Minimal Stochastic Model Research Articles

Minimal stochastic variability model of blazars in turbulent cascade

Minimal stochastic variability model of blazars in turbulent cascade

A Dimensionless Parameter for Predicting Convective Self‐Aggregation Onset in a Stochastic Reaction‐Diffusion Model of Tropical Radiative‐Convective Equilibrium

AbstractWe introduce a minimal stochastic lattice model for the column relative humidity (R) in the tropics, which incorporates convective moistening, horizontal transport and subsidence drying. The probability of convection occurring in a location increases with R, based on Tropical Rainfall Measuring Mission observations, providing a positive feedback that could lead to aggregation. We show that the simple model reproduces many aspects of full‐physics cloud resolving model experiments. Depending on model parameter settings and domain size and resolution choices, it can produce both random and aggregated equilibrium states. Clustering occurs more readily with larger domains and coarser resolutions, in agreement with full‐physics models. Using dimensional arguments and fits from empirical data, we derive a dimensionless parameter which we call the aggregation number, Nag, that predicts whether a specific model and experiment setup will result in an aggregated or random state. The parameter includes the moistening feedback strength, the horizontal moisture transport efficiency, the subsidence timescale, the domain size and spatial resolution. Using large ensembles of experiments, we show that the transition between random and aggregated states occurs at a critical value of Nag. We argue that Nag could help to understand the differences in aggregation states between full‐physics, cloud resolving models, which show little consensus about the robustness of self‐organized patterns, whose emergence appears to be sensitive to the model setup, physics and parameterizations.

Stochastic processes for pulsar timing noise: fluctuations in the internal and external torques

AbstractYoung pulsars deviate from a perfectly regular spin-down by two non-deterministic phenomena: impulsive glitches and timing noise. Both phenomena are interesting per se and may provide insights into the superfluid properties of neutron stars, but they also act as a barrier to high-precision pulsar timing and gravitational wave experiments. We study a minimal stochastic model to describe the spin-down of a multicomponent neutron star, with fluctuations in both the internal and external torques. The power spectral density and timing noise strength of this kind of model can be obtained analytically and compared with known results from pulsar timing observational campaigns. In particular, the presence of flat regions of the power spectral density can be interpreted as a signature of the presence of internal superfluid components. We also derive the expected scaling of the timing noise strength with the pulsar’s rotational parameters (or characteristic age). Therefore, the present framework offers a theoretical guideline to interpret the observed features of timing noise in both single pulsars and across the pulsar population.

Stiffening and dynamics of a two-dimensional active elastic solid.

This work deals with the mechanical properties and dynamics of an active elastic solid defined as a two-dimensional network of active stochastic particles interacting by nonlinear hard springs. By proposing a discrete model, it is numerically found that when activity in the system is turned on, the active solid stiffens as a function of propulsion forces, thus deviating from equilibrium mechanics. To understand this effect a minimal stochastic model is offered, and a physical explanation based on spatial symmetry-breaking is put forward. In addition, the dynamics of the active solid in the absence of an external stress is also studied. From this, three main features are observed to emerge, namely, a collective behavior within the active solid, a time-density fluctuation, and oscillating dynamics of the internal stresses towards a steady state.

Shear induced diffusion of platelets revisited.

The transport of platelets in blood is commonly assumed to obey an advection-diffusion equation with a diffusion constant given by the so-called Zydney-Colton theory. Here we reconsider this hypothesis based on experimental observations and numerical simulations including a fully resolved suspension of red blood cells and platelets subject to a shear. We observe that the transport of platelets perpendicular to the flow can be characterized by a non-trivial distribution of velocities with and exponential decreasing bulk, followed by a power law tail. We conclude that such distribution of velocities leads to diffusion of platelets about two orders of magnitude higher than predicted by Zydney-Colton theory. We tested this distribution with a minimal stochastic model of platelets deposition to cover space and time scales similar to our experimental results, and confirm that the exponential-powerlaw distribution of velocities results in a coefficient of diffusion significantly larger than predicted by the Zydney-Colton theory.

Leading edge maintenance in migrating cells is an emergent property of branched actin network growth.

Animal cell migration is predominantly driven by the coordinated, yet stochastic, polymerization of thousands of nanometer-scale actin filaments across micron-scale cell leading edges. It remains unclear how such inherently noisy processes generate robust cellular behavior. We employed high-speed imaging of migrating neutrophil-like HL-60 cells to explore the fine-scale shape fluctuations that emerge and relax throughout the process of leading edge maintenance. We then developed a minimal stochastic model of the leading edge that reproduces this stable relaxation behavior. Remarkably, we find lamellipodial stability naturally emerges from the interplay between branched actin network growth and leading edge shape - with no additional feedback required - based on a synergy between membrane-proximal branching and lateral spreading of filaments. These results thus demonstrate a novel biological noise-suppression mechanism based entirely on system geometry. Furthermore, our model suggests that the Arp2/3-mediated ~70-80° branching angle optimally smooths lamellipodial shape, addressing its long-mysterious conservation from protists to mammals.

Fluctuations in T cell receptor and pMHC interactions regulate T cell activation.

Adaptive immune responses depend on interactions between T cell receptors (TCRs) and peptide major histocompatibility complex (pMHC) ligands located on the surface of T cells and antigen presenting cells (APCs), respectively. As TCRs and pMHCs are often only present at low copy numbers their interactions are inherently stochastic, yet the role of stochastic fluctuations on T cell function is unclear. Here, we introduce a minimal stochastic model of T cell activation that accounts for serial TCR-pMHC engagement, reversible TCR conformational change and TCR aggregation. Analysis of this model indicates that it is not the strength of binding between the T cell and the APC cell per se that elicits an immune response, but rather the information imparted to the T cell from the encounter, as assessed by the entropy rate of the TCR-pMHC binding dynamics. This view provides an information-theoretic interpretation of T cell activation that explains a range of experimental observations. Based on this analysis, we propose that effective T cell therapeutics may be enhanced by optimizing the inherent stochasticity of TCR-pMHC binding dynamics.

Thermal modulation of Zebrafish exploratory statistics reveals constraints on individual behavioral variability

BackgroundVariability is a hallmark of animal behavior. It contributes to survival by endowing individuals and populations with the capacity to adapt to ever-changing environmental conditions. Intra-individual variability is thought to reflect both endogenous and exogenous modulations of the neural dynamics of the central nervous system. However, how variability is internally regulated and modulated by external cues remains elusive. Here, we address this question by analyzing the statistics of spontaneous exploration of freely swimming zebrafish larvae and by probing how these locomotor patterns are impacted when changing the water temperatures within an ethologically relevant range.ResultsWe show that, for this simple animal model, five short-term kinematic parameters — interbout interval, turn amplitude, travelled distance, turn probability, and orientational flipping rate — together control the long-term exploratory dynamics. We establish that the bath temperature consistently impacts the means of these parameters, but leave their pairwise covariance unchanged. These results indicate that the temperature merely controls the sampling statistics within a well-defined kinematic space delineated by this robust statistical structure. At a given temperature, individual animals explore the behavioral space over a timescale of tens of minutes, suggestive of a slow internal state modulation that could be externally biased through the bath temperature. By combining these various observations into a minimal stochastic model of navigation, we show that this thermal modulation of locomotor kinematics results in a thermophobic behavior, complementing direct gradient-sensing mechanisms.ConclusionsThis study establishes the existence of a well-defined locomotor space accessible to zebrafish larvae during spontaneous exploration, and quantifies self-generated modulation of locomotor patterns. Intra-individual variability reflects a slow diffusive-like probing of this space by the animal. The bath temperature in turn restricts the sampling statistics to sub-regions, endowing the animal with basic thermophobicity. This study suggests that in zebrafish, as well as in other ectothermic animals, ambient temperature could be used to efficiently manipulate internal states in a simple and ethological way.

Principles of low dissipation computing from a stochastic circuit model

We introduce a thermodynamically consistent, minimal stochastic model for complementary logic gates built with field-effect transistors. We characterize the performance of such gates with tools from information theory and study the interplay between accuracy, speed, and dissipation of computations. With a few universal building blocks, such as the NOT and NAND gates, we are able to model arbitrary combinatorial and sequential logic circuits, which are modularized to implement computing tasks. We find generically that high accuracy can be achieved provided sufficient energy consumption and time to perform the computation. However, for low-energy computing, accuracy and speed are coupled in a way that depends on the device architecture and task. Our work bridges the gap between the engineering of low dissipation digital devices and theoretical developments in stochastic thermodynamics, and provides a platform to study design principles for low dissipation digital devices.

Transitional Channel Flow: A Minimal Stochastic Model.

In line with Pomeau’s conjecture about the relevance of directed percolation (DP) to turbulence onset/decay in wall-bounded flows, we propose a minimal stochastic model dedicated to the interpretation of the spatially intermittent regimes observed in channel flow before its return to laminar flow. Numerical simulations show that a regime with bands obliquely drifting in two stream-wise symmetrical directions bifurcates into an asymmetrical regime, before ultimately decaying to laminar flow. The model is expressed in terms of a probabilistic cellular automaton of evolving von Neumann neighborhoods with probabilities educed from a close examination of simulation results. It implements band propagation and the two main local processes: longitudinal splitting involving bands with the same orientation, and transversal splitting giving birth to a daughter band with an orientation opposite to that of its mother. The ultimate decay stage observed to display one-dimensional DP properties in a two-dimensional geometry is interpreted as resulting from the irrelevance of lateral spreading in the single-orientation regime. The model also reproduces the bifurcation restoring the symmetry upon variation of the probability attached to transversal splitting, which opens the way to a study of the critical properties of that bifurcation, in analogy with thermodynamic phase transitions.

A stochastic model for dynamics of FtsZ filaments and the formation of Z -ring.

Understanding the mechanisms responsible for the formation and growth of FtsZ polymers and their subsequent formation of the Z -ring is important for gaining insight into the cell division in prokaryotic cells. In this work, we present a minimal stochastic model that qualitatively reproduces in vitro observations of polymerization, formation of dynamic contractile ring that is stable for a long time and depolymerization shown by FtsZ polymer filaments. In this stochastic model, we explore different mechanisms for ring breaking and hydrolysis. In addition to hydrolysis, which is known to regulate the dynamics of other tubulin polymers like microtubules, we find that the presence of the ring allows for an additional mechanism for regulating the dynamics of FtsZ polymers. Ring breaking dynamics in the presence of hydrolysis naturally induce rescue and catastrophe events in this model irrespective of the mechanism of hydrolysis.

A Minimal Stochastic Model of Transcriptional and Splicing Regulation.

In the past decades an extensive mathematical literature was developed to model and analyze gene networks under both deterministic and stochastic formalisms. However, such literature is predominantly focused to deal with the modeling of transcriptional and translational regulation, but results related to post-transcriptional regulation and its connection with transcriptional regulation are poorly investigated. However, it is becoming of paramount importance the need for modeling post-transcriptional regulation via splicing especially for minor organisms or viruses.The aim of this study is to propose a first general basic modeling scheme for modeling gene expression via alternative splicing and investigating the basic deterministic and stochastic features of the pre-mRNA, mRNAs and proteins under different biological conditions.This first study showed the dynamical properties of alternative splicing, the faster kinetics of the pre-mRNA compared to the mRNA and the importance to stochastically model gene networks when considering the post-transcriptional regulation.

Single rod-shaped cell fluctuations from stochastic surface and volume growth rates.

Growing rod-shaped bacterial cells need to modulate the production rates of different surface and bulk components. Population data show that the balance between these rates is central for cell physiology and affects cell shape, but we still know little about these processes in single cells. We study a minimal stochastic model where single cells grow by two fluctuating volume-specific surface and volume growth rates, solving for the steady-state distributions and the correlation functions of the main geometric features. Our predictions allow us to address the detectability of different scenarios for the intrinsic coupling between the allocation of resources to surface and bulk growth.

First passage times and minimum actions for a stochastic minimal bistable system

Bistability is an ubiquitous phenomenon in biological systems, and always plays important roles in cell division, differentiation, cancer onset, apoptosis and so on. However, stochastic fluctuations in bistable systems are still hard to understand. To address this issue, we propose a chemical master equation model for a minimal bistable system, which underlies generally bistable systems. For this master equation model, we mainly focus on the mean first passage times (MFPTs) by respectively using Gillespie algorithm and an approximation method of the large deviation theory, and does on minimum actions along optimal transition paths from OFF to ON states by the large deviation theory. Further, we find that for this stochastic system the MFPTs have different change tendencies compared to the corresponding minimum actions. Our results of this minimal stochastic model can also well understand more general bistable systems.

Globulelike Conformation and Enhanced Diffusion of Active Polymers.

We study the dynamics and conformation of polymers composed by active monomers. By means of Brownian dynamics simulations we show that, when the direction of the self-propulsion of each monomer is aligned with the backbone, the polymer undergoes a coil-to-globulelike transition, highlighted by a marked change of the scaling exponent of the gyration radius. Concurrently, the diffusion coefficient of the center of mass of the polymer becomes essentially independent of the polymer size for sufficiently long polymers or large magnitudes of the self-propulsion. These effects are reduced when the self-propulsion of the monomers is not bound to be tangent to the backbone of the polymer. Our results, rationalized by a minimal stochastic model, open new routes for activity-controlled polymers and, possibly, for a new generation of polymer-based drug carriers.

Stochastic sequestration dynamics: a minimal model with extrinsic noise for bimodal distributions and competitors correlation

Many biological processes are known to be based on molecular sequestration. This kind of dynamics involves two types of molecular species, namely targets and sequestrants, that bind to form a complex. In the simple framework of mass-action law, key features of these systems appear to be threshold-like profiles of the amounts of free molecules as a function of the parameters determining their possible maximum abundance. However, biochemical processes are probabilistic and take place in stochastically fluctuating environments. How these different sources of noise affect the final outcome of the network is not completely characterised yet. In this paper we specifically investigate the effects induced by a source of extrinsic noise onto a minimal stochastic model of molecular sequestration. We analytically show how bimodal distributions of the targets can appear and characterise them as a result of noise filtering mediated by the threshold response. We then address the correlations between target species induced by the sequestrant and discuss how extrinsic noise can turn the negative correlation caused by competition into a positive one. Finally, we consider the more complex scenario of competitive inhibition for enzymatic kinetics and discuss the relevance of our findings with respect to applications.

Role of weakest links and system-size scaling in multiscale modeling of stochastic plasticity

Plastic deformation of crystalline and amorphous matter often involves intermittent local strain burst events. To understand the physical background of the phenomenon a minimal stochastic mesoscopic model was introduced, where microstructural details are represented by a fluctuating local yielding threshold. In the present paper, we propose a method for determining this yield stress distribution by lower scale discrete dislocation dynamics simulations and using a weakest link argument. The success of scale-linking is demonstrated on the stress-strain curves obtained by the resulting mesoscopic and the discrete dislocation models. As shown by various scaling relations they are statistically equivalent and behave identically in the thermodynamic limit. The proposed technique is expected to be applicable for different microstructures and amorphous materials, too.

The adaptation rate of a quantitative trait in an environmental gradient

The spatial range of a species habitat is generally determined by the ability of the species to cope with biotic and abiotic variables that vary in space. Therefore, the species range is itself an evolvable property. Indeed, environmental gradients permit a mode of evolution in which range expansion and adaptation go hand in hand. This process can contribute to rapid evolution of drug resistant bacteria and viruses, because drug concentrations in humans and livestock treated with antibiotics are far from uniform. Here, we use a minimal stochastic model of discrete, interacting organisms evolving in continuous space to study how the rate of adaptation of a quantitative trait depends on the steepness of the gradient and various population parameters. We discuss analytical results for the mean-field limit as well as extensive stochastic simulations. These simulations were performed using an exact, event-driven simulation scheme that can deal with continuous time-, density- and coordinate-dependent reaction rates and could be used for a wide variety of stochastic systems. The results reveal two qualitative regimes. If the gradient is shallow, the rate of adaptation is limited by dispersion and increases linearly with the gradient slope. If the gradient is steep, the adaptation rate is limited by mutation. In this regime, the mean-field result is highly misleading: it predicts that the adaptation rate continues to increase with the gradient slope, whereas stochastic simulations show that it in fact decreases with the square root of the slope. This discrepancy underscores the importance of discreteness and stochasticity even at high population densities; mean-field results, including those routinely used in quantitative genetics, should be interpreted with care.

Limited by sensing - A minimal stochastic model of the lag-phase during diauxic growth

Many microbes when grown on a mixture of two carbon sources utilise first and exclusively the preferred sugar, before switching to the less preferred carbon source. This results in two distinct exponential growth phases, often interrupted by a lag-phase of reduced growth termed the lag-phase. While the lag-phase appears to be an evolved feature, it is not clear what drives its evolution, as it comes with a substantial up-front fitness penalty due to lost growth. In this article a minimal mathematical model based on a master-equation approach is proposed. This model can explain many empirically observed phenomena. It suggests that the lag-phase can be understood as a manifestation of the trade-off between switching speed and switching efficiency. Moreover, the model predicts heterogeneity of the population during the lag-phase. Finally, it is shown that the switch from one carbon source to another one is a sensing problem and the lag-phase is a manifestation of known fundamental limitations of biological sensors.

Bacteria differently deploy type-IV pili on surfaces to adapt to nutrient availability

The structure of bacterial biofilms depends on environmental conditions, such as availability of nutrients, during biofilm formation. In turn, variations in biofilm structure in part reflect differences in bacterial motility during early biofilm formation. Pseudomonas aeruginosa deprived of nutrients remain dispersed on a surface, whereas cells supplemented with additional nutrients cluster and form microcolonies. At the single-cell scale, how bacteria modify their motility to favour distinct life cycle outcomes remains poorly understood. High-throughput algorithms were used to track thousands of P. aeruginosa moving using type-IV pili (TFP) on surfaces in varying nutrient conditions and hence identify four distinct motility types. A minimal stochastic model was used to reproduce the TFP-driven motility types. We report that P. aeruginosa cells differently deploy TFP to alter the distribution of motility types under different nutrient conditions. Bacteria preferentially crawl unidirectionally under nutrient-limited conditions, but preferentially stall under nutrient-supplemented conditions. Motility types correlate with subcellular localisation of FimX, a protein required for TFP assembly and implicated in environmental response. The subcellular distribution of FimX is asymmetric for unidirectional crawling, consistent with TFP assembled primarily at the leading pole, whereas for non-translational types FimX expression is symmetric or non-existent. These results are consistent with a minimal stochastic model that reproduces the motility types from the subcellular average concentration and asymmetry of FimX. These findings reveal that P. aeruginosa deploy TFP symmetrically or asymmetrically to modulate motility behaviours in different nutrient conditions and thereby form biofilms only where nutrients are sufficient, which greatly enhances their competitive capacity in diverse environments.