Learning spatio-temporal patterns with Neural Cellular Automata

Neural Cellular Automata (NCA) are a powerful combination of machine learning and mechanistic modelling. We train NCA to learn complex dynamics from time series of images and Partial Differential Equation (PDE) trajectories. Our method is designed to identify underlying local rules that govern large scale dynamic emergent behaviours. Previous work on NCA focuses on learning rules that give stationary emergent structures. We extend NCA to capture both transient and stable structures within the same system, as well as learning rules that capture the dynamics of Turing pattern formation in nonlinear PDEs. We demonstrate that NCA can generalise very well beyond their PDE training data, we show how to constrain NCA to respect given symmetries, and we explore the effects of associated hyperparameters on model performance and stability. Being able to learn arbitrary dynamics gives NCA great potential as a data driven modelling framework, especially for modelling biological pattern formation.

Sequential mutations in exponentially growing populations

Stochastic models of sequential mutation acquisition are widely used to quantify cancer and bacterial evolution. Across manifold scenarios, recurrent research questions are: how many cells are there with n alterations, and how long will it take for these cells to appear. For exponentially growing populations, these questions have been tackled only in special cases so far. Here, within a multitype branching process framework, we consider a general mutational path where mutations may be advantageous, neutral or deleterious. In the biologically relevant limiting regimes of large times and small mutation rates, we derive probability distributions for the number, and arrival time, of cells with n mutations. Surprisingly, the two quantities respectively follow Mittag-Leffler and logistic distributions regardless of n or the mutations' selective effects. Our results provide a rapid method to assess how altering the fundamental division, death, and mutation rates impacts the arrival time, and number, of mutant cells. We highlight consequences for mutation rate inference in fluctuation assays.

Interference with HIV infection of the first cell is essential for viral clearance at sub-optimal levels of drug inhibition

HIV infection can be cleared with antiretroviral drugs if they are administered before exposure, where exposure occurs at low viral doses which infect one or few cells. However, infection clearance does not happen once infection is established, and this may be because of the very early formation of a reservoir of latently infected cells. Here we investigated whether initial low dose infection could be cleared with sub-optimal drug inhibition which allows ongoing viral replication, and hence does not require latency for viral persistence. We derived a model for infection clearance with inputs being drug effects on ongoing viral replication and initial number of infected cells. We experimentally tested the model by inhibiting low dose infection with the drug tenofovir, which interferes with initial infection, and atazanavir, which reduces the cellular virion burst size and hence inhibits replication only after initial infection. Drugs were used at concentrations which allowed infection to expand. Under these conditions, tenofovir dramatically increased clearance while atazanavir did not. Addition of latency to the model resulted in a minor decrease in clearance probability if the drug inhibited initial infection. If not, latency strongly decreased clearance even at low latent cell frequencies. Therefore, the ability of drugs to clear initial but not established infection can be recapitulated without latency and depends only on the ability to target initial infection. The presence of latency can dramatically decrease infection clearance, but only if the drug is unable to interfere with infection of the first cells.

A feature of viral infections such as HIV is that successful transmission occurs with low probability and is preventable by administration of drugs before exposure to the virus. Yet, once established, the infection is difficult or impossible to eradicate within its host. In the case of HIV, this may be explained by the establishment of a latent reservoir of infected cells insensitive to antiretroviral drugs. Here we use a combined modelling and experimental approach to determine whether low dose HIV infection can be cleared at drug concentrations which allow the expansion of HIV infection once established. We show that such sub-optimal drug levels are effective at clearing infection, provided they target the virus before it infects the first set of cells. The difference in the effect of drugs before and after the initial cells are infected does not require the establishment of viral latency. Rather, it is a quantitative effect, where the low infection dose can be cleared before amplifying viral numbers by infecting the first cells.

Spatial evolution of tumors with successive driver mutations

We study the influence of driver mutations on the spatial evolutionary dynamics of solid tumors. We start with a cancer clone that expands uniformly in three dimensions giving rise to a spherical shape. We assume that cell division occurs on the surface of the growing tumor. Each cell division has a chance to give rise to a mutation that activates an additional driver gene. The resulting clone has an enhanced growth rate, which generates a local ensemble of faster growing cells, thereby distorting the spherical shape of the tumor. We derive formulas for the abundance and diversity of additional driver mutations as function of time. Our model is semi-deterministic: the spatial growth of cancer clones is deterministic, while mutants arise stochastically.

Evolutionary dynamics of cancer in response to targeted combination therapy

In solid tumors, targeted treatments can lead to dramatic regressions, but responses are often short-lived because resistant cancer cells arise. The major strategy proposed for overcoming resistance is combination therapy. We present a mathematical model describing the evolutionary dynamics of lesions in response to treatment. We first studied 20 melanoma patients receiving vemurafenib. We then applied our model to an independent set of pancreatic, colorectal, and melanoma cancer patients with metastatic disease. We find that dual therapy results in long-term disease control for most patients, if there are no single mutations that cause cross-resistance to both drugs; in patients with large disease burden, triple therapy is needed. We also find that simultaneous therapy with two drugs is much more effective than sequential therapy. Our results provide realistic expectations for the efficacy of new drug combinations and inform the design of trials for new cancer therapeutics.

DOI:http://dx.doi.org/10.7554/eLife.00747.001

As medicine becomes increasingly personalized, more and more emphasis is being placed on the development of therapies that target specific cancer-causing mutations. But while many of these drugs are effective in the short term, and do extend patient lives, tumors tend to evolve resistance to them within a few months.

The key problem is that large tumors are genetically diverse. This means that for any given treatment, there is likely to be a small population of cells within the tumor that is resistant to the effects of the drug. When the drug is given to a patient, these cells will survive and multiply and this will lead ultimately to treatment failure. Given that a single drug is therefore highly unlikely to eradicate a tumor, combinations of two or more drugs may offer a higher chance of cure. This approach has been effective in the treatment of HIV as well as certain forms of leukemia.

Here, Bozic et al. present a mathematical model designed to predict the effects of combination targeted therapies on tumors, based on the data obtained from 20 melanoma (skin cancer) patients. Their model revealed that if even 1 of the 6.6 billion base pairs of DNA present in a human diploid cell has undergone a mutation that confers resistance to each of two drugs, treatment with those drugs will not lead to sustained improvement for the majority of patients. This confirms the need to develop drugs that target distinct pathways.

The model also reveals that combination therapy with two drugs given simultaneously is far more effective than sequential therapy where the drugs are used one after the other. Indeed, the model of Bozic et al. indicates that sequential treatment offers no chance of a cure, even when there are no cross-resistance mutations present, whereas combination therapy offers some hope of a cure, even in the presence of cross-resistance mutations.

By emphasizing the need to develop drugs that target distinct pathways, and to administer them in combination rather than sequentially, the study by Bozic et al. offers valuable advice for drug development and the design of clinical trials, as well as for clinical practice.

DOI:http://dx.doi.org/10.7554/eLife.00747.002

Molecular spiders on a plane

Synthetic biomolecular spiders with "legs" made of single-stranded segments of DNA can move on a surface covered by single-stranded segments of DNA called substrates when the substrate DNA is complementary to the leg DNA. If the motion of a spider does not affect the substrates, the spider behaves asymptotically as a random walk. We study the diffusion coefficient and the number of visited sites for spiders moving on the square lattice with a substrate in each lattice site. The spider's legs hop to nearest-neighbor sites with the constraint that the distance between any two legs cannot exceed a maximal span. We establish analytic results for bipedal spiders, and investigate multileg spiders numerically. In experimental realizations legs usually convert substrates into products (visited sites). The binding of legs to products is weaker, so the hopping rate from the substrates is smaller. This makes the problem non-Markovian and we investigate it numerically. We demonstrate the emergence of a counterintuitive behavior-the more spiders are slowed down on unvisited sites, the more motile they become.

Distant metastasis occurs late during the genetic evolution of pancreatic cancer

Metastasis, the dissemination and growth of neoplastic cells in an organ distinct from that in which they originated, is the most common cause of death in cancer patients. This is particularly true for pancreatic cancers, where most patients are diagnosed with metastatic disease and few show a sustained response to chemotherapy or radiation therapy. Whether the dismal prognosis of patients with pancreatic cancer compared to patients with other types of cancer is a result of late diagnosis or early dissemination of disease to distant organs is not known. Here we rely on data generated by sequencing the genomes of seven pancreatic cancer metastases to evaluate the clonal relationships among primary and metastatic cancers. We find that clonal populations that give rise to distant metastases are represented within the primary carcinoma, but these clones are genetically evolved from the original parental, non-metastatic clone. Thus, genetic heterogeneity of metastases reflects that within the primary carcinoma. A quantitative analysis of the timing of the genetic evolution of pancreatic cancer was performed, indicating at least a decade between the occurrence of the initiating mutation and the birth of the parental, non-metastatic founder cell. At least five more years are required for the acquisition of metastatic ability and patients die an average of two years thereafter. These data provide novel insights into the genetic features underlying pancreatic cancer progression and define a broad time window of opportunity for early detection to prevent deaths from metastatic disease.

Distant metastasis occurs late during the genetic evolution of pancreatic cancer

Metastasis, the dissemination and growth of neoplastic cells in an organ distinct from that in which they originated, is the most common cause of death in cancer patients. This is particularly true for pancreatic cancers, where most patients are diagnosed with metastatic disease and few show a sustained response to chemotherapy or radiation therapy. Whether the dismal prognosis of patients with pancreatic cancer compared to patients with other types of cancer is a result of late diagnosis or early dissemination of disease to distant organs is not known. Here we rely on data generated by sequencing the genomes of seven pancreatic cancer metastases to evaluate the clonal relationships among primary and metastatic cancers. We find that clonal populations that give rise to distant metastases are represented within the primary carcinoma, but these clones are genetically evolved from the original parental, non-metastatic clone. Thus, genetic heterogeneity of metastases reflects that within the primary carcinoma. A quantitative analysis of the timing of the genetic evolution of pancreatic cancer was performed, indicating at least a decade between the occurrence of the initiating mutation and the birth of the parental, non-metastatic founder cell. At least five more years are required for the acquisition of metastatic ability and patients die an average of two years thereafter. These data provide novel insights into the genetic features underlying pancreatic cancer progression and define a broad time window of opportunity for early detection to prevent deaths from metastatic disease.

Effect of stalling after mismatches on the error catastrophe in nonenzymatic nucleic acid replication

The frequency of errors during genome replication limits the amount of functionally important information that can be passed on from generation to generation. During the origin of life, mutation rates are thought to have been quite high, raising a classic chicken-and-egg paradox: could nonenzymatic replication propagate sequences accurately enough to allow for the emergence of heritable function? Here we show that the theoretical limit on genomic information content may increase substantially as a consequence of dramatically slowed polymerization after mismatches. As a result of postmismatch stalling, accurate copies of a template tend to be completed more rapidly than mutant copies and the accurate copies can therefore begin a second round of replication more quickly. To quantify this effect, we characterized an experimental model of nonenzymatic, template-directed nucleic acid polymerization. We found that most mismatches decrease the rate of primer extension by more than 2 orders of magnitude relative to a matched (Watson-Crick) control. A chemical replication system with this property would be able to propagate sequences long enough to have function. Our study suggests that the emergence of functional sequences during the origin of life would be possible even in the face of the high intrinsic error rates of chemical replication.

Evolutionary dynamics in structured populations

Evolutionary dynamics shape the living world around us. At the centre of every evolutionary process is a population of reproducing individuals. The structure of that population affects evolutionary dynamics. The individuals can be molecules, cells, viruses, multicellular organisms or humans. Whenever the fitness of individuals depends on the relative abundance of phenotypes in the population, we are in the realm of evolutionary game theory. Evolutionary game theory is a general approach that can describe the competition of species in an ecosystem, the interaction between hosts and parasites, between viruses and cells, and also the spread of ideas and behaviours in the human population. In this perspective, we review the recent advances in evolutionary game dynamics with a particular emphasis on stochastic approaches in finite sized and structured populations. We give simple, fundamental laws that determine how natural selection chooses between competing strategies. We study the well-mixed population, evolutionary graph theory, games in phenotype space and evolutionary set theory. We apply these results to the evolution of cooperation. The mechanism that leads to the evolution of cooperation in these settings could be called ‘spatial selection’: cooperators prevail against defectors by clustering in physical or other spaces.

Strategy selection in structured populations

Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix(abcd). We study a mutation and selection process. For weak selection strategy A is favored over B if and only if sigma a+b>c+sigma d. This means the effect of population structure on strategy selection can be described by a single parameter, sigma. We present the values of sigma for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a sigma, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between sigma and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, sigma, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.

Mutation-selection equilibrium in games with mixed strategies

We develop a new method for studying stochastic evolutionary game dynamics of mixed strategies. We consider the general situation: there are n pure strategies whose interactions are described by an nxn payoff matrix. Players can use mixed strategies, which are given by the vector (p(1),...,p(n)). Each entry specifies the probability to use the corresponding pure strategy. The sum over all entries is one. Therefore, a mixed strategy is a point in the simplex S(n). We study evolutionary dynamics in a well-mixed population of finite size. Individuals reproduce proportional to payoff. We consider the case of weak selection, which means the payoff from the game is only a small contribution to overall fitness. Reproduction can be subject to mutation; a mutant adopts a randomly chosen mixed strategy. We calculate the average abundance of every mixed strategy in the stationary distribution of the mutation-selection process. We find the crucial conditions that specify if a strategy is favored or opposed by selection. One condition holds for low mutation rate, another for high mutation rate. The result for any mutation rate is a linear combination of those two. As a specific example we study the Hawk-Dove game. We prove general statements about the relationship between games with pure and with mixed strategies.

Mutation-selection equilibrium in games with multiple strategies

In evolutionary games the fitness of individuals is not constant but depends on the relative abundance of the various strategies in the population. Here we study general games among n strategies in populations of large but finite size. We explore stochastic evolutionary dynamics under weak selection, but for any mutation rate. We analyze the frequency dependent Moran process in well-mixed populations, but almost identical results are found for the Wright-Fisher and Pairwise Comparison processes. Surprisingly simple conditions specify whether a strategy is more abundant on average than 1/n, or than another strategy, in the mutation-selection equilibrium. We find one condition that holds for low mutation rate and another condition that holds for high mutation rate. A linear combination of these two conditions holds for any mutation rate. Our results allow a complete characterization of nxn games in the limit of weak selection.

Evolution of cooperation by phenotypic similarity

The emergence of cooperation in populations of selfish individuals is a fascinating topic that has inspired much work in theoretical biology. Here, we study the evolution of cooperation in a model where individuals are characterized by phenotypic properties that are visible to others. The population is well mixed in the sense that everyone is equally likely to interact with everyone else, but the behavioral strategies can depend on distance in phenotype space. We study the interaction of cooperators and defectors. In our model, cooperators cooperate with those who are similar and defect otherwise. Defectors always defect. Individuals mutate to nearby phenotypes, which generates a random walk of the population in phenotype space. Our analysis brings together ideas from coalescence theory and evolutionary game dynamics. We obtain a precise condition for natural selection to favor cooperators over defectors. Cooperation is favored when the phenotypic mutation rate is large and the strategy mutation rate is small. In the optimal case for cooperators, in a one-dimensional phenotype space and for large population size, the critical benefit-to-cost ratio is given by b/c = 1 + 2/square root(3). We also derive the fundamental condition for any two-strategy symmetric game and consider high-dimensional phenotype spaces.

Progenitor cell self-renewal and cyclic neutropenia

OBJECTIVES

Cyclic neutropenia (CN) is a rare genetic disorder where patients experience regular cycling of numbers of neutrophils and various other haematopoietic lineages. The nadir in neutrophil count is the main source of problems due to risk of life-threatening infections. Patients with CN benefit from granulocyte colony stimulating factor therapy, although cycling persists. Mutations in neutrophil elastase gene (ELA2) have been found in more than half of patients with CN. However, neither connection between phenotypic expression of ELA2 and CN nor the mechanism of cycling is known.

MATERIALS AND METHODS

Recently, a multicompartment model of haematopoiesis that couples stem cell replication with marrow output has been proposed. In the following, we couple this model of haematopoiesis with a linear feedback mechanism via G-CSF.

RESULTS

We propose that the phenotypic effect of ELA2 mutations leads to reduction in self-renewal of granulocytic progenitors. The body responds by overall relative increase of G-CSF and increasing progenitor cell self-renewal, leading to cell count cycling.

CONCLUSION

The model is compatible with available experimental data and makes testable predictions.

Strategy abundance in 2x2 games for arbitrary mutation rates

We study evolutionary game dynamics in a well-mixed populations of finite size, N. A well-mixed population means that any two individuals are equally likely to interact. In particular we consider the average abundances of two strategies, A and B, under mutation and selection. The game dynamical interaction between the two strategies is given by the 2x2 payoff matrix (acbd). It has previously been shown that A is more abundant than B, if a(N-2)+bN>cN+d(N-2). This result has been derived for particular stochastic processes that operate either in the limit of asymptotically small mutation rates or in the limit of weak selection. Here we show that this result holds in fact for a wide class of stochastic birth-death processes for arbitrary mutation rate and for any intensity of selection.

Cyclic neutropenia in mammals

Cyclic neutropenia (CN) has been well documented in humans and the gray collie. A recent model of the architecture and dynamics of hematopoiesis has been used to provide insights into the mechanism of cycling of this disorder. It provides a link between the cycling period and the cells where the mutated ELA2 is expressed. Assuming that the biologic defect in CN is the same in dogs, and the observation that the structure of hematopoiesis is invariant across mammals, we use allometric scaling techniques to correctly predict the period of cycling in the gray collie and extend it to other mammals from mice to elephants. This work provides additional support for the relevance of animal models to understand disease but cautions that disease dynamics in model animals are different and this has to be taken into consideration when planning experiments.

Logarithmic current fluctuations in nonequilibrium quantum spin chains

We study zero-temperature quantum spin chains, which are characterized by a nonvanishing current. For the XX model starting from the initial state mid R:cdots, three dots, centered upward arrow upward arrow upward arrow downward arrow downward arrow downward arrowcdots, three dots, centered we derive an exact expression for the variance of the total spin current. We show that asymptotically the variance exhibits an anomalously slow logarithmic growth; we also extract the subleading constant term. We then argue that the logarithmic growth remains valid for the XXZ model in the critical region.

Exciting hard spheres

We investigate the collision cascade that is generated by a single moving particle in a static and homogeneous hard-sphere gas. We argue that the number of moving particles at time t grows as t;{xi} and the number collisions up to time t grows as t;{eta} , with xi=2d(d+2) , eta=2(d+1)(d+2) , and d the spatial dimension. These growth laws are the same as those from a hydrodynamic theory for the shock wave emanating from an explosion. Our predictions are verified by molecular dynamics simulations in d=1 and 2. For a particle incident on a static gas in a half-space, the resulting backsplatter ultimately contains almost all the initial energy.

In vivo analysis of chlorophyll a fluorescence induction

Quantitative characteristics of photosynthetic electron transport were evaluated in vivo on the basis of the multi-exponential analysis of OJIP fluorescence transients induced by saturating actinic light. The OJIP fluorescence curve F(t), measured in Chlamydomonas reinhardtii cells, was transformed into the (1 - F(O)/F(t)) x (F(V)/F(M))(-1) transient, which is shown to relate to PS 2 closure. We assumed that kinetics of PS 2 closure during OJIP rise reflects time-separated processes related to the establishment of redox equilibrium at the PS 2 acceptor side (OJ), PQ pool (JI), and beyond Cyt b/f (IP). Three-exponential fitting was applied to (1 - F(O)/F(t)) x (F(V)/F(M))(-1) transient to obtain lifetimes and amplitudes of the OJ, JI, and IP components of PS 2 closure, which were used to calculate overall rates of reduction and re-oxidation of the PS 2 acceptor side, PQ pool, and intermediates beyond Cyt b/f complex. The results, obtained in the presence of inhibitors, oxidative reagents, and under different stress conditions prove the suggested model and characterize the introduced parameters as useful indicators of photosynthetic function.

Voter models on heterogeneous networks

We study simple interacting particle systems on heterogeneous networks, including the voter model and the invasion process. These are both two-state models in which in an update event an individual changes state to agree with a neighbor. For the voter model, an individual "imports" its state from a randomly chosen neighbor. Here the average time TN to reach consensus for a network of N nodes with an uncorrelated degree distribution scales as N mu1 2/mu2, where mu k is the kth moment of the degree distribution. Quick consensus thus arises on networks with broad degree distributions. We also identify the conservation law that characterizes the route by which consensus is reached. Parallel results are derived for the invasion process, in which the state of an agent is "exported" to a random neighbor. We further generalize to biased dynamics in which one state is favored. The probability for a single fitter mutant located at a node of degree k to overspread the population-the fixation probability--is proportional to k for the voter model and to 1k for the invasion process.

Genetic progression and the waiting time to cancer

Cancer results from genetic alterations that disturb the normal cooperative behavior of cells. Recent high-throughput genomic studies of cancer cells have shown that the mutational landscape of cancer is complex and that individual cancers may evolve through mutations in as many as 20 different cancer-associated genes. We use data published by Sjöblom et al. (2006) to develop a new mathematical model for the somatic evolution of colorectal cancers. We employ the Wright-Fisher process for exploring the basic parameters of this evolutionary process and derive an analytical approximation for the expected waiting time to the cancer phenotype. Our results highlight the relative importance of selection over both the size of the cell population at risk and the mutation rate. The model predicts that the observed genetic diversity of cancer genomes can arise under a normal mutation rate if the average selective advantage per mutation is on the order of 1%. Increased mutation rates due to genetic instability would allow even smaller selective advantages during tumorigenesis. The complexity of cancer progression can be understood as the result of multiple sequential mutations, each of which has a relatively small but positive effect on net cell growth.

Aging and immortality in a cell proliferation model

We investigate a model of cell division in which the length of telomeres within a cell regulates its proliferative potential. At each division, telomeres undergo a systematic length decrease as well as a superimposed fluctuation due to exchange of telomere DNA between the two daughter cells. A cell becomes senescent when one or more of its telomeres become shorter than a critical length. We map this telomere dynamics onto a biased branching-diffusion process with an absorbing boundary condition whenever any telomere reaches the critical length. Using first-passage ideas, we find a phase transition between finite lifetime and immortality (infinite proliferation) of the cell population as a function of the influence of telomere shortening, fluctuations, and cell division.

Dynamics of an idealized model of microtubule growth and catastrophe

We investigate a simple dynamical model of a microtubule that evolves by attachment of guanosine triphosphate (GTP) tubulin to its end, irreversible conversion of GTP to guanosine diphosphate (GDP) tubulin by hydrolysis, and detachment of GDP at the end of a microtubule. As a function of rates of these processes, the microtubule can grow steadily or its length can fluctuate wildly. In the regime where detachment can be neglected, we find exact expressions for the tubule and GTP cap length distributions, as well as power-law length distributions of GTP and GDP islands. In the opposite limit of instantaneous detachment, we find the time between catastrophes, where the microtubule shrinks to zero length, and determine the size distribution of avalanches (sequence of consecutive GDP detachment events). We obtain the phase diagram for general rates and verify our predictions by numerical simulations.

Guiding fields for phase separation: controlling Liesegang patterns

Liesegang patterns emerge from precipitation processes and may be used to build bulk structures at submicrometer length scales. Thus they have significant potential for technological applications provided adequate methods of control can be devised. Here we describe a simple, physically realizable pattern control based on the notion of driven precipitation, meaning that the phase separation is governed by a guiding field such as, for example, a temperature or pH field. The phase separation is modeled through a nonautonomous Cahn-Hilliard equation whose spinodal is determined by the evolving guiding field. Control over the dynamics of the spinodal gives control over the velocity of the instability front that separates the stable and unstable regions of the system. Since the wavelength of the pattern is largely determined by this velocity, the distance between successive precipitation bands becomes controllable. We demonstrate the above ideas by numerical studies of a one-dimensional system with a diffusive guiding field. We find that the results can be accurately described by employing a linear stability analysis (pulled-front theory) for determining the velocity-local-wavelength relationship. From the perspective of the Liesegang theory, our results indicate that the so-called revert patterns may be naturally generated by diffusive guiding fields.

Molecular Spiders in One Dimension

Molecular spiders are synthetic bio-molecular systems which have "legs" made of short single-stranded segments of DNA. Spiders move on a surface covered with single-stranded DNA segments complementary to legs. Different mappings are established between various models of spiders and simple exclusion processes. For spiders with simple gait and varying number of legs we compute the diffusion coefficient; when the hopping is biased we also compute their velocity.

Molecular spiders with memory

Synthetic biomolecular spiders with "legs" made of single-stranded segments of DNA can move on a surface which is also covered by single-stranded segments of DNA complementary to the leg DNA. In experimental realizations, when a leg detaches from a segment of the surface for the first time it alters that segment, and legs subsequently bind to these altered segments more weakly. Inspired by these experiments, we investigate spiders moving along a one-dimensional substrate, whose legs leave newly visited sites at a slower rate than revisited sites. For a random walk (one-leg spider), the slowdown does not affect the long time behavior. For a bipedal spider, however, the slowdown generates an effective bias toward unvisited sites, and the spider behaves similarly to the excited walk. Surprisingly, the slowing down of the spider at new sites increases the diffusion coefficient and accelerates the growth of the number of visited sites.

Exact steady-state velocity of ratchets driven by random sequential adsorption

We solve the problem of discrete translocation of a polymer through a pore, driven by the irreversible, random sequential adsorption of particles on one side of the pore. Although the kinetics of the wall motion and the deposition are coupled, we find the exact steady-state distribution for the gap between the wall and the nearest deposited particle. This result enables us to construct the mean translocation velocity demonstrating that translocation is faster when the adsorbing particles are smaller. Monte-Carlo simulations also show that smaller particles gives less dispersion in the ratcheted motion. We also define and compare the relative efficiencies of ratcheting by deposition of particles with different sizes and we describe an associated "zone-refinement" process.

Dynamics of Microtubule Instabilities

We investigate an idealized model of microtubule dynamics that involves: (i) attachment of guanosine triphosphate (GTP) at rate λ, (ii) conversion of GTP to guanosine diphosphate (GDP) at rate 1, and (iii) detachment of GDP at rate μ. As a function of these rates, a microtubule can grow steadily or its length can fluctuate wildly. For μ = 0, we find the exact tubule and GTP cap length distributions, and power-law length distributions of GTP and GDP islands. For μ = ∞, we argue that the time between catastrophes, where the microtubule shrinks to zero length, scales as e(λ). We also discuss the nature of the phase boundary between a growing and shrinking microtubule.

The emergence of tumor metastases

The appearance of metastases is an ominous sign in the natural history of any malignant tumor. Their presence implies a high tumor burden and greatly decreases the probability of a cure. Metastasis development requires the evolution of tumor cells that can survive in an environment that is normally not supportive to their growth and such cells must leave the tumor to establish tumor niches elsewhere. The interactions between the appearance of cells with metastatic ability in the primary tumor and their exit from the tumor lead to complex dynamics that can be either beneficial or detrimental to the tumor. We develop a simple mathematical model to illustrate how the interplay between mutation rate and export probability affects the intratumoral dynamics of metastasis-enabled cells and the rate of metastases formation.

Flows on graphs with random capacities

We investigate flows on graphs whose links have random capacities. For binary trees we derive the probability distribution for the maximal flow from the root to a leaf, and show that for infinite trees it vanishes beyond a certain threshold that depends on the distribution of capacities. We then examine the maximal total flux from the root to the leaves. Our methods generalize to simple graphs with loops, e.g., to hierarchical lattices and to complete graphs.

Fixation of Strategies for an Evolutionary Game in Finite Populations

A stochastic evolutionary dynamics of two strategies given by 2x 2 matrix games is studied in finite populations. We focus on stochastic properties of fixation: how a strategy represented by a single individual wins over the entire population. The process is discussed in the framework of a random walk with site dependent hopping rates. The time of fixation is found to be identical for both strategies in any particular game. The asymptotic behavior of the fixation time and fixation probabilities in the large population size limit is also discussed. We show that fixation is fast when there is at least one pure evolutionary stable strategy (ESS) in the infinite population size limit, while fixation is slow when the ESS is the coexistence of the two strategies.

Evolutionary dynamics on degree-heterogeneous graphs

The evolution of two species with different fitness is investigated on degree-heterogeneous graphs. The population evolves either by one individual dying and being replaced by the offspring of a random neighbor (voter model dynamics) or by an individual giving birth to an offspring that takes over a random neighbor node (invasion process dynamics). The fixation probability for one species to take over a population of N individuals depends crucially on the dynamics and on the local environment. Starting with a single fitter mutant at a node of degree k, the fixation probability is proportional to k for voter model dynamics and to 1/k for invasion process dynamics.

"Burnt-bridge" mechanism of molecular motor motion

Motivated by a biased diffusion of molecular motors with the bias dependent on the state of the substrate, we investigate a random walk on a one-dimensional lattice that contains weak links (called "bridges") which are affected by the walker. Namely, a bridge is destroyed with probability when p the walker crosses it; the walker is not allowed to cross it again and this leads to a directed motion. The velocity of the walker is determined analytically for equidistant bridges. The special case of p = 1 is more tractable--both the velocity and the diffusion constant are calculated for uncorrelated locations of bridges, including periodic and random distributions.

Dynamics of social balance on networks

We study the evolution of social networks that contain both friendly and unfriendly pairwise links between individual nodes. The network is endowed with dynamics in which the sense of a link in an imbalanced triad--a triangular loop with one or three unfriendly links--is reversed to make the triad balanced. With this dynamics, an infinite network undergoes a dynamic phase transition from a steady state to "paradise"--all links are friendly--as the propensity p for friendly links in an update event passes through 1/2 . A finite network always falls into a socially balanced absorbing state where no imbalanced triads remain. If the additional constraint that the number of imbalanced triads in the network not increase in an update is imposed, then the network quickly reaches a balanced final state.

Weight-driven growing networks

We study growing networks in which each link carries a certain weight (randomly assigned at birth and fixed thereafter). The weight of a node is defined as the sum of the weights of the links attached to the node, and the network grows via the simplest weight-driven rule: A newly added node is connected to an already existing node with the probability which is proportional to the weight of that node. We show that the node weight distribution n (w) has a universal tail, that is, it is independent of the link weight distribution: n (w) approximately w(-3) as w-->infinity . Results are particularly neat for the exponential link weight distribution when n (w) is algebraic over the entire weight range.

Exponential velocity tails in a driven inelastic Maxwell model

The problem of the steady-state velocity distribution in a driven inelastic Maxwell model of shaken granular material is revisited. Numerical solution of the master equation and analytical arguments show that the model has bilateral exponential velocity tails [P(v) approximately e(-|v|/sqrt[D])], where D is the amplitude of the noise. Previous study of this model predicted Gaussian tails [P(v) approximately e(-av(2))].

Roughness distributions for 1/f alpha signals

The probability density function (PDF) of the roughness, i.e., of the temporal variance, of 1/f(alpha) noise signals is studied. Our starting point is the generalization of the model of Gaussian, time periodic, 1/f noise, discussed in our recent Letter [Phys. Rev. Lett. 87, 240601 (2001)], to arbitrary power law. We investigate three main scaling regions (alpha < or = 1/2, 1/2 < alpha < or = 1, and 1< alpha), distinguished by the scaling of the cumulants in terms of the microscopic scale and the total length of the period. Various analytical representations of the PDF allow for a precise numerical evaluation of the scaling function of the PDF for any alpha. A simulation of the periodic process makes it possible to study also nonperiodic, thus experimentally more relevant, signals on relatively short intervals embedded in the full period. We find that for alpha < or = 1/2 the scaled PDFs in both the periodic and the nonperiodic cases are Gaussian, but for alpha > 1/2 they differ from the Gaussian and from each other. Both deviations increase with growing alpha. That conclusion, based on numerics, is reinforced by analytic results for alpha = 2 and alpha-->infinity, in the latter limit the scaling function of the PDF being finite for periodic signals, but developing a singularity for the aperiodic ones. Finally, an overview is given for the scaling of cumulants of the roughness and the various scaling regions in arbitrary dimensions. We suggest that our theoretical and numerical results open a different perspective on the data analysis of 1/f(alpha) processes.

1/f noise and extreme value statistics

We study finite-size scaling of the roughness of signals in systems displaying Gaussian 1/f power spectra. It is found that one of the extreme value distributions, the Fisher-Tippett-Gumbel (FTG) distribution, emerges as the scaling function when boundary conditions are periodic. We provide a realistic example of periodic 1/f noise, and demonstrate by simulations that the FTG distribution is a good approximation for the case of nonperiodic boundary conditions as well. Experiments on voltage fluctuations in GaAs films are analyzed and excellent agreement is found with the theory.

Critical behavior of a lattice prey-predator model

The critical properties of a simple prey-predator model are revisited. For some values of the control parameters, the model exhibits a line of directed percolationlike transitions to a single absorbing state. For other values of the control parameters one finds a second line of continuous transitions toward an infinite number of absorbing states, and the corresponding steady-state exponents are mean-field-like. The critical behavior of the special point T (bicritical point), where the two transition lines meet, belongs to a different universality class. A particular strategy for preparing the initial states used for the dynamical Monte Carlo method is devised to correctly describe the physics of the system near the second transition line. Relationships with a forest fire model with immunization are also discussed.

Phase transitions and oscillations in a lattice prey-predator model

A coarse grained description of a two-dimensional prey-predator system is given in terms of a simple three-state lattice model containing two control parameters: the spreading rates of prey and predator. The properties of the model are investigated by dynamical mean-field approximations and extensive numerical simulations. It is shown that the stationary state phase diagram is divided into two phases: a pure prey phase and a coexistence phase of prey and predator in which temporal and spatial oscillations can be present. Besides the usual directed percolationlike transition, the system exhibits an unexpected, different type of transition to the prey absorbing phase. The passage from the oscillatory domain to the nonoscillatory domain of the coexistence phase is described as a crossover phenomena, which persists even in the infinite size limit. The importance of finite size effects are discussed, and scaling relations between different quantities are established. Finally, physical arguments, based on the spatial structure of the model, are given to explain the underlying mechanism leading to local and global oscillations.

Asymmetric exclusion process with next-nearest-neighbor interaction: some comments on traffic flow and a nonequilibrium reentrance transition

We study the steady-state behavior of a driven nonequilibrium lattice gas of hard-core particles with next-nearest-neighbor interaction. We calculate the exact stationary distribution of the periodic system and for a particular line in the phase diagram of the system with open boundaries where particles can enter and leave the system. For repulsive interactions the dynamics can be interpreted as a two-speed model for traffic flow. The exact stationary distribution of the periodic continuous-time system turns out to coincide with that of the asymmetric exclusion process (ASEP) with discrete-time parallel update. However, unlike in the (single-speed) ASEP, the exact flow diagram for the two-speed model resembles in some important features the flow diagram of real traffic. The stationary phase diagram of the open system obtained from Monte Carlo simulations can be understood in terms of a shock moving through the system and an overfeeding effect at the boundaries, thus confirming theoretical predictions of a recently developed general theory of boundary-induced phase transitions. In the case of attractive interaction we observe an unexpected reentrance transition due to boundary effects.

Spatial evolutionary prisoner's dilemma game with three strategies and external constraints

The emergency of mutual cooperation is studied in a spatially extended evolutionary prisoner's dilemma game in which the players are located on the sites of cubic lattices for dimensions d=1, 2, and 3. Each player can choose one of the three following strategies: cooperation (C), defection (D) or "tit for tat" (T). During the evolutionary process the randomly chosen players adopt one of their neighboring strategies if the chosen neighbor has a higher payoff. Moreover, an external constraint imposes that the players always cooperate with probability p. The stationary state phase diagram is computed by both using generalized mean-field approximations and Monte Carlo simulations. Nonequilibrium second-order phase transitions associated with the extinction of one of the possible strategies are found and the corresponding critical exponents belong to the directed percolation universality class. It is shown that externally forcing the collaboration does not always produce the desired result.

Transport in the XX chain at zero temperature: emergence of flat magnetization profiles

We study the connection between magnetization transport and magnetization profiles in zero-temperature XX chains. The time evolution of the transverse magnetization m(x,t) is calculated using an inhomogeneous initial state that is the ground state at fixed magnetization but with m reversed from -m(0) for x<0 to m(0) for x>0. In the long-time limit, the magnetization evolves into a scaling form m(x,t)=Phi(x/t) and the profile develops a flat part (m=Phi=0) in the (x/t)<or=c(m(0)) region. The flat region shrinks to zero if m(0)-->1/2 while it expands with the maximum velocity c(0)=1 for m(0)-->0. The states emerging in the scaling limit are compared to those of a homogeneous system where the same magnetization current is driven by a bulk field, and we find that the expectation values of various quantities (energy, occupation number in the fermionic representation) agree in the two systems.