New Journal of Physics

Under both physiological (development, regeneration) and pathological conditions (cancer metastasis), cells migrate while sensing environmental cues in the form of mechanical, chemical or electrical stimuli. Although it is known that osteoblasts respond to exogenous electric fields, the underlying mechanism of electrotactic collective movement of human osteoblasts is unclear. Theoretical approaches to study electrotactic cell migration until now mainly used reaction-diffusion models, and did not consider the effect of electric field on single-cell motility, or incorporate spatially dependent cell-to-cell interactions. Here, we present a computational model that takes into account cell interactions and describes cell migration in direct current electric field. We compare this model with in vitro experiments in which human primary osteoblasts are exposed to direct current electric field of varying field strength. Our results show that cell-cell interactions and fluctuations in the migration direction together lead to anode-directed collective migration of osteoblasts. Author summaryElectrotactic migration of cells involves directed movement of a large number of single cells under the influence of external electric field. Influencing the migration behaviour of osteoblasts by external direct current electric field offers a promising approach towards building highly effective implants for bone regeneration. We present a computational model for electrotactic migration of osteoblasts subject to external direct current electric field. Our model considers individual cells that interact with each other and the external electric field, and, replicates the experimental observations, based on single-cell analysis, of the response of osteoblasts to electrical stimulation of varying strengths for 7 hours. Our results suggest that tracking trajectories of individual cells provide a way of determining the role of various interactions of a cell in collective migration. Our model provides a framework that links single cell response to the large scale collective dynamics.

Embryonic development in multicellular organisms exhibits diverse morphogenetic patterns, which can generally be categorized into fundamental types such as monolayer and multilayer spheres, as well as cell masses. Furthermore, we identify two distinct processes for the formation of spherical structures. These basic patterns are thought to be governed by the microscopic properties of intercellular adhesion. However, the specific mechanisms linking the microscopic factors to the emergence of distinct macroscopic morphogenetic patterns remain poorly understood. In this study, we explore how different morphogenetic patterns arise by employing a computational model that incorporates intercellular adhesion and polarity. Our results demonstrate that all fundamental morphogenetic patterns can be generated through the interplay of two key parameters: the strength of cell polarity and the regulation of polarity via mechanical signals. Furthermore, analytical discussions reveal partial mechanisms underlying the formation of these patterns. These findings highlight the critical role of physical constraints in morphogenesis and suggest potential applications in the design principles for artificial tissues and organoids. Author summaryLiving organisms build their bodies through morphogenesis, during which cells autonomously arrange themselves into functional structures such as sheets, tubes, and spheres. From simple monolayered spheres to complex multilayered tissues organized by adhesion, it remains unclear how such diverse forms arise. Here, we mathematically modeled a population of proliferating cells governed only by two microscopic factors: the strength of polarity-dependent adhesion and the time scale at which polarity is regulated by cell-cell contact. Surprisingly, we found that this minimal model reproduces five basic morphological types observed in living embryos, including monolayer/multilayer structures and two distinct modes of cavity formation: by wrapping around or by inflating from the inside. Systematic simulations revealed that these macroscale outcomes are determined solely by two parameters controlling polarity strength and its regulation, suggesting that simple physical rules underlie diverse developmental architectures. Analysis of the model uncovers phase transitions between the five morphogenetic types and reveals how varying polarity and adhesion can recapitulate features of real embryogenesis. Our work proposes a unified framework that connects microscopic polarity mechanics to diverse developmental morphologies and provides a foundation for future applications in organoid design and tissue engineering.

Variation in the chromosome numbers can arise from the erroneous mitosis or fusion and fission of chromosomes. While the mitotic errors lead to an increase or decrease in the overall chromosomal substance in the daughter cells, fission and fusion keep this conserved. Variations in chromosome numbers are assumed to be a crucial driver of speciation. For example, the members of the muntjac species are known to have very different karyotypes with the chromosome numbers varying from 2n = 70 + 3B in the brown brocket deer to 2n = 46 in the Chinese muntjac and 2n = 6/7 in the Indian muntjac. The chromosomal content in the nucleus of these closely related mammals is roughly the same and various chromosome fusion and fission pathways have been suggested as the evolution process of these karyotypes. Similar trends can also be found in lepidoptera and yeast species which show a wide variation of chromosome numbers. The effect of chromosome number variation on the spindle assembly time and accuracy is still not properly addressed. We computationally investigate the effect of conservation of the total chromosomal substance on the spindle assembly during prometaphase. Our results suggest that chromosomal fusion pathways aid the microtubule-driven Search and Capture of the kinetochore in cells with monocentric chromosomes. We further report a comparative analysis of the site and percentage of amphitelic captures, dependence on cell shape, position of the kinetochore in respect of chromosomal volume partitioning.

Phenotypic decision-making is a process of determining important phenotypes in accordance with the available microenvironmental information. Although phenotypic decision at the level of a single cell has been precisely studied, but the knowledge is still imperceptible at the multicellular level. How cells sense their environment and adapt? How single cells change their phenotype in a multicellular complex environment (without knowing the interactions among the cells), is still a rheotorical question. To unravel the fragmental story of multicellular decision-making, Least microEnvironmental Uncertainty Principle (LEUP) was refined and applied in this context. To address this set of questions, we use variational principle to grasp the role of sensitivity, build a LEUP driven agent-based model on a lattice which solely hinges on microenvironmental information and investigate the parallels in a well-known biological system, viz., Notch-Delta-Jagged signaling pathway. The analyses of this model led us to interesting spatiotemporal patterns in a population of cells, responsive to the sensitivity parameter and the radius of interaction. This resembles the tissue-level pattern of a population of cells interacting via Notch-Delta-Jagged signaling pathway in some parameter regimes.

Rotating excitation waves and electrical turbulence in excitable cardiac tissue are associated with arrhythmias such as life-threatening ventricular fibrillation. Experimental studies (S. Luther et al., Nature 475, 235-239 (2011)). have shown that a time-periodic sequence of low-energy electrical far-field pulses is able to terminate fibrillation more gently than a single high-energy pulse. During this so called low-energy antifibrillation pacing (LEAP), only tissue near sufficiently large conduction heterogeneities, such as large coronary arteries, is activated. Based on extensive simulations and simple theoretical reasoning, we present a comprehensive unified mechanism for successful LEAP in two spatial dimensional systems, which is able to explain both the termination of stable spirals and of spatiotemporal chaos. We carried out extensive simulations (more than 500000 runs for each considered model) varying pacing periods, pacing field strength and initial conditions using a model of cardiac tissue perforated by blood vessels, which was found earlier to reproduce the behavior seen in the LEAP experiments for different dynamical regimes and different cellular models (P. Buran et al., Chaos 27, 113110 (2017) and New J. Phys. 24 083024 (2022)). We studied altogether three different cellular models to capture qualitatively different kinds of fibrillatory states like stable spirals and spatiotemporal chaos. To achieve a mechanistic understanding of the simulation results, we have investigated a variety of macroscopic observables characterizing an excitable medium with respect to their correlation with the success of an individual low-energy pulse during LEAP. We found in all considered cases that the refractory boundary length LRB, the total length of the borders between refractory and excitable parts of the tissue, displays the strongest correlation with the success of the pacing and thus predicts best the success of an individual LEAP pulse. Furthermore, we found the success probability PL decays exponentially with this length according to PL = exp(- k(E)LRB), where E is the strength of the electrical field in pacing and k(E) is a monotonically decreasing function of E. A closer look at the spatiotemporal dynamics in the simulations reveals that actually each pulse during LEAP annihilates practically all defects and excitation fronts, however, also induces new pairs of defects and associated excitation fronts at the refractory boundaries. The success probability of each individual pulse can thus be simply interpreted as the probability that no new rotor pair gets created by the shock, while all existing defects get annihilated. This assumption allows to derive the observed exponential dependence of the success probability on the refractory boundary length, where the prefactor k(E) in the exponent is equal (for stable spirals) or proportional (for spatiotemporal chaos) to the probability{lambda} (E) that a new rotor pairs is created by the shock along a segment of unit length along the refractory boundary. Our findings are in conformity with the upper limit of vulnerability (ULV) hypothesis, which states that the single pulse defibrillation threshold is simply given by the lowest field strength, where no new rotor pairs arise as a result of the shock. LEAP operates at field strengths (and energies) below this ULV limit. Successful LEAP protocols are characterized by a coordinated interplay between the pulses, that gradually decreases the refractory boundary length and therefore simultaneously increases the success probability until complete defibrillation is achieved.

Population dynamics of different species across food web models of diverse spatial scales have been extensively investigated over the past three decades. Chaotic fluctuations in species populations are generally associated with increased extinction risk due to frequent periods of low species density leading to cascading effects. The existence of a food web where the cohabiting species population oscillates chaotically has been largely attributed to chaos control or death of chaos, mediated either by internal mechanisms, such as the food web favoring parameter values that keep the species population out of the chaotic region, or external mechanisms like environmental forcing. Yet, this picture is not complete, as food webs do not exist in isolation, but are generally connected to each other through dispersal of species, forming a metacommunity. A metacommunity of food webs whose dynamics is that of chaotic attractors can only persist through control and particularly the the quenching of chaos. We claim that habitat heterogeneity in such a metacommunity fulfills this purpose. We address this question by analyzing the dynamics of a drive-response metacommunity composed of five chaotic food webs, each located in a distinct patch. Each patch contains an inherently chaotic tritrophic food web, with habitat heterogeneity present among patches. The first patch functions as the drive, exerting external influence on the dynamics of the remaining four response patches. Our results establish that in a drive-response metacommunity, strong influence from the drive quenches dynamical chaos in both drive and response patches, often leading to steady states. This phenomenon is observed in two distinct metacommunity network structures. The heterogeneity of the response systems (food web models) and the dissimilarity between drive and response systems are found to play significant roles in suppressing chaotic population dynamics. These findings strongly imply that the persistence of such inherently chaotic metacommunities may result from the quenching of chaos through the interplay of chaos, habitat heterogeneity, and to some degree network structure shaped by dispersal. Furthermore, we illustrate that these metacommunities are also susceptible to extinction due to dispersal-induced synchronization. Accordingly, this study also investigates dispersal-induced complete synchronization among the constituent patches.

Marine infectious diseases are more prevalent in recent times due to climate change and other anthropogenic pressures, posing a substantial threat to marine ecosystems and the conservation of their biodiversity. An important subset of marine organisms are sessile, for which the most common mechanism for disease transmission is direct contact with waterborne parasites. Only recently, some deterministic compartmental models have been proposed to describe this kind of epidemics, being these models based on non-spatial descriptions where space is homogenised and parasite mobility is not explicitly accounted for. However, in realistic situations, epidemic transmission is conditioned by the spatial distribution of hosts and the parasites mobility patterns. Thus, the interplay between these factors is expected to have a crucial effect in the evolution of the epidemic, so calling for a explicit description of space. In this work we develop a spatially-explicit individual-based model to study disease transmission by waterborne parasites in sessile marine populations. We investigate the impact of spatial disease transmission, performing extensive numerical simulations and analytical approximations. Specifically, the effects of parasite mobility into the epidemic threshold and the temporal evolution of the epidemic are assessed. We show that larger values of pathogen mobility have two main implications: more severe epidemics, as the number of infections increases, and shorter time-scales to extinction. Moreover, an analytical expression for the basic reproduction number of the spatial model, [Formula], is derived as function of the non-spatial counterpart, R0, which characterises a transition between a disease-free and a propagation phase, in which the disease propagates over a large fraction of the system. This allows to determine a phase diagram for the epidemic model as function of the parasite mobility and the basic reproduction number of the non-spatial model.

Local stresses in a tissue, a collective property, regulate cell division and apoptosis. In turn, cell growth and division induce active stresses in the tissue. As a consequence, there is a feed-back between cell growth and local stresses. However, how the cell dynamics depend on local stress-dependent cell division and the feedback strength is not fully understood. Here, we probe the consequences of stress-mediated growth and cell division on cell dynamics using agent-based simulations of a two-dimensional growing tissue. We discover a rich dynamical behavior of individual cells, ranging from jamming (mean square displacement, {Delta}(t) [~] t with less than unity), to hyperdiffusion ( > 2) depending on cell division rate and the strength of the mechanical feedback. Strikingly, {Delta}(t) is determined by the tissue growth law, which quantifies cell proliferation (number of cells N (t) as a function of time). The growth law (N (t) [~] t{lambda} at long times) is regulated by the critical pressure that controls the strength of the mechanical feedback and the ratio between cell division-apoptosis rates. We show that{lambda} [~] , which implies that higher growth rate leads to a greater degree of cell migration. The variations in cell motility are linked to the emergence of highly persistent forces extending over several cell cycle times. Our predictions are testable using cell-tracking imaging techniques.

A variety of living and non-living systems exhibit collective motion. From swarm robotics to bacterial swarms, and tissue wound healing to human crowds, examples of collective motion are highly diverse but all of them share the common necessary ingredient of moving and interacting agents. While collective motion has been extensively studied in non-proliferating systems, how the proliferation of constituent agents affects their collective behavior is not well understood. Here, we focus on growing active agents as a model for cells and study how the interplay between noise in their direction of movement and proliferation determines the overall spatial pattern of collective motion. In this agent-based model, motile cells possess the ability to adhere to each other through cell-cell adhesion, grow in size and divide. Cell-cell interactions influence not only the direction of cell movement but also cell growth through a force-dependent mechanical feedback process. We show that noise in the direction of a cells motion has striking effects on the emergent spatial distribution of cell collectives and proliferation. While higher noise strength leads to a random spatial distribution of cells, we also observe increased cell proliferation. On the other hand, low noise strength leads to a ring-like spatial distribution of cell collectives together with lower proliferation. Our findings provide insight into how noise in the direction of cell motion determines the local spatial organization of cells with consequent mechanical feedback on cell division impacting cell proliferation due to the formation of cell clusters.

Arrangement of variant phenotypes in ordered spatial assemblies during division of stem cells is essential for the self-organization of cell tissues. The cellular patterns of phenotypes competing for space and resources against one another are mostly driven by secreted diffusible chemical signaling clues. This complex process is carried out within a chronological framework of interplaying intracellular and intercellular events. This includes receiving external stimulants-whether secreted by other individuals or provided by the environment-interpreting these environmental signals and incorporating the information to designate cell fate. An enhanced understanding of the building blocks of this framework would be of help to set the scene for promising regenerative therapies. In this study, by proposing a designative computational map, we show that there is a correspondence between signaling and developmental patterns that are produced by competing cells. That is, the model provides an appropriate prediction for the final structure of the differentiated cells in a competitive environment. Besides, given that the final state of the cellular organization is known, the corresponding regressive signaling patterns are partly predictable following the proposed map. Author SummaryMulticellular organisms are made of repeated divisions of single cells and aggregation of their offspring together. However, the aggregated formations are not colony-like accumulations of piled-up cells. Instead, they are "emergent" spatiotemporal structures of developmentally differentiated cell types. The functionally integrated structures remain relatively constant throughout the life of the organisms, despite the death and production of new cells. The question is: How differentiated cells are capable of making variant patterns without any predefined templates? It is shown that with a variety of differentiated cell types, emergence of complex patterns is feasible through the interplay of intercellular interactions and intracellular decision-making switches. Such conceptual understanding has the potential to generate a multitude of novel and precisely controlled cellular behaviors.

Cell motility represents one of the most fundamental function in mechanobiology. Cell motility is directly implicated in development, cancer or tissue regeneration, but it also plays a key role in the future of tissue and biomedical engineering. Here, we derived a computational model of cell motility that incorporates the most important mechanisms toward cell motility: cell protrusion, polarization and retrograde flow. We first validate our model to explain two important types of cell migration, i.e. confined and ameboid cell migration, as well as all phases of the latter cell migration type, i.e. symmetric cell spreading, cell polarization and latter migration. Then, we use our model to investigate durotaxis and chemotaxis. The model predicts that chemotaxis alone induces larger migration velocities than durotaxis and that durotaxis is activated in soft matrices but not in stiff ones. More importantly, we analyze the competition between chemical and mechanical signals. We show that chemotaxis rules over durotaxis in most situations although durotaxis diminishes chemotaxis. Moreover, we show that inhibiting the effect of GTPases in actin polymerization at the cell front may allow durotaxis to take control over chemotaxis in soft substrates. Understanding how the main forces in cell motility cooperate, and how a precise manipulation of external cues may control directed cell migration is not only key for a fundamental comprehension of cell biology but also to engineer better biomimetic tissues. To this end, we provide a freely-available platform to predict all phases and modes of cell motility analyzed in this work.

AO_SCPLOWBSTRACTC_SCPLOWCell migration is of major importance for the understanding of phenomena such as morphogenesis, cancer metastasis, or wound healing. In many of these situations cells are under external confinement. In this work we show by means of computer simulations with a Cellular Potts Model (CPM) that the presence of a bottleneck in an otherwise straight channel has a major influence on the internal organisation of an invading cellular monolayer and the motion of individual cells therein. Comparable to a glass or viscoelastic material, the cell sheet is found to exhibit features of both classical solids and classical fluids. The local ordering on average corresponds to a regular hexagonal lattice, while the relative motion of cells is unbounded. Compared to an unconstricted channel, we observe that a bottleneck perturbs the formation of regular hexagonal arrangements in the epithelial sheet and leads to pile-ups and backflow of cells near the entrance to the constriction, which also affects the overall invasion speed. The scale of these various phenomena depends on the dimensions of the different channel parts, as well as the shape of the funnel domain that connects wider to narrower regions.

Aging is a complex process involving multiple factors and subcellular processes, ultimately leading to the death of an organism. The microscopic processes that cause aging are relatively well understood and effective macroscopic theories help explain the universality of aging in complex systems. However, these theories fail to explain the diversity of aging observed for various lifeforms. As such, more complete "mesoscopic" theories of aging are needed, combining the biophysical details of microscopic failure and the macroscopic structure of complex systems. Here we explore two models: (1) a network theoretic model, and (2) a convection diffusion model emphasizing the biophysical details of communicated signals. The first model allows us to explore the effects of connectivity structures on aging. In our second model, cells interact through cooperative and antagonistic factors. We find by varying the ratio at which these factors affect cell death, as well as the reaction kinetics, diffusive and flow parameters, we obtain a wide diversity of mortality curves. As such, the connectivity structures as well as the biophysical details of how various factors are transported in an organism may explain the diversity of aging observed across different lifeforms.

The tumor stroma is a tissue composed primarily of extracellular matrix, fibroblasts, immune cells, and vasculature. Its structure and functions, such as nutrient support and waste removal, are altered during malignancy. Tumor cells transform fibroblasts into cancer-associated fibroblasts, which have important immunosuppressive activity on which growth, invasion, and metastasis depend. These activated fibroblasts prevent immune cell infiltration into the tumor nest, thereby promoting cancer progression and inhibiting T-cell-based immunotherapy. To understand these complex interactions, we measure the density of different cell types in the stroma using immunohistochemistry techniques on tumor samples from lung cancer patients. We incorporate these data, and also known information on cell proliferation rates and relevant biochemical interactions, into a minimal dynamical system with few parameters. A spatio-temporal approach to the inhomogeneous environment explains the cell distribution and fate of lung carcinomas. The model reproduces that cancer-associated fibroblasts act as a barrier to tumor growth, but also reduce the efficiency of the immune response. The final outcome depends on the parameter values for each patient and leads to either tumor invasion, persistence, or eradication as a result of the interplay between cancer cell growth, T-cell cytotoxic activity, and fibroblast attraction, activation, and spatial dynamics. Our conclusion is that a wide spectrum of scenarios exists as a result of the competition between the characteristic times of cancer cell growth and the activity rates of the other species. Nevertheless, distinct trajectories and patterns allow quantitative predictions that may help in the selection of new therapies and personalized protocols. We conclude with different options for further modeling. O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=118 SRC="FIGDIR/small/575824v3_ufig1.gif" ALT="Figure 1"> View larger version (44K): org.highwire.dtl.DTLVardef@156ac4forg.highwire.dtl.DTLVardef@14a9a19org.highwire.dtl.DTLVardef@cafebeorg.highwire.dtl.DTLVardef@11aa3bf_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOGraphical AbstractC_FLOATNO C_FIG

Turing patterns are a fundamental concept in developmental biology, describing how homogeneous tissues develop into self-organized spatial patterns. However, the classical Turing mechanism, which relies on linear stability analysis, often fails to capture the complexities of real biological systems, such as multistability, non-linearities, growth, and boundary conditions. Here, we explore the impact of these factors on Turing pattern formation, contrasting linear stability analysis with numerical simulations based on a simple reaction-diffusion model, motivated by synthetic gene-regulatory pathways. We demonstrate how non-linearities introduce multistability, leading to unexpected pattern outcomes not predicted by the traditional Turing theory. The study also examines how growth and realistic boundary conditions influence pattern robustness, revealing that different growth regimes and boundary conditions can disrupt or stabilize pattern formation. Our findings are critical for understanding pattern formation in both natural and synthetic biological systems, providing insights into engineering robust patterns for applications in synthetic biology. Author summaryDuring development, tissues self-organize to go from a single cell to a structured organism. In this process, simple chemical reactions lead to the emergence of the intricate designs we see in nature, like the stripes on a zebra or the labyrinths on a brain cortex. Although multiple theories have been proposed to model this phenomenon, one of the most simple and popular ones was introduced in the 1950s by the mathematician Alan Turing. However, his theory oversimplifies the biological conditions and ignores properties such as non-linearities, boundary effects, or growth in the tissue. In this work, we used a combination of mathematical models and computer simulations to investigate how these real-world factors influence pattern formation. Our findings show that when we account for these realistic effects, the patterns that emerge can be very different from what Turings theory would predict. Thus, this work may help us better understand the laws behind pattern formation and could have practical applications in tissue engineering for medical or environmental applications.

We investigate a spatially explicit metapopulation model consisting of one predator and two hierarchically competing prey species on a discrete lattice. Each local population follows stochastic rules for extinction, colonisation, competition, and predation. From the master equation of this individual-based model, we rigorously derive the corresponding mean-field equations. The analysis of these first-principles mean-field equations reveals the existence of a rich phase diagram with different coexistence regimes depending on parameter values. We identify both stable and spiral nodes, in agreement with the damped oscillations observed in Monte Carlo simulations in the latter case. We find qualitative agreement between the mean-field results and Monte Carlo simulations for the three-species coexistence and for the coexistence of the two prey species. However, the mean-field description fails to reproduce the coexistence between the predator and the inferior prey at high predator coverages. We argue that this discrepancy arises from spatial prey aggregation, which the mean-field approach cannot capture since it neglects correlations. In the stochastic model, spatial clustering acts as a crucial protective mechanism against predation, particularly for the best coloniser. Our findings suggest that prey aggregation contributes to system stability when colonisation and predation operate at comparable spatial scales. The combination of first-principles mean-field equations and stochastic simulations constitutes a powerful framework for clarifying the roles of hierarchical interactions, predation, colonisation, spatial organisation and stochasticity in multi-species coexistence.

Motile cells inside living tissues often encounter junctions, where their path branches into several alternative directions of migration. We present a theoretical model of cellular polarization for cells migrating along one-dimensional lines, arriving at a symmetric Y-junction and extending protrusions along the different paths that emanate from the junction. The model predicts the spontaneous emergence of deterministic oscillations between competing protrusions, whereby the cellular polarization and growth alternates between the competing protrusions. The oscillations are modified by cellular noise, but remain as a dominant feature which affects the time it takes the cell to migrate across the junction. These predicted oscillations in the cellular polarization during the directional decision making process at the junction are found experimentally for two different cell types, non-cancerous endothelial and cancerous glioma cells, migrating on patterned network of thin adhesive lanes with junctions.

Self-organized patterns in the actin cytoskeleton are essential for eukaryotic cellular life. They are the building blocks of many functional structures that often operate simultaneously to facilitate, for example, nutrient uptake and movement of cells. However, to identify how qualitatively distinct actin patterns can coexist remains a challenge. Here, we use bifurcation theory to reveal a generic mechanism of pattern coexistence, showing that different types of wave patterns can simultaneously emerge in the actin system. Our theoretical analysis is complemented by live-cell imaging experiments revealing that narrow, planar, and fast-moving excitable pulses may indeed coexist with ring-shaped macropinocytic actin waves in the cortex of motile amoeboid cells.

Temporal environmental variations may promote diversity in communities of competing populations. Here we compare the effect of environmental stochasticity with the effect of periodic (e.g., seasonal) cycles, using analytic solutions and individual-based Monte-Carlo simulations. Even when stochasticity facilitates coexistence it still allows for rare sequences of bad years that may drive a population to extinction, therefore the stabilizing effect of periodic variations is stronger. Correspondingly, the mean time to extinction grows exponentially with community size in periodic environment and switch to power-law dependence under stochastic fluctuations. On the other hand, the number of temporal niches in periodic environment is typically lower, so as diversity increases stochastic temporal variations may support higher species richness.

The thermodynamics feature of systems that are driven out of equilibrium is explored for M Brownian ratchets that are arranged in a complex network. The exact time-dependent solution depicts that as the network size increases, the entropy S, entropy production ep(t), and entropy extraction hd(t) of the system step up which is feasible since these thermodynamic quantities exhibit an extensive property. In other words, as the number of lattice size increases, the entropy S, entropy production ep(t), and entropy extraction hd(t) step up revealing that these complex networks can not be reduced into the corresponding one-dimensional lattice. On the contrary, the rate for thermodynamic relations such as the velocity V, entropy production rate[e] p(t) and entropy extraction rate [Formula] become independent of the network size in the long time limit. The exact analytic result also shows that the free energy decreases with the system size. The model system is further analyzed by including heat transfer via kinetic energy. Since the heat exchange via kinetic energy does not affect the energy extraction rate, the heat dumped to the cold reservoirs contributes only to the internal entropy production. As the result, such systems exhibit a higher degree of irreversibility. The thermodynamic features of a system that operates between hot and cold baths are also compared and contrasted with a system that operates in a heat bath where its temperature varies linearly along the reaction coordinate. Regardless of the network arrangements, the entropy, entropy production, and extraction rates are considerably larger for the linearly varying temperature case than a system that operates between hot and cold baths. PACS numbersValid PACS appear here