: 5 figures. arXiv admin note: text overlap with arXiv:2202.04806

We compute the free energy of confinement ${\cal{F}}$ for a wormlike chain (WLC), with persistence length $l_p$, that is confined to the surface of a cylinder of radius $R$ under an external tension $f$ using a mean field variational approach. For long chains, we analytically determine the behavior of the chain in a variety of regimes, which are demarcated by the interplay of $l_p$, the Odijk deflection length ($l_d=(R^2l_p)^{1/3}$), and the Pincus length ($l_f = {k_BT}/{f}$, with $k_BT$ being the thermal energy). The theory accurately reproduces the Odijk scaling for strongly confined chains at $f=0$, with ${\cal{F}}\sim Ll_p^{-1/3}R^{-2/3}$. For moderate values of $f$, the Odijk scaling is discernible only when ${l_p}\gg R$ for strongly confined chains. Confinement does not significantly alter the scaling of the mean extension for sufficiently high tension. The theory is used to estimate unwrapping forces for DNA from nucleosomes., Comment: 17 pages, 4 figures

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 heterochromatin (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 $\textit{without any parameter}$, 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 the imaging data of the inverted nuclei found in photoreceptor rods in nocturnal mammals. The proposed parameter free method and applications show that compartment formation in conventional and inverted nuclei is best explained by the inequality between the effective interaction energies.

The well known phenomenon of phase separation in synthetic polymers and proteins has become a major topic in biophysics because it has been invoked as a mechanism of compartment formation in cells, without the need for membranes. Most of the coacervates (or condensates) are composed of Intrinsically Disordered Proteins (IDPs) or regions that are structureless, often in interaction with RNA and DNA. One of the more intriguing IDPs is the 526-residue RNA binding protein, Fused In Sarcoma (FUS), whose monomer conformations and condensates exhibit unusual behavior that are sensitive to solution conditions. By focussing principally on the N-terminus low complexity domain (FUS-LC comprising residues 1-214) and other truncations, we rationalize the findings of solid state NMR experiments, which show that FUS-LC adopts a non-polymorphic fibril (core-1) involving residues 39-95, flanked by fuzzy coats on both the N- and C- terminal ends. An alternate structure (core-2), whose free energy is comparable to core-1, emerges only in the truncated construct (residues 110-214). Both core-1 and core-2 fibrils are stabilized by a Tyrosine ladder as well as hydrophilic interactions. The morphologies (gels, fibrils, and glass-like behavior) adopted by FUS seem to vary greatly, depending on the experimental conditions. The effect of phosphorylation is site specific and affects the stability of the fibril depending on the sites that are phosphorylated. Many of the peculiarities associated with FUS may also be shared by other IDPs, such as TDP43 and hnRNPA2. We outline a number of problems for which there is no clear molecular understanding.

Folding of ribozymes into well-defined tertiary structures usually requires divalent cations. How Mg$^{2+}$ 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 Mg$^{2+}$ 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 Mg$^{2+}$ 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., Comment: Main text + Supplementary Information

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, $C_v(t) \sim t^{-1/2}$, which has the same structure as in classical 2D fluids but with a different scaling exponent., Comment: Main text, 23 pages, 4 figures

A recent experiment on zebrafish blastoderm morphogenesis showed that the viscosity of the non-confluent embryonic tissue grows sharply until a critical cell density, and is a constant as the density is increased further. The increase in the viscosity as a function of the cell area fraction ({\phi}) follows the behavior familiar in glass forming systems but the saturation at high {\phi} is a surprise, especially in a living system. To determine the mechanism of this unexpected behavior of viscosity, we perform 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 available free area per cell to saturate beyond a critical density. Saturation in the free space not only explains the viscosity plateau but also links equilibrium geometrical packing to dramatic increase in the relaxation dynamics., Comment: 19 pages, 5 figures

Chaperonins are biological nanomachines that help newly translated proteins to fold by rescuing them from kinetically trapped misfolded states. Protein folding assistance by the chaperonin machinery is obligatory in vivo for a subset of proteins in the bacterial proteome. Chaperonins are large oligomeric complexes, with unusual seven fold symmetry (group I) or eight/nine fold symmetry (group II), that form double-ring constructs, enclosing a central folding chamber. Dramatic large-scale conformational changes, that take place during ATP-driven cycles, allow chaperonins to bind misfolded proteins, encapsulate them into the expanded cavity and release them back into the cellular environment, regardless of whether they are folded or not. The theory associated with the iterative annealing mechanism, which incorporated the conformational free energy landscape description of protein folding, \textit{quantitatively} explains most, if not all, the available data. Misfolded conformations are associated with low energy minima in a rugged energy landscape. Random disruptions of these low energy conformations result in higher free energy, less folded, conformations that can stochastically partition into the native state. Group I chaperonins (GroEL homologues in eubacteria and endosymbiotic organelles), recognize a large number of misfolded proteins non-specifically and operate through highly coordinated cooperative motions. By contrast, the less well understood group II chaperonins (CCT in Eukarya and thermosome/TF55 in Archaea), assist a selected set of substrate proteins. Chaperonins are implicated in bacterial infection, autoimmune disease, as well as protein aggregation and degradation diseases. Understanding the chaperonin mechanism and their substrates is important not only for the fundamental aspect of cellular protein folding, but also for effective therapeutic strategies., Comment: 26 pages, 4 figures, to be published in Frontiers in Molecular Biosciences

The organization of interphase chromosomes in a number of species is starting to emerge thanks to advances in a variety of experimental techniques. However, much less is known about the dynamics, especially in the functional states of chromatin. Some experiments have shown that the motility of individual loci in human interphase chromosome decreases during transcription, and increases upon inhibiting transcription. This is a counter-intuitive finding because it is thought that the active mechanical force ($F$) on the order of ten pico-newtons, generated by RNA polymerase II (RNAPII) that is presumably transmitted to the gene-rich region of the chromatin, would render it more open, thus enhancing the mobility. We developed a minimal active copolymer model for interphase chromosomes to investigate how $F$ affects the dynamical properties of chromatin. The movements of the loci in the gene-rich region are suppressed in an intermediate range of $F$, and are enhanced at small $F$ values, which has also been observed in experiments. In the intermediate $F$, the bond length between consecutive loci increases, becoming commensurate with the distance at the minimum of the attractive interaction between non-bonded loci. This results in a transient disorder-to-order transition, leading to a decreased mobility during transcription. Strikingly, the $F$-dependent change in the locus dynamics preserves the organization of the chromosome at $F=0$. Transient ordering of the loci, which is not found in the polymers with random epigenetic profiles, in the gene-rich region might be a plausible mechanism for nucleating a dynamic network involving transcription factors, RNAPII, and chromatin., Comment: 12 pages, 7 figures, Supplemental Material

The principles that govern the organization of genomes, which are needed for a deeper understanding of how chromosomes are packaged and function in eukaryotic cells, could be deciphered if the three-dimensional (3D) structures are known. Recently, single-cell imaging experiments have determined the 3D coordinates of a number of loci in a chromosome. Here, we introduce a computational method (Distance Matrix to Ensemble of Structures, DIMES), based on the maximum entropy principle, with experimental pair-wise distances between loci as constraints, to generate a unique ensemble of 3D chromatin structures. Using the ensemble of structures, we quantitatively account for the distribution of pair-wise distances, three-body co-localization and higher-order interactions. We demonstrate that the DIMES method can be applied to both small length-scale and chromosome-scale imaging data to quantify the extent of heterogeneity and fluctuations in the shapes on various length scales. We develop a perturbation method that is used in conjunction with DIMES to predict the changes in 3D structures from structural variations. Our method also reveals quantitative differences between the 3D structures inferred from Hi-C and the ones measured in imaging experiments. Finally, the physical interpretation of the parameters extracted from DIMES provides insights into the origin of phase separation between euchromatin and heterochromatin domains.

Measurements of local stresses on the cancer cells (CCs), inferred by embedding inert compressible tracer particles (TPs) in a growing multicellular spheroid (MCS), show that pressure decreases monotonically as the distance from the core of the MCS increases. How faithfully do the TPs report the local stresses in the CCs is an important question because pressure buildup in the MCS is dynamically generated due to CC division, which implies that the CC dynamics should be minimally altered by the TPs. Here using theory and simulations, we show that although the TP dynamics is unusual, exhibiting sub-diffusive behavior on times less than the CC division times and hyper-diffusive dynamics on in the long-time limit, they do not affect the long-time CC dynamics or the local CC stress distributions. The CC pressure profile within the MCS, which decays from a high value at the core to the periphery, is almost identical with and without the TPs. That the TPs have insignificant effect on the local stresses in the MCS implies that they are reliable reporters of the CC microenvironment., Comment: 11 pages, 5 figures. arXiv admin note: text overlap with arXiv:2003.12941

The growth of a tissue, which depends on cell-cell interactions and biologically relevant process such as cell division and apoptosis, is regulated by a mechanical feedback mechanism. We account for these effects in a minimal two-dimensional model in order to investigate the consequences of mechanical feedback, which is controlled by a critical pressure, $p_c$. A cell can only grow and divide if the pressure it experiences, due to interaction with its neighbors, is less than $p_c$. Because temperature is an irrelevant variable in the model, the cell dynamics is driven by self-generated active forces (SGAFs) that are created by cell division. It is shown that even in the absence of intercellular interactions, cells undergo diffusive behavior. The SGAF driven diffusion is indistinguishable from the well-known dynamics of a free Brownian particle at a fixed finite temperature. When intercellular interactions are taken into account, we find persistent temporal correlations in the force-force autocorrelation function ($FAF$) that extends over timescale of several cell division times. The time-dependence of the $FAF$ reveals memory effects, which increases as pc increases. The observed non-Markovian effects emerge due to the interplay of cell division and mechanical feedback, and is inherently a non-equilibrium phenomenon., Comment: 6 pages, 4 figures

Molecular motors belonging to the kinesin and myosin super family hydrolyze ATP by cycling through a sequence of chemical states. These cytoplasmic motors are dimers made up of two linked identical monomeric globular proteins. Fueled by the free energy generated by ATP hydrolysis, the motors walk on polar tracks (microtubule or filamentous actin) processively, which means that only one head detaches and executes a mechanical step while the other stays bound to the track. Thus, the one motor head must regulate chemical state of the other, referred to as "gating", a concept that is not fully understood. Inspired by experiments, showing that only a fraction of the energy from ATP hydrolysis is used to advance the kinesin motors against load, we demonstrate that additional energy is used for coordinating the chemical cycles of the two heads in the dimer - a feature that characterizes gating. To this end, we develop a general framework based on information theory and stochastic thermodynamics, and establish that gating could be quantified in terms of information flow between the motor heads. Applications of the theory to kinesin-1 and Myosin V show that information flow occurs, with positive cooperativity, at external resistive loads that are less than a critical value, $F_c$. When force exceeds $F_c$, effective information flow ceases. Interestingly, $F_c$, which is independent of the input energy generated through ATP hydrolysis, coincides with force at which the probability of backward steps starts to increase. Our findings suggest that transport efficiency is optimal only at forces less than $F_c$, which implies that these motors must operate at low loads under $\textit{in vivo}$ conditions.

Inevitably, almost all cancer patients develop resistance to targeted therapy. Intratumor heterogeneity (ITH) is a major cause of drug resistance. Mathematical models that explain experiments quantitatively is useful in understanding the origin of ITH, which then could be used to explore scenarios for efficacious therapy. Here, we develop a mathematical model to investigate ITH in breast cancer by exploiting the observation that HER2+ and HER2- cells could divide symmetrically or asymmetrically. Our predictions for the evolution of cell fractions are in quantitative agreement with single-cell experiments. Remarkably, the colony size of HER2+ cells emerging from a single HER2- cell (or vice versa), which occurs in about four cell doublings, agrees perfectly with experimental results, without tweaking any parameter in the model. The theory quantitatively explains experimental data on the responses of breast cancer tumor under different treatment protocols. We then used the model to predict that, not only the order of two drugs, but also the treatment period for each drug and the tumor cell plasticity could be manipulated to improve the treatment efficacy. Mathematical models, when integrated with data on patients, make possible exploration of a broad range of parameters readily, which might provide insights in devising effective therapies., Comment: 23 pages, 5 figures in the main text

The beautiful structures of single and multi-domain proteins are clearly ordered in some fashion but cannot be readily classified using group theory methods that are successfully used to describe periodic crystals. For this reason, protein structures are considered to be aperiodic, and may have evolved this way for functional purposes, especially in instances that require a combination of softness and rigidity within the same molecule. By analyzing the solved protein structures, we show that orientational symmetry is broken in the aperiodic arrangement of the secondary structural elements (SSEs), which we deduce by calculating the nematic order parameter, $P_{2}$. We find that the folded structures are nematic droplets with a broad distribution of $P_{2}$. We argue that non-zero values of $P_{2}$, leads to an arrangement of the SSEs that can resist mechanical forces, which is a requirement for allosteric proteins. Such proteins, which resist mechanical forces in some regions while being flexible in others, transmit signals from one region of the protein to another (action at a distance) in response to binding of ligands (oxygen, ATP or other small molecules).

Deviations from linearity in the dependence of the logarithm of protein unfolding rates, $\log k_u(f)$, as a function of mechanical force, $f$, measurable in single molecule experiments, can arise for many reasons. In particular, upward curvature in $\log k_u(f)$ as a function of $f$ implies that the underlying energy landscape must be multidimensional with the possibility that unfolding ensues by parallel pathways. Here, simulations using the SOP-SC model of a wild type $\beta$-sandwich protein and several mutants, with immunoglobulin folds, show upward curvature in the unfolding kinetics. There are substantial changes in the structures of the transition state ensembles as force is increased, signaling a switch in the unfolding pathways. Our results, when combined with previous theoretical and experimental studies, show that parallel unfolding of structurally unrelated single domain proteins can be determined from the dependence of $\log k_u(f)$ as a function of force (or $\log k_u[C]$ where $[C]$ is the denaturant concentration)., Comment: 15 pages, 5 figures

Experiments and theories have shown that when steric interactions between crowding particles and proteins are dominant, which give rise to Asakura-Oosawa depletion forces, then the stabilities of the proteins increase compared to the infinite dilution case. We show using theoretical arguments that the crowder volume fraction ($\Phi_C$) dependent increase in the melting temperature of globular proteins, $\Delta T_m(\Phi_C) \approx \Phi_C^{\alpha}$ where $\alpha = \frac{1}{(3 \nu_{eff} - 1)}$. The effective Flory exponent, $\nu_{eff}$, relates the radius of gyration in the unfolded state to the number of amino acid residues in the protein. We derive the bound 1.25 $\le \alpha \le$ 2.0. The theoretical predictions are confirmed using molecular simulations of $\lambda$ repressor in the presence of spherical crowding particles. Analyses of previous simulations and experiments confirm the predicted theoretical bound for $\alpha$. We show that the non-specific attractions between crowding particles and amino acid residues have to be substantial to fully negate the enhanced protein stabilities due to intra protein attractive Asakura-Oosawa (AO) depletion potential. Using the findings, we provide an alternate explanation for the very modest (often less than 0.5 Kcal/mol) destabilization in certain proteins in the cellular milieu. Cellular environment is polydisperse containing large and small crowding agents. AO arguments suggest that proteins would be localized between large (sizes exceeding that of the proteins) crowders, which are predicted to have negligible effect on stability. {\it In vitro} experiments containing mixtures of crowding particles could validate or invalidate the predictions., Comment: 20 pages, 4 figures

The number of compact structures of a single condensed polymer (SCP), with similar free energies, grows exponentially with the degree of polymerization. In analogy with structural glasses (SGs), we expect that at low temperatures chain relaxation should occur by activated transitions between the compact metastable states. By evolving the states of the SCP that is linearly coupled to a reference state, we show that, below a dynamical transition temperature ($T_d$), the SCP is trapped in a metastable state leading to slow dynamics. At a lower temperature, $T_K \ne 0$, the configurational entropy vanishes, resulting in a thermodynamic random first order ideal glass transition. The relaxation time obeys the Vogel-Fulcher-Tamman law, diverging at $T=T_0 \approx T_K$. These findings, accord well with the random first order transition theory, establishing that SCP and SG exhibit similar universal characteristics.

A single solid tumor, composed of nearly identical cells, exhibits heterogeneous dynamics. Cells dynamics in the core is glass-like whereas those in the periphery undergo diffusive or super-diffusive behavior. Quantification of heterogeneity using the mean square displacement or the self-intermediate scattering function, which involves averaging over the cell population, hides the complexity of the collective movement. Using the t-distributed stochastic neighbor embedding (t-SNE), a popular unsupervised machine learning dimensionality reduction technique, we show that the phase space structure of an evolving colony of cells, driven by cell division and apoptosis, partitions into nearly disjoint sets composed principally of core and periphery cells. The non-equilibrium phase separation is driven by the differences in the persistence of self-generated active forces induced by cell division. Extensive heterogeneity revealed by t-SNE paves way towards understanding the origins of intratumor heterogeneity using experimental imaging data., Comment: 6 pages, 4 figures

The probability of two loci, separated by a certain genome length, being in contact can be inferred using the Chromosome Conformation Capture (3C) method and related Hi-C experiments. How to go from the contact map, a matrix listing the mean contact probabilities between a large number of pairs of loci, to an ensemble of three-dimensional structures is an open problem. A solution to this problem, without assuming an assumed energy function, would be the first step in understanding the way nature has solved the packaging of chromosomes in tight cellular spaces. We created a theory, based on polymer physics characteristics of chromosomes and the maximum entropy principles, referred to as HIPPS (Hi-C-Polymer-Physics-Structures) method, that allows us to calculate the 3D structures solely from Hi-C contact maps. We created an ensemble of 3D structures for the 23 chromosomes from lymphoblastoid cells using the measured contact maps as inputs. The HIPPS method shows that conformations of chromosomes are heterogeneous even in a single cell type. The differences in the conformational heterogeneity of the same chromosome in different cell types (normal as well as cancerous cells) can also be quantitatively discerned using our theory. We validate the method by showing that the calculated volumes of the 23 chromosomes from the predicted 3D structures are in good agreement with experimental estimates. Because the method is general, the 3D structures for any species may be calculated directly from the contact map without the need to assume a specific polymer model, as is customarily done.

By embedding inert tracer particles (TPs) in a growing multicellular spheroid the local stresses on the cancer cells (CCs) can be measured. In order for this technique to be effective the unknown effect of the dynamics of the TPs on the CCs has to be elucidated to ensure that the TPs do not greatly alter the local stresses on the CCs. We show, using theory and simulations, that the self-generated (active) forces arising from proliferation and apoptosis of the CCs drive the dynamics of the TPs far from equilibrium. On time scales less than the division times of the CCs, the TPs exhibit sub-diffusive dynamics (the mean square displacement, $\Delta_{TP}(t) \sim t^{\beta_{TP}}$ with $\beta_{TP}<1$), similar to glass-forming systems. Surprisingly, in the long-time limit, the motion of the TPs is hyper-diffusive ($\Delta_{TP}(t) \sim t^{\alpha_{TP}}$ with $\alpha_{TP}>2$) due to persistent directed motion for long times. In comparison, proliferation of the CCs randomizes their motion leading to superdiffusive behavior with $\alpha_{CC}$ exceeding unity. Most importantly, $\alpha_{CC}$ is not significantly affected by the TPs. Our predictions could be tested using \textit{in vitro} imaging methods where the motion of the TPs and the CCs can be tracked., Comment: 5 pages, 5 figures

Heterogeneity is a hallmark of all cancers. Tumor heterogeneity is found at different levels -- interpatient, intrapatient, and intratumor heterogeneity. All of them pose challenges for clinical treatments. The latter two scenarios can also increase the risk of developing drug resistance. Although the existence of tumor heterogeneity has been known for two centuries, a clear understanding of its origin is still elusive, especially at the level of intratumor heterogeneity (ITH). The coexistence of different subpopulations within a single tumor has been shown to play crucial roles during all stages of carcinogenesis. Here, using concepts from evolutionary game theory and public goods game, often invoked in the context of the tragedy of commons, we explore how the interactions among subclone populations influence the establishment of ITH. By using an evolutionary model, which unifies several experimental results in distinct cancer types, we develop quantitative theoretical models for explaining data from {\it in vitro} experiments involving pancreatic cancer as well as {\it vivo} data in glioblastoma multiforme. Such physical and mathematical models complement experimental studies, and could optimistically also provide new ideas for the design of efficacious therapies for cancer patients., Comment: 29 pages, 8 figures, 1 table

Collective cell movement, characterized by multiple cells that are in contact for substantial periods of time and undergo correlated motion, plays a central role in cancer and embryogenesis. Recent imaging experiments have provided time-dependent traces of individual cells, thus providing an unprecedented picture of tumor spheroid growth. By using simulations of a minimal cell model, we analyze the experimental data that map the movement of cells in fibrosarcoma tumor spheroid embedded in a collagen matrix. Both simulations and experiments show that cells in the core of the spheroid exhibit subdiffusive glassy dynamics (mean square displacement, $\Delta(t) \approx t^{\alpha}$ with $\alpha < 1$), whereas cells in the periphery exhibit superdiffusive motion, $\Delta(t) \approx t^{\alpha}$ with $\alpha>1)$. The motion of most of the cells near the periphery undergo highly persistent and correlated directional motion, thus explaining the observed superdiffusive behavior. The $\alpha$ values for cells in the core and periphery, extracted from simulations and experiments are in near quantitative agreement with each other, which is surprising given that no parameter in the model was used to fit the measurements. The qualitatively different dynamics of cells in the core and periphery is captured by the fourth order susceptibility, introduced to characterize metastable states in glass forming systems. Analyses of the velocity autocorrelation of individual cells show remarkable spatial heterogeneity with no two cells exhibiting similar behavior. The prediction that $\alpha$ should depend on the location of the cells in the tumor is amenable to experimental test. The highly heterogeneous dynamics of cells in the tumor spheroid provides a plausible mechanism for the origin of intratumor heterogeneity., Comment: 30 pages, 6 figures (19 panels)

Molecular chaperones are ATP-consuming biological machines, which facilitate the folding of proteins and RNA molecules that are kinetically trapped in misfolded states for long times. Unassisted folding occurs by the kinetic partitioning mechanism according to which folding to the native state, with low probability as well as misfolding to one of the many metastable states, with high probability, occur rapidly on similar time scales. GroEL is an all-purpose stochastic machine that assists misfolded substrate proteins (SPs) to fold. The RNA chaperones (CYT-19) help the folding of ribozymes that readily misfold. GroEL does not interact with the folded proteins but CYT-19 disrupts both the folded and misfolded ribozymes. Despite this major difference, the Iterative Annealing Mechanism (IAM) quantitatively explains all the available experimental data for assisted folding of proteins and ribozymes. Driven by ATP binding and hydrolysis and GroES binding, GroEL undergoes a catalytic cycle during which it samples three allosteric states, referred to as T (apo), R (ATP bound), and R'' (ADP bound). In accord with the IAM predictions, analyses of the experimental data shows that the efficiency of the GroEL-GroES machinery and mutants is determined by the resetting rate $k_{R''\rightarrow T}$, which is largest for the wild type GroEL. Generalized IAM accurately predicts the folding kinetics of Tetrahymena ribozyme and its variants. Chaperones maximize the product of the folding rate and the steady state native state fold by driving the substrates out of equilibrium. Neither the absolute yield nor the folding rate is optimized., Comment: 39 pages, 11 figures

Colloidal particles, which are ubiquitous, have become ideal testing grounds for the structural glass transition (SGT) theories. In these systems glassy behavior is manifested as the density of the particles is increased. Thus, soft colloidal particles with varying degree of softness capture diverse glass forming properties, observed normally in molecular glasses. By performing Brownian dynamics simulations for a binary mixture of micron-sized charged colloidal suspensions, known to form Wigner glasses, we show that by tuning the softness of the potential, achievable by changing the monovalent salt concentration, there is a continuous transition between fragile to strong behavior. Remarkably, this is found in a system where the well characterized potential between the colloidal particles is isotropic. We also show that the predictions of the random first order transition (RFOT) theory quantitatively describes the universal features such as the growing correlation length, $\xi\sim (\phi_K/\phi - 1)^{-\nu}$ with $\nu = 2/3$ where $\phi_K$, the analogue of the Kauzmann temperature, depends on the salt concentration. As anticipated by the RFOT predictions, we establish a causal relationship between the growing correlation length and a steep increase in the relaxation time and dynamic heterogeneity. The broad range of fragility observed in Wigner glasses is used to draw analogies with molecular glasses. The large variations in the fragility is found only when the temperature dependence of the viscosity is examined for a large class of diverse glass forming materials. In sharp contrast, this is vividly illustrated in a single system that can be experimentally probed. Our work also shows that the RFOT predictions are accurate in describing the dynamics over the entire density range, regardless of the fragility of the glasses, implying that the physics describing the SGT is universal.

Many biological functions are executed by molecular machines, which consume energy and convert it into mechanical work. Biological machines have evolved to transport cargo, facilitate folding of proteins and RNA, remodel chromatin and replicate DNA. A common aspect of these machines is that their functions are driven out of equilibrium. It is a challenge to provide a general framework for understanding the functions of biological machines, such as molecular motors, molecular chaperones, and helicases. Using these machines as prototypical examples, we describe a few general theoretical methods providing insights into their functions. Although the theories rely on coarse-graining of these complex systems they have proven useful in not only accounting for many in vitro experiments but also addressing questions such as how the trade-off between precision, energetic costs and optimal performances are balanced. However, many complexities associated with biological machines will require one to go beyond current theoretical methods. Simple point mutations in the enzyme could drastically alter functions, making the motors bi-directional or result in unexpected diseases or dramatically restrict the capacity of molecular chaperones to help proteins fold. These examples are reminders that while the search for principles of generality in biology is intellectually stimulating, one also ought to keep in mind that molecular details must be accounted for to develop a deeper understanding of processes driven by biological machines. Going beyond generic descriptions of in vitro behavior to making genuine understanding of in vivo functions will likely remain a major challenge for some time to come. The combination of careful experiments and the use of physical principles will be useful in elucidating the rules governing the workings of biological machines.

Dimeric molecular motors walk on polar tracks by binding and hydrolyzing one ATP per step. Despite tremendous progress, the waiting state for ATP binding in the well-studied kinesin that walks on microtubule (MT), remains controversial. One experiment suggests that in the waiting state both heads are bound to the MT, while the other shows that ATP binds to the leading head after the partner head detaches. To discriminate between these two scenarios, we developed a theory to calculate accurately several experimentally measurable quantities as a function of ATP concentration and resistive force. In particular, we predict that measurement of the randomness parameter could discriminate between the two scenarios for the waiting state of kinesin, thereby resolving this standing controversy.

It is known that mechanical interactions couple a cell to its neighbors, enabling a feedback loop to regulate tissue growth. However, the interplay between cell-cell adhesion strength, local cell density and force fluctuations in regulating cell proliferation is poorly understood. Here, we show that spatial variations in the tumor growth rates, which depend on the location of cells within tissue spheroids, are strongly influenced by cell-cell adhesion. As the strength of the cell-cell adhesion increases, intercellular pressure initially decreases, enabling dormant cells to more readily enter into a proliferative state. We identify an optimal cell-cell adhesion regime where pressure on a cell is a minimum, allowing for maximum proliferation. We use a theoretical model to validate this novel collective feedback mechanism coupling adhesion strength, local stress fluctuations and proliferation.Our results, predicting the existence of a non-monotonic proliferation behavior as a function of adhesion strength, are consistent with experimental results. Several experimental implications of the proposed role of cell-cell adhesion in proliferation are quantified, making our model predictions amenable to further experimental scrutiny. We show that the mechanism of contact inhibition of proliferation, based on a pressure-adhesion feedback loop, serves as a unifying mechanism to understand the role of cell-cell adhesion in proliferation., Comment: 18 pages, 5 figures (Main Text) and 6 figures (Appendix); 14 pages, 4 figures (SI)

Hi-C experiments are used to infer the contact probabilities between loci separated by varying genome lengths. Contact probability should decrease as the spatial distance between two loci increases. However, studies comparing Hi-C and FISH data show that in some cases the distance between one pair of loci, with larger Hi-C readout, is paradoxically larger compared to another pair with a smaller value of the contact probability. Here, we show that the FISH-Hi-C paradox can be resolved using a theory based on a Generalized Rouse Model for Chromosomes (GRMC). The FISH-Hi-C paradox arises because the cell population is highly heterogeneous, which means that a given contact is present in only a fraction of cells. Insights from the GRMC is used to construct a theory, without any adjustable parameters, to extract the distribution of subpopulations from the FISH data, which quantitatively reproduces the Hi-C data. Our results show that heterogeneity is pervasive in genome organization at all length scales, reflecting large cell-to-cell variations.

The envelopes covering bacterial cytoplasm possess remarkable elastic properties. They are rigid enough to resist large pressures while being flexible enough to adapt to growth under environmental constraints. Similarly, the virus shells play an important role in their functions. However, the effects of mechanical properties of the outer shell in controlling and maintaining the sizes of bacteria or viruses are unknown. Here, we present a hydrodynamic "bubbles with shell" model, motivated by the study of bubble stability in fluids, to demonstrate that shell rigidity and turgor pressure control the sizes of bacteria and viruses. A dimensionless compliance parameter, expressed in terms of the elastic modulus of the shell, its thickness and the turgor pressure, determines membrane response to deformation and the size of the organisms. By analyzing experiment data, we show that bacterial and viral sizes correlate with shell elasticity, which plays a critical role in regulating size., Comment: 29 pages, 7 figures in Main Text and Appendix

The highly charged RNA molecules, with each phosphate carrying a single negative charge, cannot fold into well-defined architectures with tertiary interactions, in the absence of ions. For ribozymes, divalent cations are known to be more efficient than monovalent ions in driving them to a compact state although Mg$^{2+}$ ions are needed for catalytic activities. Therefore, how ions interact with RNA is relevant in understanding RNA folding. It is often thought that most of the ions are territorially and non-specifically bound to the RNA, as predicted by the counterion condensation (CIC) theory. Here, we show using simulations of ${\it Azoarcus}$ ribozyme, based on an accurate coarse-grained Three Site Interaction (TIS) model, with explicit divalent and monovalent cations, that ion condensation is highly specific and depends on the nucleotide position. The regions with high coordination between the phosphate groups and the divalent cations are discernible even at very low Mg$^{2+}$ concentrations when the ribozyme does not form tertiary interactions. Surprisingly, these regions also contain the secondary structural elements that nucleate subsequently in the self-assembly of RNA, implying that ion condensation is determined by the architecture of the folded state. These results are in sharp contrast to interactions of ions (monovalent and divalent) with rigid charged rods in which ion condensation is uniform and position independent. The differences are explained in terms of the dramatic non-monotonic shape fluctuations in the ribozyme as it folds with increasing Mg$^{2+}$ or Ca$^{2+}$ concentration., Comment: 22 pages including 9 figures, 5 SI figures, and 1 SI table

Allosteric signaling in biological molecules, which may be viewed as specific action at a distance due to localized perturbation upon binding of ligands or changes in environmental cues, is pervasive in biology. Phenomenological MWC and KNF models galvanized research for over five decades, and these models continue to be the basis for describing the allostery in an array of systems. However, understanding allosteric signaling and the associated dynamics between distinct allosteric states is challenging. In this review, we first describe symmetry and rigidity as essential requirements for allosteric proteins. The general features, with MWC and KNF as two extreme scenarios, emerge when allosteric signaling is viewed from an energy landscape perspective. To go beyond the general theories, we describe computational tools to predict the network of residues that carry allosteric signals. Methods to obtain molecular insights into the dynamics of allosteric transitions are briefly mentioned. The utility of the methods is illustrated by applications to systems ranging from monomeric proteins to multisubunit proteins. Finally, the role allostery plays in the functions of ATP-consuming molecular machines, bacterial chaperonin GroEL and molecular motors, is described. Although universal molecular principles governing allosteric signaling do not exist, we can draw the following general conclusions. (1) Multiple pathways connecting allosteric states are highly heterogeneous. (2) Allosteric signaling is exquisitely sensitive to the architecture of the system, which implies that the capacity for allostery is encoded in the structure itself. (3) The mechanical modes that connect distinct allosteric states are robust to sequence variations. (4) Extensive investigations of allostery in Hemoglobin and more recently GroEL, show that a network of salt-bridge rearrangements serves as allosteric switches., Comment: 93 pages, 20 figures

Intratumor heterogeneity (ITH), referring to coexistence of different cell subpopulations in a single tumor, has been a major puzzle in cancer research for almost half a century. The lack of understanding of the underlying mechanism of ITH hinders progress in developing effective therapies for cancers. Based on the findings in a recent quantitative experiment on pancreatic cancer, we developed a general evolutionary model for one type of cancer, accounting for interactions between different cell populations through paracrine or juxtacrine factors. We show that the emergence of a stable heterogeneous state in a tumor requires an unequal allocation of paracrine growth factors ("public goods") between cells that produce them and those that merely consume them. Our model provides a quantitative explanation of recent {\it in vitro} experimental studies in pancreatic cancer in which insulin growth factor (IGF-II) plays the role of public goods. The calculated phase diagrams as a function of exogenous resources and fraction of growth factor producing cells show ITH persists only in a narrow range of concentration of exogenous IGF-II. Remarkably, maintenance of ITH requires cooperation among tumor cell subpopulations in harsh conditions, specified by lack of exogenous IGF-II, whereas surplus exogenous IGF-II elicits competition. Our theory also quantitatively accounts for measured {\it in vivo} tumor growth in glioblastoma multiforme (GBM). The predictions for GBM tumor growth as a function of the fraction of tumor cells are amenable to experimental tests. The mechanism for ITH also provides hints for devising efficacious therapies., Comment: 27 pages, 7 figures

Experiments and simulations have established that dynamics in a class of living and abiotic systems that are far from equilibrium exhibit super diffusive behavior at long times, which in some cases (for example evolving tumor) is preceded by slow glass-like dynamics. By using the evolution of a collection of tumor cells, driven by mechanical forces and subject to cell birth and apoptosis, as a case study we show theoretically that on short time scales the mean square displacement is sub-diffusive due to jamming, whereas at long times it is super diffusive. The results obtained using stochastic quantization method, which is needed because of the absence of fluctuation-dissipation theorem (FDT), show that the super-diffusive behavior is universal and impervious to the nature of cell-cell interactions. Surprisingly, the theory also quantitatively accounts for the non-trivial dynamics observed in simulations of a model soap foam characterized by creation and destruction of spherical bubbles, which suggests that the two non-equilibrium systems belong to the same universality class. The theoretical prediction for the super diffusion exponent is in excellent agreement with simulations for collective motion of tumor cells and dynamics associated with soap bubbles.

We investigated frictional effects on the folding rates of a human telomerase hairpin (hTR HP) and H-type pseudoknot from the Beet Western Yellow Virus (BWYV PK) using simulations of the Three Interaction Site (TIS) model for RNA. The heat capacity from TIS model simulations, calculated using temperature replica exchange simulations, reproduces nearly quantitatively the available experimental data for the hTR HP. The corresponding results for BWYV PK serve as predictions. We calculated the folding rates ($k_\mathrm{F}$) from more than 100 folding trajectories for each value of the solvent viscosity ($\eta$) at a fixed salt concentration of 200 mM. By using the theoretical estimate ($\propto$$\sqrt{N}$ where $N$ is the number of nucleotides) for folding free energy barrier, $k_\mathrm{F}$ data for both the RNAs are quantitatively fit using one-dimensional Kramers' theory with two parameters specifying the curvatures in the unfolded basin and the barrier top. In the high-friction regime ($\eta\gtrsim10^{-5}\,\textrm{Pa\ensuremath{\cdot}s}$), for both HP and PK, $k_\mathrm{F}$s decrease as $1/\eta$ whereas in the low friction regime, $k_\mathrm{F}$ values increase as $\eta$ increases, leading to a maximum folding rate at a moderate viscosity ($\sim10^{-6}\,\textrm{Pa\ensuremath{\cdot}s}$), which is the Kramers turnover. From the fits, we find that the speed limit to RNA folding at water viscosity is between 1 and 4 $\mathrm{\mu s}$, which is in accord with our previous theoretical prediction as well as results from several single molecule experiments. Both the RNA constructs fold by parallel pathways. Surprisingly, we find that the flux through the pathways could be altered by changing solvent viscosity, a prediction that is more easily testable in RNA than in proteins.

Random polyampholytes (PAs) contain positively and negatively charged monomers that are distributed randomly along the polymer chain. The interaction between charges is assumed to be given by the Debye-Huckel potential. We show that the size of the PA is determined by an interplay between electrostatic interactions, giving rise to the polyelectrolyte (PE) effect due to net charge per monomer ($\sigma$), and an effective attractive PA interaction due to charge fluctuations, $\delta \sigma$. The interplay between these terms gives rise to non-monotonic dependence of the radius of gyration, $R_g$ on the inverse Debye length, $\kappa$ when PA effects are important (${\frac{\delta \sigma}{\sigma}} > 1$). In the opposite limit, $R_g$ decreases monotonically with increasing $\kappa$. Simulations of PA chains, using a charged bead-spring model, further corroborates our theoretical predictions. The simulations unambiguously show that conformational heterogeneity manifests itself among sequences that have identical PA parameters. A clear implication is that the phases of PA sequences, and by inference IDPs, cannot be determined using only the bare PA parameters ($\sigma$ and $\delta \sigma$). The theory is used to calculate the changes in $R_g$ on $N$, the number of residues for a set of Intrinsically Disordered Proteins (IDPs). For a certain class of IDPs, with $N$ between 24 to 441, the size grows as $R_g \sim N^{0.6}$, which agrees with data from Small Angle X-ray Scattering (SAXS) experiments., Comment: J. Chem. Phys. 2018