Physical Biology

Trigger waves are positive feedback-mediated traveling waves that propagate without slowing down or diminishing in amplitude, enabling biochemical coordination over distances where diffusion alone would be insufficient. Apoptosis can spread as trigger waves of caspase activity, as demonstrated in Xenopus eggs and egg extracts. We measured apoptotic wave speeds in extracts at two cell cycle states and found that waves propagated more slowly in M-phase arrested cytostatic factor (CSF) extracts than in interphase extracts (21 versus 27 m min-1). Luther's equation, v=2{surd}D{kappa}, predicts wave speed from the diffusion coefficient D and the positive-feedback doubling rate {kappa}. However, D and {kappa} were comparable between the two extracts and could not explain the speed difference. This discrepancy highlighted a key limitation of the classical equation: it does not account for an activation threshold in trigger wave initiation, and in fact the threshold required to trigger apoptosis was found to differ between cell cycle states. We therefore extended Luther's equation to include such a threshold, obtaining v{propto}{surd}(AD/Uc ), where A is the reaction rate and Uc is the activation threshold. Perturbing apoptotic sensitivity confirmed the predicted threshold-speed relationship. These findings establish the activation threshold as a key determinant of trigger wave speed and provide a broadly applicable scaling law for predicting the spread of switch-like biochemical signals.

Non-genetic adaptation enables cancer cells to alter their phenotype under stress without requiring new mutations. However, the mechanisms by which electrical, mechanical, and hypoxic cues combine to shape this process in 3D tissues remain poorly understood. This work presents an agent-based tumor model that integrates vascular oxygen supply, a globally imposed electric field, mechanically mediated crowding and compression cues, phenotype transitions, cell growth, mitosis, death, and inheritance of adaptive memory across division. The simulated tumors exhibit a three-stage trajectory consisting of necrosis onset, transient collapse of live mass, and partial regrowth accompanied by progressive accumulation of adapted cells. Continuous electrical stimulation produces a dose-dependent reduction in live mass while markedly increasing the adapted fraction, with comparatively limited changes in final necrotic burden. This response is strongly conditioned by mechanics and reshapes (and is reshaped by) adaptive capacity. Pulsed stimulation further shows that, in the model, electric field amplitude and temporal schedule jointly determine memory phenomena, phenotypic diversification, and growth recovery. These results show that coupling local oxygen availability, mechanical constraints, electrical forcing, and history-dependent phenotype transitions can generate distinct tissue-level patterns of phenotypic heterogeneity. Both stimulus magnitude and temporal protocol influenced the resulting population structure, suggesting that the history of physical stress may be an important determinant of adaptive dynamics in spatially organized tumor models.

Understanding the relationship between gene expression dynamics and cellular identity remains a central challenge in single cell biology. Here, we introduce a novel computational and mathematical framework that integrates information geometry, fuzzy topology, and UMAP analysis to model gene expression landscapes derived from single cell RNA sequencing data. We formalize gene expression data as a fuzzy topological space, where interactions between expression points are governed by probabilistic distributions inspired by manifold learning approaches such as UMAP. Within this framework, we define an information geometric structure through a Fisher metric induced by these distributions, enabling the computation of geodesic trajectories that capture cellular differentiation processes. A key contribution of this work is the derivation of analytical conditions, expressed as expression radius formulas, that characterize local neighborhoods in gene expression space. These conditions allow for the identification of genes associated with stem cell states and predictions in transitional cell types in future work. Application of the proposed framework to single cell datasets reveals biologically meaningful gene sets enriched in key regulatory pathways and transcription factors, demonstrating the capacity of our approach to uncover latent structure in complex gene expression data. Our results suggest that integrating differential geometry with statistical learning theory offers a powerful paradigm for modeling genotype and phenotype relationships and cellular state transitions, with potential implications for precision medicine and systems biology.

The dominant paradigm for temperature robustness in biomolecular circuits is for the parameters to be tuned to have matching temperature dependencies so that their overall effect cancels out. This contrasts with the robustness due to circuit structure, typically operative in circuits where robustness to a single input parameter is desired. The importance of the circuit structure in temperature robustness is generally unclear. We addressed this issue in a benchmark negative feedback circuit using a combination of theoretical modelling and experimental measurements. We found that the response to a temperature perturbation in a model of negative feedback was qualitatively different from the response in a model without feedback. We experimentally measured the response of the negative feedback circuit to a temperature perturbation and found that it was smaller than that of the circuit without feedback, in line with the theoretical finding. We confirmed this theoretical prediction experimentally. The initial response of the negative feedback circuit, paradoxically, was larger than the circuit without feedback. The resolution of this paradox was in accounting for the faster dynamics in the negative feedback circuit. These results show a simple design principle of temperature robustness that can operate in a widespread circuit motif and may also apply to other perturbations which, like temperature, affect multiple parameters simultaneously.

Based on our experimental observation of activation state oscillations of different frequencies used to communicate information by the master human stress response regulator protein p38 MAPK 1, we develop a simple graphical approach for understanding and predicting the behavior of complex biological networks acting upon signals carrying information as different frequency waves of chemicals. This approach uses the same techniques used for analyzing and understanding information transmission using waves of electrical currents and fields used in electrical alternating current (AC) circuits. We show how biological components can be organized to behave as standard components found in electronic telecommunications circuits. Finally, we demonstrate how such components can be organized into complex biological signaling cascades whose behavior can be qualitatively and quantitatively understood, and whose output resembles that experimentally observed in p38.

Cell motility is powered by the bacterial flagellar motor, a rotary nanomachine whose activity is dynamically modulated by external stimuli. In Magnetospirillum gryphiswaldense, magnetic and chemical inputs are thought to converge at the level of motility control, yet the mechanisms underlying chemotactic regulation remain poorly understood. Here, we seek to establish a quantitative reference-state of a model strain of magnetotactic bacteria through bacterial flagellar motor dynamics. To achieve this goal, magnetotaxis was kept as a natural factor by keeping the earth's magnetic field as the only source of magnetism. To avoid aerotaxis bias, oxygen gradients were removed by implementing two different constant oxygen conditions: environmental oxygen exposure, and limited oxygen exposure. Together, the tethered-cell bacterial flagellar rotational assay and the free-swimming assay in the absence of external stimuli presented in this paper establish a framework for investigating motor and cellular swimming adaptation to magnetic, aerotactic, and chemical signals. Under reference-state conditions, the bacterial flagellar motor showed a tendency to exhibit log-normal distributions for the time spent in each motor state: runs in different directions (counter-clockwise and clockwise), and pause. A semi-Markov reference-state model was developed to provide a quantitative description of bacterial flagellar motor dynamics. The model revealed that cell magnetic polarity modulates the transition pathways leading to the paused state, whereas oxygen, in the absence of a gradient, primarily regulates residence in each motility state.

A fundamental biological process of transcription occurs in the cell nucleus, which is a complex medium that also contains multiple heterogeneous structures known as biomolecular condensates. Interestingly, some of these condensates contain mRNA molecules in addition to proteins, suggesting an important cellular role in transcription that is not yet well understood. In this work, we develop a minimal theoretical framework for quantitative investigation of the role of reversible mRNA condensation in transcription. Our discrete-state stochastic approach accounts for the most relevant processes, allowing us to explicitly evaluate the properties of the system and clarify the effects of condensation. Analytical calculations supported by computer simulations suggest that reversible mRNA condensation influences the transcription processes by maintaining a constant level of free mRNA in the nucleoplasm while lowering the degree of stochastic noise and increasing the robustness against external perturbations. Physicochemical arguments are presented to explain these observations. The proposed theoretical framework elucidates important microscopic aspects of transcription, providing a convenient quantitative tool for investigating complex biological phenomena.

Understanding noise propagation in gene regulatory circuits requires accounting for both model and resource constraints. In this work, we investigated the role of model order in influencing stochastic behaviour by deriving and analytically comparing reduced protein-only models with higher-order models that include mRNA and molecular complexes, and found that protein-based models can exhibit higher noise levels in the gene expression. Through frequency-response analysis, we explained that the higher-order models provide additional noise-filtering effects. We also analyzed a one-dimensional constrained model and showed that the Fano factor decreases as the strength of resource constraint increases. Finally, we considered larger circuit motifs, such as toggle switches and incoherent feed-forward loops, and found that resource limitations can minimise stochastic switching in a bistable circuit, whereas in an incoherent feed-forward loop, resource constraints can make the adaptation faster. Our results highlight that both mechanistic detail and shared resource constraints play a central role in determining fluctuation levels in biomolecular circuits.

Spatial proofreading is a mechanism that can enhance molecular discrimination by exploiting nonequilibrium diffusive transport between spatially separated source and readout regions. Here, we introduce a thermodynamically consistent reaction-diffusion model in which a gradient of active substrates is generated self-consistently by a reversible kinase-phosphatase switch coupled to nucleotide chemostats. The resulting chemical-potential gradient explicitly controls the nonequilibrium driving and allows the discrimination potential to be related to microscopic chemical rates and diffusive transport. A simple one-dimensional analysis shows that the ability of an enzyme to discriminate between wrong and right substrates is governed by a subtle balance between substrate-gradient confinement, controlled by phosphatase activity, the diffusive crossing time across the source-readout domain, and selective complex dissociation. We then extend the model to two-dimensional domains, showing that source, i.e. kinase, localization modulates spatial specificity by shaping the effective diffusive paths to the readout boundary. Finally, stochastic simulations reveal that molecular fluctuations generate intermittent wrong-readout events, which can be characterized through an event-weighted measure of specificity. Interestingly, we find that fluctuations promote frequent transitions to long-residence states in which discrimination is better than predicted by the deterministic estimate. Overall, our work highlights the importance of thermodynamically consistent descriptions of spatial proofreading and clarifies how energy input, transport, and spatial organization jointly shape biochemical discrimination.

Quantitative characterization of cell proliferation is central to preclinical drug discovery. Here, we evaluated Random Forest (RF) regression for predicting confluence-based cell growth trends using data from human cancer cell lines and benchmarked its performance against the widely used logistic and Gompertz models. The RF model achieved higher predictive accuracy within the evaluated dataset, particularly at higher initial seeding densities. We further extended the RF framework to predict growth curves across seeding densities. Collectively, these findings highlight the potential of data-driven modeling to complement conventional mathematical approaches and support more informed cell-culture design.

Oncogenic mutations are known to occur at non-random rates and specific hotspots, yet the structural and thermodynamic factors underlying these hotspots remain poorly understood. This study presents molecular dynamics simulations of 22 cancer hotspots across 10 oncogene families, simulated as histone-DNA complexes. Across 18 of 22 mutations, thermodynamic effects were predominantly localized to within 6 [A] of the DNA-histone interface (mean capture 104%), with van der Waals interactions driving the effect at the thermodynamic extremes, indicating that oncogenic mutations alter nucleosome binding through precise local contact changes rather than global structural rearrangements. Further, analysis revealed a gene-family correlated pattern in thermodynamic stability. RAS family mutations showed a consistent trend toward nucleosome stabilization (mean {Delta}{Delta}G = -4.82 kcal mol-1), while kinase domain mutations trended toward destabilization (mean {Delta}{Delta}G = +40.12 kcal mol-1), a difference reaching statistical significance in this exploratory analysis (Mann-Whitney U, p = 0.008). This interface-specific mechanism, combined with the gene-family-correlated thermodynamic pattern, provides a biophysical framework for understanding nucleosome-level contributions to cancer hotspot mutation biology.

There is strong evidence of tumors manipulating their resting membrane potential (Vmem). While most fully-differentiated cells have a Vmem of roughly -70mV, tumor cells are generally depolarized, with Vmem {approx}-30mV, which more closely resembles the Vmem of stem cells. This is often believed to serve the purpose of accelerating the cell cycle and hence advantaging tumor proliferation. But when the tumor becomes invasive, its cells sometimes revert to a hyperpolarized Vmem with no obvious reason why. Separately, it is well accepted that solid tumors that are not yet invasive greatly underexpress connexins relative to healthy tissue; connexins, for our purpose, form gap junctions (GJs), small connecting tubes between nearby cells. Tumors that are invasive, by contrast, overexpress connexins. There is very little explanation for the paradox that connexins are first underexpressed and then overexpressed. However, it has long been known that Vmem electrically gates GJs; specifically, that homotypic GJs conduct best when the two cells they connect have a similar Vmem. Our in-silico model results explain this phenomenon, showing that when considered together, tumors' electrical and connexin-expression behaviors form a unified and effective strategy to control communication between the tumor and its healthy neighbor cells. This has implications for the emerging field of cancer bioelectrics, potentially leading to more precisely-targeted therapies.

Haptotaxis is the directed migration of cells along gradients of substrate adhesion, and is an important guidance mechanism in biology. Recent experiments on fibronectin-patterned tracks show that cells typically migrate toward higher adhesion, then oscillate around the adhesion maximum, with substantial variability in speed and cell length. To explain these cellular shape and migration dynamics we present a minimal model for confined haptotaxis, built around one key physical ingredient: local adhesion strength directly enhances the recruitment of protrusive actin polymerization activity at the cell edge. This coupling creates a front-rear difference in protrusive activity, biasing the cell towards polarization in the direction of the higher adhesion. Quantitative comparison with experiment shows strong agreement at multiple levels: cell population-level directionality statistics and position-dependent changes in cell length and velocity. At the single-cell level, the model reproduces the full diversity of experimentally observed trajectories, including haptotactic bias and its dependence on adhesion-gradient strength, initial position, length and speed variations, myosin II inhibition, and migration on inverted adhesion gradients. Variability in experimental trajectories and migration speed is explained by differences in intrinsic actin polymerization activity and stochastic fluctuations, accounting for the full spectrum of observed migration patterns. Together, our results identify adhesion-dependent amplification of protrusive activity as a minimal and sufficient physical mechanism for understanding haptotaxis.

It is important to determine whether native state dynamics drive the misfolding and oligomerization of the prion protein, which are important events in prion disease, and how they are modulated by conformational conversion. Native (N) mouse prion protein (moPrP) is known to form small (OS) and large (OL) oligomers rich in {beta}-sheet, and in this study, photoinduced electron transfer-fluorescence correlation spectroscopy (PET-FCS) has been used to characterize intramolecular dynamics within individual monomeric units in both isolated OS and OL, as well as the diffusion properties of the oligomers. It is estimated that OS and OL comprise of about 15 and 55 monomeric units, respectively. Microsecond dynamics at each of the two regions that are the 1-3 and 2-3 interfaces of native protein are distinct in N, OS and OL, although they occur on very similar timescales. Analysis of the evolution of the distribution of diffusion times, determined using the maximum entropy method, indicates heterogeneity in the oligomerization reaction. Analysis of the change in the fluctuations which occur in two different timescales in the native state ensemble shows that they are damped more at the erstwhile 1-3 interface than the erstwhile 2-3 interface. The difference in the extent of damping at the erstwhile 1-3 and 2-3 interfaces can be explained on the basis of the structural changes known to occur across each region. The changes in dynamics occur concurrently in both regions, indicating that the structural changes accompanying conformational conversion also occur simultaneously during the oligomerization of moPrP.

Biological processes, from molecules diffusing to their regulatory destinations to whales pursuing food or mates, are widely exploratory. Such behaviors are effective when some verifiable functional state or outcome can be reached regardless of initial conditions. In these cases, systems repeatedly undergo distinct and abortive trajectories; the process only ends when the system finds 'right' outcomes. This dominance of the terminal condition (and indifference to the initial state) provokes a very different perspective than the conventional 'dynamical systems' framework that has been a centerpiece of the quantitative sciences for centuries. We hypothesize that many of these problems defy the initial-condition driven or gradients on landscapes so useful in problems ranging from mechanics to electrodynamics to chemical kinetics to mass and heat transport. We examine several mathematical frameworks that capture and unify key aspects of exploratory dynamics. One powerful way of thinking of such processes is the geometric distribution, where repeated failures are punctuated by a successful trajectory. We develop intuitions for how random walks with resets accelerate search processes. We highlight fresh and surprising behaviors of search under drift, cues, and checkpoints. Last, appreciating the probability of trajectories conditioned on satisfying macroscopic final outcomes reveals a language for the potency of variation sculpted by selection in their broadest forms. This can give the fictitious appearance of the future making itself known in the present, but we view this as conceptually similar to the way 'fictitious forces' arise in non-inertial reference frames. These approaches stress unity, open questions, and applications across a range of biological phenomena.

Within the studies of stochastic gene expression, apart from the variability of copy number of gene products, the problems of threshold crossing of those products are biologically important as they often lead to terminal cellular events. Here, we study the threshold crossing problem of the messenger ribonucleic acid (mRNA) and present an exact probability distribution of first passage times in Laplace space. The function furnishes moments of any order and also predicts the characteristic time of the exponential tail of the distribution, which we match against Gillespie simulations. We find that all the measures of relative fluctuations of the threshold crossing times show U-shapes within this simple model of mRNA, as was found earlier in more mathematically involved models of threshold crossing time statistics of proteins. Furthermore, we extend the exact formula to include the phenomenon of DNA duplication and the corresponding doubling of transcription rate. As expected, the distribution varies considerably depending on the onset of the duplication stage within the cell cycle.

The dynamics of gene regulatory networks are governed by intrinsic noise, stemming from the random nature of biochemical reactions, and by extrinsic noise, arising from fluctuations in cellular components and environmental conditions. Together, these sources can compromise the reliability of predictive computational models if not properly accounted for, and capturing both effects within a single framework remains a non-trivial task in computational biology. In this work, we propose an uncertainty quantification framework that addresses these two contributions jointly: intrinsic stochasticity is described through a partial integro-differential equation (PIDE) for the protein probability density function, whereas extrinsic noise is represented as parametric uncertainty in the kinetic parameters. The propagation of the uncertainty is carried out via an intrusive polynomial chaos expansion (PCE), in which the PCE coefficients are obtained from a stochastic Galerkin projection of the PIDE, yielding a coupled deterministic system that is solved with standard numerical methods. We illustrate the approach on a positive autoregulatory gene network with one and two uncertain kinetic parameters. The proposed approach accurately reproduces the mean, variance, and full protein probability density function, including the bimodal distributions, at a substantially lower computational cost.

Time compaction theory is a general framework explaining how a brain can efficiently deal with dynamic situations occurring in, e.g., sports games. It involves a geometric representation of the time dimension, which enables effective learning and strategic action planning. The theory has recently received experimental support in humans. However, its current computational model has an important limitation: it does not account for deliberate waiting and speed modulation, behaviors ubiquitous in natural environments. This work substantially extends the original model formulation by a dimensional lifting of an n-D workspace into (n + 1)-D mental space, where time remains geometrically embedded. The proposed biologically inspired computational model can generate adaptive behavior across increasingly complex situations, from navigation in everyday social environments to competitive sports. Furthermore, by actively conditioning the expected responses of other agents and stabilizing future predictions, we introduce the concept of uncertainty points in sequences of generalized cognitive maps to support the generation of adaptive strategies in interactive environments, where future prediction has a limited time horizon. Thus, we provide a mechanism for chaining short-term solutions into long-term strategies, which is illustrated by simulating the behavior of a player in a real football game. Author summaryHumans often anticipate future interactions in dynamic environments. Many behaviors, such as avoiding other pedestrians, letting someone pass through a narrow corridor, or reproducing the kind of dribbling maneuvers performed by elite football players, require deciding not only where to move but also when to move. Existing theories suggest that the brain simplifies such situations by representing future interactions as static spatial maps, making them easier to learn and recall. However, current computational models cannot naturally account for common behaviors such as waiting, slowing down, or modulating speed. Here we show that these behaviors readily emerge if the model space is extended by an additional virtual coordinate that encodes accumulated waiting rather than physical time. The proposed model simultaneously admits a wide variety of behaviors, including speed modulation, multigoal decisions, and compound actions, while preserving the principles of time compaction. We illustrate the model in everyday situations and by reproducing two real football plays, comparing the observed behaviors with model simulations. Our results suggest computational principles through which the human brain may efficiently represent, memorize, and exploit dynamic situations.

Structural network representations of metastatic dissemination typically focus on static topology without resolving transport dynamics, relaxation timescales, or steady-state behaviour. Here, we formulate a discrete Markovian transport model on a directed higher-order network with transition rates derived from qualitative clinical affinity classes. By constructing a non-Hermitian row-stochastic transfer operator, we characterise the relaxation dynamics through its spectral decomposition. The system exhibits a fast-mixing regime characterised by a spectral gap of {gamma} {approx} 0.67, corresponding to a characteristic relaxation timescale of {tau} {approx} 1.49 discrete steps, with the influence of the primary tumour origin progressively attenuated during dissemination. Convergence towards a non-equilibrium steady state (NESS) is accompanied by a reduction in Shannon entropy, concentrating probability mass within specific topological sinks. This spectral relaxation delineates two distinct dynamical regimes: early transient dissemination (n < {tau}), dominated by local organ-specific transition probabilities (organotropism), and the asymptotic regime (n > {tau}), determined increasingly by the global transport architecture of the network. Comparison with independent clinical and autopsy observations across 21 primary tumours and 23 target organs indicates that the predicted stationary distribution is consistent with the observed hierarchy of metastatic organ involvement.

Social genetic effects can substantially influence phenotypes in animals and plants and the study of them is of interest. Group housing in animals for example, in pigs provide an ideal framework for studying interactions among individuals. In animals, the genetic of social effects has been studied by simplifying the underlying interactions for example, by assuming that all the individuals in the group contribute equally to the social environment. This homogeneous behaviour is not observed in real conditions. Here, we address this issue showing that identification of interacting individuals is feasible in pig behaviour during growth, and we propose a model that leads to more accurate modelling of social genetics. Our approach infers the strength of interaction via time interval between visits of animals to automatic feeders. This measure of strength was then used in a reaction norm model to evaluate the genetic determinism of social effects. We also build a framework to transform variance components estimated for a trait continuously measured, such as feed intake per visit, into variance of total or aggregated trait e.g. during the growth period. Including social effects improves the fitting of the models. Visit-based social interaction was highest at short time intervals, decreasing rapidly and becoming negligible after 80 seconds. Variance components for the aggregated trait the total feed intake support the contribution of the social interactions. Our approach allows more precise understanding of social effects through individual feeding events and a measure of interaction strength, and further translates these into in an aggregated phenotype of interest.