Universal Nature of Collapsibility in the Context of Protein Folding and Evolution.

Theory and simulations predicted that the sizes of the unfolded states of globular proteins should decrease as the denaturant concentration is reduced from a high to a low value. However, small angle X-ray scattering (SAXS) data were used to assert the opposite, while interpretation of single molecule Förster resonance energy transfer experiments (FRET) supported the theoretical predictions. The disagreement between the two experiments is the SAXS-FRET controversy. By harnessing recent advances in SAXS and FRET experiments and setting these findings in the context of a general theory and simulations, which do not rely on experimental data, we establish that compaction of unfolded states under native conditions is universal. The theory also predicts that proteins rich in ß-sheets are more collapsible than α-helical proteins. Because the extent of compaction is small, experiments have to be accurate and their interpretations should be as model-free as possible. Theory also suggests that collapsibility itself could be a physical restriction on the evolution of foldable sequences, and also provides a physical basis for the origin of multidomain proteins.

Correction: Spatially heterogeneous dynamics of cells in a growing tumor spheroid: comparison between theory and experiments.

Correction for 'Spatially heterogeneous dynamics of cells in a growing tumor spheroid: comparison between theory and experiments' by Sumit Sinha et al., Soft Matter, 2020, 16, 5294-5304, DOI: .

Spatially heterogeneous dynamics of cells in a growing tumor spheroid: comparison between theory and experiments.

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 a 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, Δ(t) ≈ tα with α < 1), whereas cells in the periphery exhibit superdiffusive motion, Δ(t) ≈ tα with α > 1. The motion of most of the cells near the periphery is highly persistent and correlated directional motion due to cell doubling and apoptosis rates, thus explaining the observed superdiffusive behavior. The α 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 α should depend on the location of the cells in the tumor is amenable to experimental testing. The highly heterogeneous dynamics of cells in the tumor spheroid provides a plausible mechanism for the origin of intratumor heterogeneity.

Charge fluctuation effects on the shape of flexible polyampholytes with applications to intrinsically disordered 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 effect due to net charge per monomer (σ) and an effective attractive PA interaction due to charge fluctuations, δσ. The interplay between these terms gives rise to non-monotonic dependence of the radius of gyration, R g , on the inverse Debye length, κ, when PA effects are important ( δ σ σ > 1 ). In the opposite limit, R g decreases monotonically with increasing κ. Simulations of PA chains, using a charged bead-spring model, further corroborate 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 intrinsically disordered proteins (IDPs), cannot be determined using only the bare PA parameters (σ and δσ). The theory is used to calculate the changes in R g on N, the number of residues for a set of IDPs. For a certain class of IDPs, with N between 24 and 441, the size grows as R g ∼ N 0.6, which agrees with data from small angle X-ray scattering experiments.

Protein collapse is encoded in the folded state architecture.

Folded states of single domain globular proteins are compact with high packing density. The radius of gyration, Rg, of both the folded and unfolded states increase as Nν where N is the number of amino acids in the protein. The values of the Flory exponent ν are, respectively, ≈⅓ and ≈0.6 in the folded and unfolded states, coinciding with those for homopolymers. However, the extent of compaction of the unfolded state of a protein under low denaturant concentration (collapsibility), conditions favoring the formation of the folded state, is unknown. We develop a theory that uses the contact map of proteins as input to quantitatively assess collapsibility of proteins. Although collapsibility is universal, the propensity to be compact depends on the protein architecture. Application of the theory to over two thousand proteins shows that collapsibility depends not only on N but also on the contact map reflecting the native structure. A major prediction of the theory is that ß-sheet proteins are far more collapsible than structures dominated by α-helices. The theory and the accompanying simulations, validating the theoretical predictions, provide insights into the differing conclusions reached using different experimental probes assessing the extent of compaction of proteins. By calculating the criterion for collapsibility as a function of protein length we provide quantitative insights into the reasons why single domain proteins are small and the physical reasons for the origin of multi-domain proteins. Collapsibility of non-coding RNA molecules is similar ß-sheet proteins structures adding support to "Compactness Selection Hypothesis".

On noise induced Poincaré-Andronov-Hopf bifurcation.

It has been numerically seen that noise introduces stable well-defined oscillatory state in a system with unstable limit cycles resulting from subcritical Poincaré-Andronov-Hopf (or simply Hopf) bifurcation. This phenomenon is analogous to the well known stochastic resonance in the sense that it effectively converts noise into useful energy. Herein, we clearly explain how noise induced imperfection in the bifurcation is a generic reason for such a phenomenon to occur and provide explicit analytical calculations in order to explain the typical square-root dependence of the oscillations' amplitude on the noise level below a certain threshold value. Also, we argue that the noise can bring forth oscillations in average sense even in the absence of a limit cycle. Thus, we bring forward the inherent general mechanism of the noise induced Hopf bifurcation naturally realisable across disciplines.

Exact solution of the Zwanzig-Lauritzen model of polymer crystallization under tension.

We solve a two-dimensional model for polymer chain folding in the presence of mechanical pulling force (f) exactly using equilibrium statistical mechanics. Using analytically derived expression for the partition function we determine the phase diagram for the model in the f-temperature (T) plane. A square root singularity in the susceptibility indicates a second order phase transition from a folded to an unfolded state at a critical force (fc) in the thermodynamic limit of infinitely long polymer chain. The temperature dependence of fc shows a reentrant phase transition, which is reflected in an increase in fc as T increases below a threshold value. As a result, for a range of f values, the unfolded state is stable at both low and high temperatures. The high temperature unfolded state is stabilized by entropy whereas the low temperature unfolded state is dominated by favorable energy. The exact calculation could serve as a benchmark for testing approximate theories that are used in analyzing single molecule pulling experiments.

Origin of superdiffusive behavior in a class of nonequilibrium systems.

Experiments and simulations have established that dynamics in a class of living and abiotic systems that are far from equilibrium exhibit superdiffusive behavior at long times, which in some cases (for example, an 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 timescales the mean-square displacement is subdiffusive due to jamming, whereas at long times it is superdiffusive. The results obtained by using a stochastic quantization method, which is needed because of the absence of the fluctuation-dissipation theorem, show that the superdiffusive behavior is universal and impervious to the nature of cell-cell interactions. Surprisingly, the theory also quantitatively accounts for the nontrivial dynamics observed in simulations of a model soap foam characterized by creation and destruction of spherical bubbles, which suggests that the two nonequilibrium systems belong to the same universality class. The theoretical prediction for the superdiffusion exponent is in excellent agreement with simulations for collective motion of tumor cells and dynamics associated with soap bubbles.

Giant Casimir Nonequilibrium Forces Drive Coil to Globule Transition in Polymers.

We develop a theory to probe the effect of nonequilibrium fluctuation-induced forces on the size of a polymer confined between two horizontal, thermally conductive plates subject to a constant temperature gradient, ∇ T. We assume that (a) the solvent is good and (b) the distance between the plates is large so that in the absence of a thermal gradient the polymer is a coil, whose size scales with the number of monomers as Nν, with ν ≈ 0.6. We find that above a critical temperature gradient, ∇ Tc ≈ N-5/4, a favorable attractive monomer-monomer interaction due to the giant Casimir force (GCF) overcomes the chain conformational entropy, resulting in a coil-globule transition. Our predictions can be verified using light-scattering experiments with polymers, such as polystyrene or polyisoprene in organic solvents in which the GCF is attractive.

Optimal information transfer in enzymatic networks: A field theoretic formulation.

Signaling in enzymatic networks is typically triggered by environmental fluctuations, resulting in a series of stochastic chemical reactions, leading to corruption of the signal by noise. For example, information flow is initiated by binding of extracellular ligands to receptors, which is transmitted through a cascade involving kinase-phosphatase stochastic chemical reactions. For a class of such networks, we develop a general field-theoretic approach to calculate the error in signal transmission as a function of an appropriate control variable. Application of the theory to a simple push-pull network, a module in the kinase-phosphatase cascade, recovers the exact results for error in signal transmission previously obtained using umbral calculus [Hinczewski and Thirumalai, Phys. Rev. X 4, 041017 (2014)2160-330810.1103/PhysRevX.4.041017]. We illustrate the generality of the theory by studying the minimal errors in noise reduction in a reaction cascade with two connected push-pull modules. Such a cascade behaves as an effective three-species network with a pseudointermediate. In this case, optimal information transfer, resulting in the smallest square of the error between the input and output, occurs with a time delay, which is given by the inverse of the decay rate of the pseudointermediate. Surprisingly, in these examples the minimum error computed using simulations that take nonlinearities and discrete nature of molecules into account coincides with the predictions of a linear theory. In contrast, there are substantial deviations between simulations and predictions of the linear theory in error in signal propagation in an enzymatic push-pull network for a certain range of parameters. Inclusion of second-order perturbative corrections shows that differences between simulations and theoretical predictions are minimized. Our study establishes that a field theoretic formulation of stochastic biological signaling offers a systematic way to understand error propagation in networks of arbitrary complexity.

Nonequilibrium statistical physics with fictitious time.

Problems in nonequilibrium statistical physics are characterized by the absence of a fluctuation dissipation theorem. The usual analytic route for treating these vast class of problems is to use response fields in addition to the real fields that are pertinent to a given problem. This line of argument was introduced by Martin, Siggia, and Rose. We show that instead of using the response field, one can, following the stochastic quantization of Parisi and Wu, introduce a fictitious time. In this extra dimension a fluctuation dissipation theorem is built in and provides a different outlook to problems in nonequilibrium statistical physics.