Evolutionary dynamics within and among competing groups.

Biological and social systems are structured at multiple scales, and the incentives of individuals who interact in a group may diverge from the collective incentive of the group as a whole. Mechanisms to resolve this tension are responsible for profound transitions in evolutionary history, including the origin of cellular life, multicellular life, and even societies. Here, we synthesize a growing literature that extends evolutionary game theory to describe multilevel evolutionary dynamics, using nested birth-death processes and partial differential equations to model natural selection acting on competition within and among groups of individuals. We analyze how mechanisms known to promote cooperation within a single group-including assortment, reciprocity, and population structure-alter evolutionary outcomes in the presence of competition among groups. We find that population structures most conducive to cooperation in multiscale systems can differ from those most conducive within a single group. Likewise, for competitive interactions with a continuous range of strategies we find that among-group selection may fail to produce socially optimal outcomes, but it can nonetheless produce second-best solutions that balance individual incentives to defect with the collective incentives for cooperation. We conclude by describing the broad applicability of multiscale evolutionary models to problems ranging from the production of diffusible metabolites in microbes to the management of common-pool resources in human societies.

Front propagation in the shadow wave-pinning model.

In this paper we consider a non-local bistable reaction-diffusion equation which is a simplified version of the wave-pinning model of cell polarization. In the small diffusion limit, a typical solution u(x, t) of this model approaches one of the stable states of the bistable nonlinearity in different parts of the spatial domain [Formula: see text], separated by an interface moving at a normal velocity regulated by the integral [Formula: see text]. In what is often referred to as wave-pinning, feedback between mass-conservation and bistablity causes the interface to slow and approach a fixed limit. In the limit of a small diffusivity [Formula: see text], we prove that for any [Formula: see text] the interface can be estimated within [Formula: see text] of the location as predicted using formal asymptotics. We also discuss the sharpness of our result by comparing the formal asymptotic results with numerical simulations.

A reduced 1D stochastic model of bleb-driven cell migration.

Blebs are pressure-driven protrusions that have been observed in cells undergoing apoptosis, cytokinesis, or migration, including tumor cells that use blebs to escape their organs of origin. Here, we present a minimal 1D model of bleb-driven cell motion that combines a simple mechanical model with turnover kinetics of the actin cortex and adhesions between the membrane and the cortex. The deterministic version of this model is used to study the properties of individual blebbing events. We further introduce stochastic turnover of the adhesions, which allows for spontaneous initiation of repeated blebbing events, thus leading to sustained cell travel. We explore how the main parameters of the system control the properties of the blebbing events and the speed of cell travel. Finally, we derive a further simplification by deriving a Langevin approximation to this stochastic model.

The importance of water and hydraulic pressure in cell dynamics.

All mammalian cells live in the aqueous medium, yet for many cell biologists, water is a passive arena in which proteins are the leading players that carry out essential biological functions. Recent studies, as well as decades of previous work, have accumulated evidence to show that this is not the complete picture. Active fluxes of water and solutes of water can play essential roles during cell shape changes, cell motility and tissue function, and can generate significant mechanical forces. Moreover, the extracellular resistance to water flow, known as the hydraulic resistance, and external hydraulic pressures are important mechanical modulators of cell polarization and motility. For the cell to maintain a consistent chemical environment in the cytoplasm, there must exist an intricate molecular system that actively controls the cell water content as well as the cytoplasmic ionic content. This system is difficult to study and poorly understood, but ramifications of which may impact all aspects of cell biology from growth to metabolism to development. In this Review, we describe how mammalian cells maintain the cytoplasmic water content and how water flows across the cell surface to drive cell movement. The roles of mechanical forces and hydraulic pressure during water movement are explored.

Long-time behavior of a PDE replicator equation for multilevel selection in group-structured populations.

In many biological systems, natural selection acts simultaneously on multiple levels of organization. This scenario typically presents an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within a group. Generalizing previous work on multilevel selection in evolutionary game theory, we consider a hyperbolic PDE model of a group-structured population, in which members within a single group compete with each other for individual-level replication; while the group also competes against other groups for group-level replication. We derive a threshold level of the relative strength of between-group competition such that defectors take over the population below the threshold while cooperation persists in the long-time population above the threshold. Under stronger assumptions on the initial distribution of group compositions, we further prove that the population converges to a steady state density supporting cooperation for between-group selection strength above the threshold. We further establish long-time bounds on the time-average of the collective payoff of the population, showing that the long-run population cannot outperform the payoff of a full-cooperator group even in the limit of infinitely-strong between-group competition. When the group replication rate is maximized by an intermediate level of within-group cooperation, individual-level selection casts a long shadow on the dynamics of multilevel selection: no level of between-group competition can erase the effects of the individual incentive to defect. We further extend our model to study the case of multiple types of groups, showing how the games that groups play can coevolve with the level of cooperation.

On the energy efficiency of cell migration in diverse physical environments.

In this work, we explore fundamental energy requirements during mammalian cell movement. Starting with the conservation of mass and momentum for the cell cytosol and the actin-network phase, we develop useful identities that compute dissipated energies during extensions of the cell boundary. We analyze 2 complementary mechanisms of cell movement: actin-driven and water-driven. The former mechanism occurs on 2-dimensional cell-culture substrate without appreciable external hydraulic resistance, while the latter mechanism is prominent in confined channels where external hydraulic resistance is high. By considering various forms of energy input and dissipation, we find that the water-driven cell-migration mechanism is inefficient and requires more energy. However, in environments with sufficiently high hydraulic resistance, the efficiency of actin-polymerization-driven cell migration decreases considerably, and the water-based mechanism becomes more efficient. Hence, the most efficient way for cells to move depends on the physical environment. This work can be extended to higher dimensions and has implication for understanding energetics of morphogenesis in early embryonic development and cancer-cell metastasis and provides a physical basis for understanding changing metabolic requirements for cell movement in different conditions.

Strong intracellular signal inactivation produces sharper and more robust signaling from cell membrane to nucleus.

For a chemical signal to propagate across a cell, it must navigate a tortuous environment involving a variety of organelle barriers. In this work we study mathematical models for a basic chemical signal, the arrival times at the nuclear membrane of proteins that are activated at the cell membrane and diffuse throughout the cytosol. Organelle surfaces within human B cells are reconstructed from soft X-ray tomographic images, and modeled as reflecting barriers to the molecules' diffusion. We show that signal inactivation sharpens signals, reducing variability in the arrival time at the nuclear membrane. Inactivation can also compensate for an observed slowdown in signal propagation induced by the presence of organelle barriers, leading to arrival times at the nuclear membrane that are comparable to models in which the cytosol is treated as an open, empty region. In the limit of strong signal inactivation this is achieved by filtering out molecules that traverse non-geodesic paths.

A computational study on the role of glutamate and NMDA receptors on cortical spreading depression using a multidomain electrodiffusion model.

Cortical spreading depression (SD) is a spreading disruption of ionic homeostasis in the brain during which neurons experience complete and prolonged depolarizations. SD is the basis of migraine aura and is increasingly associated with many other brain pathologies. Here, we study the role of glutamate and NMDA receptor dynamics in the context of an ionic electrodiffusion model. We perform simulations in one (1D) and two (2D) spatial dimension. Our 1D simulations reproduce the "inverted saddle" shape of the extracellular voltage signal for the first time. Our simulations suggest that SD propagation depends on two overlapping mechanisms; one dependent on extracellular glutamate diffusion and NMDA receptors and the other dependent on extracellular potassium diffusion and persistent sodium channel conductance. In 2D simulations, we study the dynamics of spiral waves. We study the properties of the spiral waves in relation to the planar 1D wave, and also compute the energy expenditure associated with the recurrent SD spirals.

The dual effect of ephaptic coupling on cardiac conduction with heterogeneous expression of connexin 43.

Decreased and heterogeneous expression of connexin 43 (Cx43) are common features in animal heart failure models. Ephpatic coupling, which relies on the presence of junctional cleft space between the ends of adjacent cells, has been suggested to play a more active role in mediating intercellular electrical communication when gap junctions are reduced. To better understand the interplay of Cx43 expression and ephaptic coupling on cardiac conduction during heart failure, we performed numerical simulations on our model when Cx43 expression is reduced and heterogeneous. Under severely reduced Cx43 expression, we identified three new phenomena in the presence of ephaptic coupling: alternating conduction, in which ephaptic and gap junction-mediated mechanisms alternate; instability of planar fronts; and small amplitude action potential (SAP), which has a smaller potential amplitude than the normal action potential. In the presence of heterogeneous Cx43 expression, ephaptic coupling can either prevent or promote conduction block (CB) depending on the Cx43 knockout (Cx43KO) content. When Cx43KO content is relatively high, ephaptic coupling reduces the probabilities of CB. However, ephaptic coupling promotes CB when Cx43KO and wild type cells are mixed in roughly equal proportion, which can be attributed to an increase in current-to-load mismatch.

Effects of Glia in a Triphasic Continuum Model of Cortical Spreading Depression.

Cortical spreading depression (SD) is a spreading disruption in brain ionic homeostasis during which neurons experience complete and prolonged depolarizations. SD is generally believed to be the physiological substrate of migraine aura and is associated with many other brain pathologies. Here, we perform simulations with a model of SD treating brain tissue as a triphasic continuum of neurons, glia and the extracellular space. A thermodynamically consistent incorporation of the major biophysical effects, including ionic electrodiffusion and osmotic water flow, allows for the computation of important physiological variables including the extracellular voltage (DC) shift. A systematic parameter study reveals that glia can act as both a disperser and buffer of potassium in SD propagation. Furthermore, we show that the timing of the DC shift with respect to extracellular [Formula: see text] rise is highly dependent on glial parameters, a result with implications for the identification of the propagating mechanism of SD.

Flow-Driven Cell Migration under External Electric Fields.

Electric fields influence many aspects of cell physiology, including various forms of cell migration. Many cells are sensitive to electric fields, and they can migrate toward a cathode or an anode, depending on the cell type. In this Letter, we examine an actomyosin-independent mode of cell migration under electrical fields. Our theory considers a one-dimensional cell with water and ionic fluxes at the cell boundary. Water fluxes through the membrane are governed by the osmotic pressure difference across the cell membrane. Fluxes of cations and anions across the cell membrane are determined by the properties of the ion channels as well as the external electric field. Results show that without actin polymerization and myosin contraction, electric fields can also drive cell migration, even when the cell is not polarized. The direction of migration with respect to the electric field direction is influenced by the properties of ion channels, and are cell-type dependent.

The role of short term memory and conduction velocity restitution in alternans formation.

Alternans is the periodic beat-to-beat short-long alternation in action potential duration (APD), which is considered to be a precursor of ventricular arrhythmias and sudden cardiac death. In extended cardiac tissue, electrical alternans can be either spatially concordant (SCA, all cells oscillate in phase) or spatially discordant (SDA, cells in different regions oscillate out of phase). SDA gives rise to an increase in the spatial dispersion of repolarization, which is thought to be proarrhythmic. In this paper, we investigated the effect of two aspects of short term memory (STM) (α, τ) and their interplay with conduction velocity (CV) restitution on alternans formation using numerical simulations of a mapping model with two beats of memory. Here, α quantifies the dependence of APD restitution on pacing history and τ characterizes APD accommodation, which is an exponential change of APD over time once basic cycle length (BCL) changes. Our main findings are as follows: In both single cell and spatially coupled homogeneous cable, the interplay between α and τ affects the dynamical behaviors of the system. For the case of large APD accommodation (τ ≥ 290 ms), increase in α leads to suppression of alternans. However, if APD accommodation is small (τ ≤ 250 ms), increase in α leads to appearance of additional alternans region. On the other hand, the slope of CV restitution does not change the regions of alternans in the cable. However, steep CV restitution leads to more complicated dynamical behaviors of the system. Specifically, SDA instead of SCA are observed. In addition, for steep CV restitution and sufficiently large τ, we observed formations of type II conduction block (CB2), transition from type I conduction block (CB1) to CB2, and unstable nodes.

Heart rate variability and alternans formation in the heart: The role of feedback in cardiac dynamics.

A beat-to-beat alternation in the action potential duration (APD) of myocytes, i.e. alternans, is believed to be a direct precursor of ventricular fibrillation in the whole heart. A common approach for the prediction of alternans is to construct the restitution curve, which is the nonlinear functional relationship between the APD and the preceding diastolic interval (DI). It was proposed that alternans appears when the magnitude of the slope of the restitution curve exceeds one, known as the restitution hypothesis. However, this restitution hypothesis was derived under the assumption of periodic stimulation, when there is a dependence of the DI on the immediate preceding APD (i.e. feedback). However, under physiological conditions, the heart rate exhibits substantial variations in time, known as heart rate variability (HRV), which introduces deviations from periodic stimulation in the system. In this manuscript, we investigated the role of HRV on alternans formation in isolated cardiac myocytes using numerical simulations of an ionic model of the cardiac action potential. We used this model with two different pacing protocols: a periodic pacing protocol with feedback and a protocol without feedback. We show that when HRV is incorporated in the periodic pacing protocol, it facilitated alternans formation in the isolated cell, but did not significantly change the magnitude of alternans. On the other hand, in the case of the pacing protocol without feedback, alternans formation was prevented, even in the presence of HRV.

On the role of myosin-induced actin depolymerization during cell migration.

Mammalian cell migration in open spaces requires F-actin polymerization and myosin contraction. While many studies have focused on myosin's coupling to focal adhesion and stress fibers, the indirect effect of myosin contraction on cell migration through actin depolymerization is not well studied. In this work, we quantified how cell velocity and effective power output are influenced by the rate of actin depolymerization, which is affected by myosin contraction. In addition, we derived scaling laws to provide physical insights into cell migration. Model analysis shows that the cell migration velocity displays a biphasic dependence on the rate of actin depolymerization and myosin contraction. Our model further predicts that the effective cell energy output depends not only on the cell velocity but also on myosin contractility. The work has implications on in vivo processes such as immune response and cancer metastasis, where cells overcome barriers imposed by the physical environment.

Mathematical properties of pump-leak models of cell volume control and electrolyte balance.

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.

Evolution of prosocial behaviours in multilayer populations.

Human societies include diverse social relationships. Friends, family, business colleagues and online contacts can all contribute to one's social life. Individuals may behave differently in different domains, but success in one domain may engender success in another. Here, we study this problem using multilayer networks to model multiple domains of social interactions, in which individuals experience different environments and may express different behaviours. We provide a mathematical analysis and find that coupling between layers tends to promote prosocial behaviour. Even if prosociality is disfavoured in each layer alone, multilayer coupling can promote its proliferation in all layers simultaneously. We apply this analysis to six real-world multilayer networks, ranging from the socio-emotional and professional relationships in a Zambian community, to the online and offline relationships within an academic university. We discuss the implications of our results, which suggest that small modifications to interactions in one domain may catalyse prosociality in a different domain.

Optimality of intercellular signaling: Direct transport versus diffusion.

Intercellular signaling has an important role in organism development, but not all communication occurs using the same mechanism. Here, we analyze the energy efficiency of intercellular signaling by two canonical mechanisms: Diffusion of signaling molecules and direct transport mediated by signaling cellular protrusions. We show that efficient contact formation for direct transport can be established by an optimal rate of projecting protrusions, which depends on the availability of information about the location of the target cell. The optimal projection rate also depends on how signaling molecules are transported along the protrusion, in particular the ratio of the energy cost for contact formation and molecule synthesis. Also, we compare the efficiency of the two signaling mechanisms, under various model parameters. We find that direct transport is favored over diffusion when transporting a large amount of signaling molecules. There is a critical number of signaling molecules at which the efficiencies of the two mechanisms are the same. The critical number is small when the distance between cells is far, which helps explain why protrusion-based mechanisms are observed in long-range cellular communications.

Ephaptic conduction in a cardiac strand model with 3D electrodiffusion.

We study cardiac action potential propagation under severe reduction in gap junction conductance. We use a mathematical model of cellular electrical activity that takes into account both three-dimensional geometry and ionic concentration effects. Certain anatomical and biophysical parameters are varied to see their impact on cardiac action potential conduction velocity. This study uncovers quantitative features of ephaptic propagation that differ from previous studies based on one-dimensional models. We also identify a mode of cardiac action potential propagation in which the ephaptic and gap-junction-mediated mechanisms alternate. Our study demonstrates the usefulness of this modeling approach for electrophysiological systems especially when detailed membrane geometry plays an important role.

microRNA expression profile in undifferentiated gastric cancer.

Prognosis of patients with undifferentiated gastric cancer is generally poor. The expression of various microRNAs (miRNAs) has not been comprehensively investigated in undifferentiated gastric cancer. Total RNA was extracted from the specimens of 42 undifferentiated gastric cancer tissues and paired normal gastric tissue. Quantitative reverse transcriptase-polymerase chain reaction (qRT-PCR) was performed for a set of 72 miRNAs. The expression of each miRNA relative to the internal control RNA was determined using the 2-DeltaCt method. The expression levels of 3 miRNAs (mir-34b, mir-34c and mir-128a) were significantly upregulated and those of 3 miRNAs (mir-128b, mir-129 and mir-148) were downregulated in undifferentiated gastric cancer tissue when compared with those of the paired normal tissues. The probability of survival was significantly lower in patients with high expression levels of mir-20b or 150. There was a correlation between mir-27a and lymph node metastasis. Our investigation provides a list of candidate miRNAs that may be associated with the prognosis in undifferentiated gastric cancer patients. Further study is warranted to identify the target genes of these miRNAs and their function.