Physical Biology

FRET-sensors are a well-established method to investigate protein-protein interactions. To determine how FRET-sensor can be employed for the study of switchable allosteric modulator proteins (SAMPs) I extend a previously established model for enzymatic SAMPs to include a FRET-sensor system. Using this model, I determine the prerequisites for using FRET to investigate modulator-regulator interaction. The model shows, that under saturating stimulus conditions only a trimolecular complex contributes to the measured FRET value. How the signal is relayed by the modulator can be investigated by comparing FRET values of unstimulated and signal-saturated sensor systems. Finally, to determine the allosteric mode of signal transduction the natural logarithm of the ratio of stimulated and unstimulated FRET efficiencies is a useful metric.

Potassium voltage-gated (Kv) channels need to detect and respond to rapidly changing ionic concentrations in their environment. With an essential role in regulating electric signaling, they would be expected to be optimal sensors that evolved to predict the ionic concentrations. To explore these assumptions, we use statistical mechanics in conjunction with information theory to model how animal Kv channels respond to changes in potassium concentrations in their environment. By estimating mutual information in representative Kv channel types across a variety of environments, we find two things. First, under a wide variety of environments, there is an optimal gating current that maximizes mutual information between the sensor and the environment. Second, as Kv channels evolved, they have moved towards decreasing mutual information with the environment. This either suggests that Kv channels do not need to act as sensors of their environment or that Kv channels have other functionalities that interfere with their role as sensors of their environment.

Biological systems are fundamentally containers of thermally fluctuating atoms that through unknown mechanisms are structurally layered across many thermal scales from atoms to amino acids to primary, secondary, and tertiary structures to functional proteins to functional macromolecular assemblies and up. Understanding how the irreversible kinetics (i.e., the arrow of time) of biological systems emerge from the equilibrium kinetics of constituent structures defined on smaller thermal scales is central to describing biological function. Muscles irreversible power stroke - with its mechanochemistry defined on both the thermal scale of muscle and the thermal scale of myosin motors - provides a clear solution to this problem. Individual myosin motors function as reversible force-generating switches induced by actin binding and gated by the release of inorganic phosphate, Pi. As shown in a companion article, when N individual switches thermally scale up to an ensemble of N switches in muscle, the entropy of a binary system of switches is created. We have shown in muscle that a change in state of this binary system of switches entropically drives actin-myosin binding (the switch) and muscles irreversible power stroke, and that this simple two-state model accurately accounts for most key aspects of muscle contraction. Extending this observation beyond muscle, here I show that the chemical kinetics of an ensemble of N molecules differs fundamentally from a conventional chemical analysis of N individual molecules, describing irreversible chemical reactions as being pulled into the future by the a priori defined entropy of a binary system rather than being pushed forward by the physical occupancy of chemical states (e.g., mass action).

Complex biological processes, such as cellular differentiation, require an intricate rewiring of intra-cellular signalling networks. Previous characterisations of these networks revealed that promiscuity in signalling, quantified by a raised network entropy, underlies a less differentiated and malignant cell state. A theoretical connection between entropy and Ricci curvature has led to applications of discrete curvatures to characterise biological signalling networks at distinct time points during differentiation and malignancy. However, understanding and predicting the dynamics of biological network rewiring remains an open problem. Here we construct a framework to apply discrete Ricci curvature and Ricci flow to the problem of biological network rewiring. By investigating the relationship between network entropy and Forman-Ricci curvature, both theoretically and empirically on single-cell RNA-sequencing data, we demonstrate that the two measures do not always positively correlate, as has been previously suggested, and provide complementary rather than interchangeable information. We next employ discrete normalised Ricci flow, to derive network rewiring trajectories from transcriptomes of stem cells to differentiated cells, which accurately predict true intermediate time points of gene expression time courses. In summary, we present a differential geometry toolkit for investigation of dynamic network rewiring during cellular differentiation and cancer.

Multicellular organisms develop a wide variety of highly-specialized cell types. The consistency and robustness of developmental cell fate trajectories suggests that complex gene regulatory networks effectively act as low-dimensional cell fate landscapes. A complementary set of works draws on the theory of dynamical systems to argue that cell fate transitions can be categorized into universal decision-making classes. However, the theory connecting geometric landscapes and decision-making classes to high-dimensional gene expression space is still in its infancy. Here, we introduce a phenomenological model that allows us to identify gene expression signatures of decision-making classes from single-cell RNA-sequencing time-series data. Our model combines low-dimensional gradient-like dynamical systems and high-dimensional Hopfield networks to capture the interplay between cell fate, gene expression, and signaling pathways. We apply our model to the maturation of alveolar cells in mouse lungs to show that the transient appearance of a mixed alveolar type 1/type 2 state suggests the triple cusp decision-making class. We also analyze lineage-tracing data on hematopoetic differentiation and show that bipotent neutrophil-monocyte progenitors likely undergo a heteroclinic flip bifurcation. Our results suggest it is possible to identify universal decision-making classes for cell fate transitions directly from data.

Time resolved Raman and infrared spectroscopy experiments show the basic features of information transfer between residues in proteins. Here, we present the theoretical basis of information transfer using a simple elastic net model and recently developed entropy transfer concept in proteins. Mutual information between two residues is a measure of communication in proteins which shows the maximum amount of information that may be transferred between two residues. However, it does not explain the actual amount of transfer nor the transfer rate of information between residues. For this, dynamic equations of the system are needed. We used the Schreiber theory of information transfer and the Gaussian network Model of proteins, together with the solution of the Langevin equation, to quantify allosteric information transfer. Results of the model are in perfect agreement with ultraviolet resonance Raman measurements. Analysis of the allosteric protein Human NAD-dependent isocitrate dehydrogenase shows that a multitude of paths contribute collectively to information transfer. While the peak values of information transferred are small relative to information content of residues, considering the estimated transfer rates, which are in the order of megabits per second, sustained transfer during the activity time-span of proteins may be significant.

Cells dynamically remodel their internal structures by modulating the arrangement of actin filaments (AFs). In this process, individual AFs exhibit stochastic behavior without knowing macroscopic higher-order structures they are meant to create or disintegrate. Cellular adaptation to environmental cues is accompanied with this type of self-assembly and disassembly, but the mechanism allowing for the stochastic process-driven remodeling of the cell structure remains incompletely understood. Here we employ percolation theory to explore how AFs interacting only with neighboring ones without recognizing the overall configuration can nonetheless construct stress fibers (SFs) at particular locations. To achieve this, we determine the binding and unbinding probabilities of AFs undergoing cellular tensional homeostasis, a fundamental property maintaining intracellular tension. We showed that the duration required for the assembly of SFs is shortened by the amount of preexisting actin meshwork, while the disassembly occurs independently of the presence of actin meshwork. This asymmetry between the assembly and disassembly, consistently observed in actual cells, is explained by considering the nature of intracellular tension transmission. Thus, our percolation analysis provides insights into the role of coexisting higher-order actin structures in their flexible responses during cellular adaptation.

Cooperative response is ubiquitous and vital for regulatory control and ultra-sensitivity in various cellular biophysical processes. Ligands, acting as signaling molecules, carry information which is transmitted through the elements of the biochemical network during binding processes. In this work, we address a fundamental issue regarding the link between the information content of the various states of the binding network and the observable binding statistics. Two seminal models of cooperativity, viz., the Koshland-Nemethy-Filmer (KNF) network and the Monod-Wyman-Changeux (MWC) network are considered for this purpsoe which are solved using the chemical master equation approach. Our results establish that the variation of Shannon information associated with the network states has a generic form related to the average binding number. Further, the logarithmic sensitivity of the slope of Shannon information is shown to be related to the Hill slope in terms of the variance of the binding number distributions. 1

Small molecule modifiers whence bound, allosterically, will alter the binding of a macromolecule to one- or more-cognate substrates/partners via conformational and non-conformational changes. Although allostery is inferred directly from empirical data, the mathematical basis of these models, constraints deployed and choice of parameter(s) are not clear. Here, we present and characterize a discrete-to-continuous mathematical model for ensemble distributions of a ligand-interacting macromolecular species across milieux-dependent conformational states and examine its role in the genesis and progression of cooperative binding. The premise, of our model, is a set of occupancy matrices (sparse, binary, strictly delocalized) which can be partitioned by a probability-based hyperparameter into mutually exclusive proper subsets of occupancy matrices with identical multinomial probabilities. Since each subset is canonical with a constituent occupancy matrix, it is characterized by a unique multinomial probability. The inner product of combinatorial pairs of all mutually exclusive subsets of occupancy matrices, with an expression for the summed transitional probabilities (finite differences between unique multinomial probabilities), is the differentiable matrix of strictly positive real-valued numbers for the system of ensemble distributions. Whilst the harmonic mean is presented as a generic solution for a system of ensemble distributions, the row-wise definite integral for each column is the finite union of open intervals (contiguous, strictly monotone) which in tandem with a set of interval-specific and bounded transitional probabilities constitutes a piecewise smooth curve (path-connected-, closed- and compact-set). Our discrete-to-continuous model is phenomenological and able to recapitulate the basic tenets of cooperative binding whilst offering insights into the genesis and progression of the same.

The functional activity of proteins within cells is often unequally distributed: only a small subset of molecules tends to account for the majority of cellular work. This skewed contribution pattern, reminiscent of Paretos principle, often known as the 80/20 rule, has been observed across various protein classes, yet its mechanistic origin remains poorly understood. In this study, we present a statistical mechanics-based framework that explains how such disparities naturally emerge from biologically plausible rules of interaction and regulation. By modeling proteins as elements whose activity levels and outputs evolve through mutual comparison and feedback, we demonstrate that power-law distributions can arise without assuming any intrinsic heterogeneity. The model also captures a recursive feature of disparity: even among highly active proteins, a new skewed distribution reappears when a subpopulation is isolated, reflecting the scale-invariant structure commonly observed in complex adaptive systems. We analytically derive these patterns under both positive and negative feedback scenarios and identify key conditions under which long-term functional dominance is established. Our results offer a mechanistic interpretation for the coexistence of active and inactive molecular populations and suggest that functional inequality may reflect an adaptive organizational principle of cellular systems.

Biological organisms have sensors that communicate information about the environment. Analyzing how well these biological sensors function has usually been done with mutual information between the sensor signal and the environment, but that can be computationally intractable and summarize something quite complex with just a single number. We suggest that alternatively, one may profitably analyze these biosensors using bias and variance or confusion matrices, depending on the kind of environment. Stimulus-dependent Maximum Entropy models are used to develop estimators of the environmental state given the sensor state, and these estimators in turn are then used to calculate either the bias and variance of the estimator or confusion matrices. We focus on several examples to understand the utility of non-information-based analyses: ligand-receptor binding models spanning genetic regulation to neuronal communication to bacterial chemotaxis, and spin-glass Ising models for neural activity in cultured neurons. These new computationally-efficient analyses add insight to existing analyses based on mutual information; in particular, mutual information estimates give one number to characterize responses to all environmental inputs, and this analysis method characterizes how sensors respond to each environmental input. Categorical analyses, meanwhile, indicate the presence of memory without much prediction in confusion matrix elements in cultured neural networks, adding to previous understanding from mutual information estimates. Author summaryAll living organisms use external stimuli to navigate their environment via their sensors. Because encoding information costs energy, organisms retain only a fraction of the information received from their sensors, ideally information that maximizes their ability to remember past environmental states or predict future ones, key functions that support survival. To better understand how well sensor systems absorb stimulus information, we used stimulus-dependent Maximum Entropy (MaxEnt) models with maximum likelihood estimation and typical statistical metrics, such as confusion matrices or bias and variance. This approach provides two primary benefits over previous approaches: it is more computationally efficient, and it provides a more information-rich picture on how sensors interact with stimulus.

The persistence of cell migration is a fundamental property of motile behavior, enabling cells to maintain directionality while adapting to fluctuations and external cues. This feature underlies essential processes such as development, immune responses, and cancer invasion. Classical mathematical models have offered key insights into directed migration, yet they often neglect temporal correlations arising from cellular mechanisms that stabilize polarity and protrusion dynamics, processes not well captured by simple white noise. Here, we introduce an agent-based model based on stochastic differential equations (SDEs) that integrates fractional Brownian motion (fBm) to explicitly incorporate translational autocorrelation in cell trajectories. We simulate migration as a function of angular reorientation (Dr) and the strength of correlated noise (H). In this framework, temporal correlation stabilizes trajectory features inherited from initial conditions, whereas angular reorientation introduces variability that enables transitions between erratic and directed motion. Our simulations show that, unlike models driven by white noise, positive correlation markedly enhances persistence even under strong angular reorientation. Moreover, the combination of Dr and H gives rise to emergent behaviors, particularly in the presence of taxis, where persistence and responsiveness are jointly tuned. These results identify correlated noise as a proxy for intrinsic cellular memory and provide a versatile computational framework to interpret the diversity and complexity of migratory behaviors. Significance StatementCell migration drives key biological processes such as immune surveillance, development, and cancer invasion. Most models reduce motility to random walks perturbed by white noise, overlooking temporal correlations that arise from intrinsic cellular memory. By integrating fractional Brownian motion into agent-based modeling, we show how correlated translational noise interacts with angular diffusion to produce emergent behaviors, including overshooting, exploratory loops, and persistent trajectories. Our framework unifies these outcomes under a single mechanistic description and highlights how intrinsic noise modulates taxis, exploration, and persistence. This approach provides mathematicians and cell biologists with a versatile tool to test how cells balance stability and adaptability in dynamic environments.

This study introduces a statistical mechanical framework for allosteric communication in proteins based on the spanning-tree ensemble of residue contact networks. By representing protein structures as weighted graphs, we identify each spanning tree as a topological microstate. The canonical partition function is evaluated exactly via the determinant of the reduced weighted Kirchhoff (Laplacian) matrix, allowing for the derivation of global thermodynamic functions (including Helmholtz free energy, internal energy, entropy, and heat capacity) without approximation. Allosteric channels between specific residue pairs are defined as sub-ensembles containing unique simple paths. Using the Burton-Pemantle theorem and the Moore-Penrose pseudoinverse of the graph Laplacian, we compute exact path probabilities and channel-specific thermodynamics. This methodology enables a decomposition of channel heat capacity into energetic and topological components and quantifies residue-level allosteric importance through fractional contributions to the channel partition function. The framework was applied to the G12D mutation in KRAS, comparing wild-type (PDB: 6GOD) and mutant (PDB: 6GOF) proteins. Results show that while the mutation minimally affects mean internal energy and entropy, it reduces global heat capacity by 27.3%. This indicates a topological stiffening where the mutant occupies a significantly narrower landscape of spanning-tree configurations. At the channel level, the mutation maintains distributional stability across six functional routes but triggers a substantial internal redistribution of allosteric importance. Specific residues, such as Q61 and F156, shift occupancy by up to 35.5%. These findings suggest that the G12D mutation does not destroy communication pathways but reorganizes internal information traffic to favor a catalytically impaired state. This approach provides a rigorous, parameter-free metric for understanding how point mutations perturb distal protein signaling.

Quorum sensing (QS) is a regulatory mechanism used by bacteria to coordinate group behavior in response to high cell densities. During QS, cells monitor the concentration of external signals, known as autoinducers, as a proxy for cell density. QS often involves positive feedback loops, leading to the upregulation of genes associated with QS signal production and detection. This results in distinct steady-state concentrations of QS-related molecules in QS-ON and QS-OFF states. Due to the slow decay rates of biomolecules such as proteins, even after removal of the initial stimuli, cells can retain elevated levels of QS-associated biomolecules for extended periods of time. This persistence of biomolecules after the removal of initial stimuli has the potential to impact the response to future stimuli, denoting a memory of past exposure. This phenomenon, which is a consequence of the carry-over of biomolecules rather than genetic inheritance, is known as "phenotypic" memory. This theoretical study aims to investigate the presence of phenotypic memory in QS and the conditions that influence this memory. Numerical simulations based on ordinary differential equations and analytical modeling were used to study gene expression in response to sudden changes in cell density and extracellular signal concentrations. The model examined the effect of various cellular parameters on the strength of QS memory and the impact on gene regulatory dynamics. The findings revealed that QS memory has a transient effect on the expression of QS-responsive genes. These consequences of QS memory depend strongly on how cell density was perturbed, as well as various cellular parameters, including the Fold Change in the expression of QS-regulated genes, the autoinducer synthesis rate, the autoinducer threshold required for activation, and the cell growth rate. Author summaryBacteria use a mechanism known as quorum sensing (QS) to collaborate when their numbers are high. Cells produce and detect signals that trigger the production of certain proteins and changes in cell behavior. Interestingly, the molecules produced during this process can linger even after the initial signal is gone. The persistence of these molecules is a form of "memory", as cells are temporarily changed by events in the recent past. Our theoretical study focuses on exploring this memory and the factors that influence it. To do this, we used simulations and models to examine how history of exposure to signals can affect the future response of cells. We found that the prior exposure to signals can influence how bacteria respond in the future, but this memory only has consequences under specific conditions. This research contributes to our understanding of quorum sensing and how bacteria adapt to environmental changes.

T cells must assess and choose between surveilling large areas, but also engage efficiently with the target cells. This process is translated into variations in speed and turning angle of T cells. In this study, we propose a generalized algorithm to analyze cell migration data with focus on CD8+ T cells, using clustering technique to identify the number of different migration states and Hidden Markov Model to capture the dynamical switching between them. The algorithm only requires a set of position observations in a series of times, independent of other factors. While this study focuses on CD8+ T cell migration, this approach can potentially be used broadly to study the migration of other cell types as well. For the current analysis, low and high avidity T cells in melanoma tumors were tracked ex vivo using two-photon microscopy. Our findings suggest that CD8+ T cells follow a two-state migration dynamic, with one state being faster, while the other slower and more localized. Moreover, we established a statistical methodology to analyze T cell migration to assess whether there is true variability in cell speeds as distinguished from stochastic fluctuations about a single speed, and it can be applied across different experimental platforms.

We show on the biophysical basis that the observed enzyme kinetic parameters related to the substrate binding, product turnover and catalytic efficiency follow power-law type density functions. These finding are validated with the available datasets on various enzymes across different substrates. The product turnover rates and catalytic efficiencies seems to follow a bimodal type density functions in line with the single molecule experiments on enzyme catalysis which can be explained by a sum of two different power-law type density functions. The curve geometry of the density functions is decided by the underlying biophysical factors and the location of the peak is dictated by the natural selection pressure.

Tau is a microtubule-associated protein (MAP) that is highly expressed in neurons. Recent in-vitro studies have shown that tau molecules can segregate into cohesive envelopes on paclitaxel (taxol) stabilized microtubules. These envelopes act as selective barriers gating the motor transport and protecting microtubules against the action of severing enzymes. However, the mechanisms underlying the formation of these envelopes remain unclear. Recent in vitro reconstitution assays revealed that the formation of tau envelopes induces a compaction of the underlying microtubule lattice, whereas taxol binding induces lattice expansion, indicating competitive binding between tau and taxol on microtubules. In this study, we use physical modeling to investigate how this competition between tau and taxol regulates the formation of tau envelopes. Our model includes static and dynamic floor-fields, concepts adapted from pedestrian dynamics. By comparing our simulation results with experimental data, we demonstrate that tau-taxol competition on the microtubule lattice is sufficient to explain the features of the formation and disassembly of tau envelopes. Notably, tau-tau interactions are not necessary to account for the steadiness of these envelopes.

Over the last 30 years, the hop diffusion model has been an important paradigm for interpreting both cell membrane structure and dynamics. The basic premise of the model is that the cell membrane is organized into small domain regions through a combination of multi-component phase-separation and interaction between membrane components and the proximal fibers of the intracellular cytoskeleton and extracellular matrix. The partitioned characteristics of this two-dimensional fluid are thought to impose both steric and hydrodynamic barriers that restrict the free motion of mobile membrane components, save for their occasional passage via hopping from one domain to another. Previous investigations of hop diffusion within the cell membrane have identified the potential for diffusional anisotropy [Jaqaman et al. 2011. Cell, 146(4), pp.593-606]. This work utilizes numerical simulations and develops new analytical theory to provide an approximate quantitative description of such asymmetric compartmentalization. These methods are then used to examine the physical requirements for generation of asymmetric hop diffusion within the membrane before concluding with a discussion of the potential biological consequences of such behavior.

The mechanical rigidity-flexibility architecture of protein kinases play a critical role in regulating conformational stability and inhibitor response, yet remains insufficiently quantified for Epidermal Growth Factor Receptor (EGFR). Here, we apply Constraint Network Analysis (CNA) to systematically characterize the global and local mechanical properties of wild-type EGFR in its apo state and when bound to the ATP-competitive inhibitor gefitinib. Analysis of multiple global rigidity indices reveal a well-defined rigidity percolation transition in apo EGFR at an energy cutoff of approximately -2.0 kcal mol-1, indicative of an intrinsically stable mechanical framework. Gefitinib binding shifts this transition slightly to higher energies and sharpens the percolation behavior, accompanied by enhanced long-range mechanical coupling, increased rigidity order parameters, and reduced configurational entropy. Importantly, residue-level rigidity and percolation profiles remain largely conserved between the two states, demonstrating that ligand binding does not induce large-scale reorganization of the EGFR mechanical network. Instead, inhibition arises from subtle, localized rigidification within functionally relevant regions, consistent with stabilization of an inactive conformational ensemble. Collectively, this work establishes the first CNA-based mechanical reference state for wild-type EGFR and underscores the utility of network rigidity analysis for resolving ligand-induced effects that are structurally subtle yet mechanistically significant. This framework provides a quantitative baseline for future studies of oncogenic mutations and drug-resistant EGFR variants.

Small-molecule drugs canonically act by binding to a specific site on a single protein, leading to a change in the proteins activity. In particular, allosteric agonists bind preferentially to the active form of the protein, increasing the population of the active state and increasing activity. Hence, it is not surprising that similar compounds often act in similar ways, because they naturally bind to the same sites. However, recent work has provided examples of closely related small molecules that act super-additively when co-administered, a phenomenon that is difficult to explain using this approach. Here, we derive a simple thermodynamic model that describes how a super-additive response can occur, even for two very similar ligands. We discuss its implications for the specific case of cysteine-derived compounds and the treatment of opioid withdrawal symptoms and suggest avenues by which it could be tested experimentally.