We study a stochastic spatial model of biological competition in which two species have the same birth and death rates, but different diffusion constants. In the absence of this difference, the model can be considered as an off-lattice version of the voter model and presents similar coarsening properties. We show that even a relative difference in diffusivity on the order of a few percent may lead to a strong bias in the coarsening process favoring the more agile species. We theoretically quantify this selective advantage and present analytical formulas for the average growth of the fastest species and its fixation probability.
Movement is a fundamental behaviour of organisms that not only brings about beneficial encounters with resources and mates, but also at the same time exposes the organism to dangerous encounters with predators. The movement patterns adopted by organisms should reflect a balance between these contrasting processes. This trade-off can be hypothesized as being evident in the behaviour of plankton, which inhabit a dilute three-dimensional environment with few refuges or orienting landmarks. We present an analysis of the swimming path geometries based on a volumetric Monte Carlo sampling approach, which is particularly adept at revealing such trade-offs by measuring the self-overlap of the trajectories. Application of this method to experimentally measured trajectories reveals that swimming patterns in copepods are shaped to efficiently explore volumes at small scales, while achieving a large overlap at larger scales. Regularities in the observed trajectories make the transition between these two regimes always sharper than in randomized trajectories or as predicted by random walk theory. Thus, real trajectories present a stronger separation between exploration for food and exposure to predators. The specific scale and features of this transition depend on species, gender and local environmental conditions, pointing at adaptation to state and stage-dependent evolutionary trade-offs.
We study stochastic copying schemes in which discrimination between a right and a wrong match is achieved via different kinetic barriers or different binding energies of the two matches. We demonstrate that, in single-step reactions, the two discrimination mechanisms are strictly alternative and cannot be mixed to further reduce the error fraction. Close to the lowest error limit, kinetic discrimination results in a diverging copying velocity and dissipation per copied bit. On the other hand, energetic discrimination reaches its lowest error limit in an adiabatic regime where dissipation and velocity vanish. By analyzing experimentally measured kinetic rates of two DNA polymerases, T7 and Pol?, we argue that one of them operates in the kinetic and the other in the energetic regime. Finally, we show how the two mechanisms can be combined in copying schemes implementing error correction through a proofreading pathway.
We study a stochastic community model able to interpolate from a neutral regime to a niche partitioned regime upon varying a single parameter tuning the intensity of niche stabilization, namely the difference between intraspecific and interspecific competition. By means of a self-consistent approach, we obtain an analytical expression for the species abundance distribution, in excellent agreement with stochastic simulations of the model. In the neutral limit, the Fisher log-series is recovered, while upon increasing the stabilization strength the species abundance distribution develops a maximum for species at intermediate abundances, corresponding to the emergence of a carrying capacity. Numerical studies of species extinction-time distribution show that niche-stabilization strongly affects also the dynamical properties of the system by increasing the average species lifetimes, while suppressing their fluctuations. The results are discussed in view of the niche-neutral debate and of their potential relevance to field data.
We study adaptive dynamics in games where players abandon the population at a given rate and are replaced by naive players characterized by a prior distribution over the admitted strategies. We demonstrate how such a process leads macroscopically to a variant of the replicator equation, with an additional term accounting for player turnover. We study how Nash equilibria and the dynamics of the system are modified by this additional term for prototypical examples such as the rock-paper-scissors game and different classes of two-action games played between two distinct populations. We conclude by showing how player turnover can account for nontrivial departures from Nash equilibria observed in data from lowest unique bid auctions.
We introduce a stochastic model describing aggregation of misfolded proteins and degradation by the protein quality control system in a single cell. Aggregate growth is contrasted by the cell quality control system, that attacks them at different stages of the growth process, with an efficiency that decreases with their size. Model parameters are estimated from experimental data. Two qualitatively different behaviors emerge: a homeostatic state, where the quality control system is stable and aggregates of large sizes are not formed, and an oscillatory state, where the quality control system periodically breaks down, allowing for formation of large aggregates. We discuss how these periodic breakdowns may constitute a mechanism for the development of neurodegenerative diseases.
We examine the effect of adaptive foraging behaviour within a tri-trophic food web with intra-guild predation. The intra-guild prey is allowed to adjust its foraging effort so as to achieve an optimal per capita growth rate in the face of realized feeding, predation risk and foraging cost. Adaptive fitness-seeking behaviour of the intra-guild prey has a stabilizing effect on the tri-trophic food-web dynamics provided that (i) a finite optimal foraging effort exists and (ii) the trophic transfer efficiency from resource to predator via the intra-guild prey is greater than that from the resource directly. The latter condition is a general criterion for the feasibility of intra-guild predation as a trophic mode. Under these conditions, we demonstrate rigorously that adaptive behaviour will always promote stability of community dynamics in the sense that the region of parameter space in which stability is achieved is larger than for the non-adaptive counterpart of the system.
We investigate the dynamics of an ecological system made up of one predator feeding on two different prey species. In a large range of parameter space, the system displays oscillating solutions. We show that, in the regime in which the two preys coexist, the better fit prey consistently peaks first. Further, we classify the possible oscillations of the network by a symbolic dynamics method. Our findings show that the symbolic orbits of an ecological system contain information about which of two preys is the better fit, and when one is bound to extinction.
The outcome of competition among species is influenced by the spatial distribution of species and effects such as demographic stochasticity, immigration fluxes, and the existence of preferred habitats. We introduce an individual-based model describing the competition of two species and incorporating all the above ingredients. We find that the presence of habitat preference--generating spatial niches--strongly stabilizes the coexistence of the two species. Eliminating habitat preference--neutral dynamics--the model generates patterns, such as distribution of population sizes, practically identical to those obtained in the presence of habitat preference, provided an higher immigration rate is considered. Notwithstanding the similarity in the population distribution, we show that invasibility properties depend on habitat preference in a non-trivial way. In particular, the neutral model results more invasible or less invasible depending on whether the comparison is made at equal immigration rate or at equal distribution of population size, respectively. We discuss the relevance of these results for the interpretation of invasibility experiments and the species occupancy of preferred habitats.
Many human diseases are associated with protein aggregation and fibrillation. We present experiments on in vitro glucagon fibrillation using total internal reflection fluorescence microscopy, providing real-time measurements of single-fibril growth. We find that amyloid fibrils grow in an intermittent fashion, with periods of growth followed by long pauses. The observed exponential distributions of stop and growth times support a Markovian model, in which fibrils shift between the two states with specific rates. Even if the individual rates vary considerably, we observe that the probability of being in the growing (stopping) state is very close to 1/4 (3/4) in all experiments.
We present properties of Lotka-Volterra equations describing ecological competition among a large number of interacting species. First we extend previous stability conditions to the case of a non-homogeneous niche space, i.e. that of a carrying capacity depending on the species trait. Second, we discuss mechanisms leading to species clustering and obtain an analytical solution for a state with a lumped species distribution for a specific instance of the system. We also discuss how realistic ecological interactions may result in different types of competition coefficients.
We construct a hexagonal lattice of repressing genes, such that each node represses three of the neighbors, and use it as a model for genetic regulation in spatially extended systems. Using symmetry arguments and stability analysis we argue that the repressor lattice can be in a nonfrustrated oscillating state with only three distinct phases. If the system size is not commensurate with three, oscillating solutions of several different phases are possible. As the strength of the interactions between the nodes increases, the system undergoes many transitions, breaking several symmetries. Eventually dynamical frustrated states appear, where the temporal evolution is chaotic, even though there are no built-in frustrations. Applications of the repressor lattice to real biological systems are discussed.
In Parkinsons disease (PD), there is evidence that alpha-synuclein (alphaSN) aggregation is coupled to dysfunctional or overburdened protein quality control systems, in particular the ubiquitin-proteasome system. Here, we develop a simple dynamical model for the on-going conflict between alphaSN aggregation and the maintenance of a functional proteasome in the healthy cell, based on the premise that proteasomal activity can be titrated out by mature alphaSN fibrils and their protofilament precursors. In the presence of excess proteasomes the cell easily maintains homeostasis. However, when the ratio between the available proteasome and the alphaSN protofilaments is reduced below a threshold level, we predict a collapse of homeostasis and onset of oscillations in the proteasome concentration. Depleted proteasome opens for accumulation of oligomers. Our analysis suggests that the onset of PD is associated with a proteasome population that becomes occupied in periodic degradation of aggregates. This behavior is found to be the general state of a proteasome/chaperone system under pressure, and suggests new interpretations of other diseases where protein aggregation could stress elements of the protein quality control system.
We formulate general rules for a coarse graining of the dynamics, which we term "symbolic dynamics," of feedback networks with monotonic interactions, such as most biological modules. Networks which are more complex than simple cyclic structures can exhibit multiple different symbolic dynamics. Nevertheless, we show several examples where the symbolic dynamics is dominated by a single pattern that is very robust to changes in parameters and is consistent with the dynamics being dictated by a single feedback loop. Our analysis provides a method for extracting these dominant loops from short time series, even if they only show transient trajectories.
The general tendency for species number (S) to increase with sampled area (A) constitutes one of the most robust empirical laws of ecology, quantified by species-area relationships (SAR). In many ecosystems, SAR curves display a power-law dependence, S proportional, variantA(z). The exponent z is always less than one but shows significant variation in different ecosystems. We study the multitype voter model as one of the simplest models able to reproduce SAR similar to those observed in real ecosystems in terms of basic ecological processes such as birth, dispersal and speciation. Within the model, the species-area exponent z depends on the dimensionless speciation rate nu, even though the detailed dependence is still matter of controversy. We present extensive numerical simulations in a broad range of speciation rates from nu=10(-3) down to nu=10(-11), where the model reproduces values of the exponent observed in nature. In particular, we show that the inverse of the species-area exponent linearly depends on the logarithm of nu. Further, we compare the model outcomes with field data collected from previous studies, for which we separate the effect of the speciation rate from that of the different species lifespans. We find a good linear relationship between inverse exponents and logarithm of species lifespans. However, the slope sets bounds on the speciation rates that can hardly be justified on evolutionary basis, suggesting that additional effects should be taken into account to consistently interpret the observed exponents.
We derive a simple expression for the probability of trajectories of a master equation. The expression is particularly useful when the number of states is small and permits the calculation of observables that can be defined as functionals of whole trajectories. We illustrate the method with a two-state master equation, for which we calculate the distribution of the time spent in one state and the distribution of the number of transitions, each in a given time interval. These two expressions are obtained analytically in terms of modified Bessel functions.
In many developing tissues, neighboring cells enter different developmental pathways, resulting in a fine-grained pattern of different cell states. The most common mechanism that generates such patterns is lateral inhibition, for example through Delta-Notch coupling. In this work, we simulate growth of tissues consisting of a hexagonal arrangement of cells laterally inhibiting their neighbors. We find that tissue growth by cell division and cell migration tends to produce ordered patterns, whereas lateral growth leads to disordered, patchy patterns. Ordered patterns are very robust to mutations (gene silencing or activation) in single cells. In contrast, mutation in a cell of a disordered tissue can produce a larger and more widespread perturbation of the pattern. In tissues where ordered and disordered patches coexist, the perturbations spread mostly at boundaries between patches. If cell division occurs on time scales faster than the degradation time, disordered patches will appear. Our work suggests that careful experimental characterization of the disorder in tissues could pinpoint where and how the tissue is susceptible to large-scale damage even from single cell mutations.
As a model for cell-to-cell communication in biological tissues, we construct repressor lattices by repeating a regulatory three-node motif on a hexagonal structure. Local interactions can be unidirectional, where a node either represses or activates a neighbor that does not communicate backwards. Alternatively, they can be bidirectional where two neighboring nodes communicate with each other. In the unidirectional case, we perform stability analyses for the transitions from stationary to oscillating states in lattices with different regulatory units. In the bidirectional case, we investigate transitions from oscillating states to ordered patterns generated by local switches. Finally, we show how such stable patterns in two-dimensional lattices can be generalized to three-dimensional systems.
We analyze a class of network motifs in which a short, two-node positive feedback motif is inserted in a three-node negative feedback loop. We demonstrate that such networks can undergo a bifurcation to a state where a stable fixed point and a stable limit cycle coexist. At the bifurcation point the period of the oscillations diverges. Further, intrinsic noise can make the system switch between oscillatory state and the stationary state spontaneously. We find that this switching also occurs in previous models of circadian clocks that use this combination of positive and negative feedbacks. Our results suggest that real-life circadian systems may need specific regulation to prevent or minimize such switching events.
Understanding factors that shape biodiversity and species coexistence across scales is of utmost importance in ecology, both theoretically and for conservation policies. Species-area relationships (SARs), measuring how the number of observed species increases upon enlarging the sampled area, constitute a convenient tool for quantifying the spatial structure of biodiversity. While general features of species-area curves are quite universal across ecosystems, some quantitative aspects can change significantly. Several attempts have been made to link these variations to ecological forces. Within the framework of spatially explicit neutral models, here we scrutinize the effect of varying the local population size (i.e. the number of individuals per site) and the level of habitat saturation (allowing for empty sites). We conclude that species-area curves become shallower when the local population size increases, while habitat saturation, unless strongly violated, plays a marginal role. Our findings provide a plausible explanation of why SARs for microorganisms are flatter than those for larger organisms.
We study competition between two biological species advected by a compressible velocity field. Individuals are treated as discrete Lagrangian particles that reproduce or die in a density-dependent fashion. In the absence of a velocity field and fitness advantage, number fluctuations lead to a coarsening dynamics typical of the stochastic Fisher equation. We investigate three examples of compressible advecting fields: a shell model of turbulence, a sinusoidal velocity field and a linear velocity sink. In all cases, advection leads to a striking drop in the fixation time, as well as a large reduction in the global carrying capacity. We find localization on convergence zones, and very rapid extinction compared to well-mixed populations. For a linear velocity sink, one finds a bimodal distribution of fixation times. The long-lived states in this case are demixed configurations with a single interface, whose location depends on the fitness advantage.
In lowest unique bid auctions, N players bid for an item. The winner is whoever places the lowest bid, provided that it is also unique. We use a grand canonical approach to derive an analytical expression for the equilibrium distribution of strategies. We then study the properties of the solution as a function of the mean number of players, and compare them with a large data set of internet auctions. The theory agrees with the data with striking accuracy for small population-size N, while for larger N a qualitatively different distribution is observed. We interpret this result as the emergence of two different regimes, one in which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game.
Related JoVE Video
Journal of Visualized Experiments
What is Visualize?
JoVE Visualize is a tool created to match the last 5 years of PubMed publications to methods in JoVE's video library.
How does it work?
We use abstracts found on PubMed and match them to JoVE videos to create a list of 10 to 30 related methods videos.
Video X seems to be unrelated to Abstract Y...
In developing our video relationships, we compare around 5 million PubMed articles to our library of over 4,500 methods videos. In some cases the language used in the PubMed abstracts makes matching that content to a JoVE video difficult. In other cases, there happens not to be any content in our video library that is relevant to the topic of a given abstract. In these cases, our algorithms are trying their best to display videos with relevant content, which can sometimes result in matched videos with only a slight relation.