Minimal frustration underlies the usefulness of incomplete regulatory network models in biology.

Regulatory networks as large and complex as those implicated in cell-fate choice are expected to exhibit intricate, very high-dimensional dynamics. Cell-fate choice, however, is a macroscopically simple process. Additionally, regulatory network models are almost always incomplete and/or inexact, and do not incorporate all the regulators and interactions that may be involved in cell-fate regulation. In spite of these issues, regulatory network models have proven to be incredibly effective tools for understanding cell-fate choice across contexts and for making useful predictions. Here, we show that minimal frustration-a feature of biological networks across contexts but not of random networks-can compel simple, low-dimensional steady-state behavior even in large and complex networks. Moreover, the steady-state behavior of minimally frustrated networks can be recapitulated by simpler networks such as those lacking many of the nodes and edges and those that treat multiple regulators as one. The present study provides a theoretical explanation for the success of network models in biology and for the challenges in network inference.

Phenomenological Approach to Cancer Cell Persistence.

Drug persistence is a phenomenon by which a small percentage of cancer cells survive the presentation of targeted therapy by transitioning to a quiescent state. Eventually some of these persister cells can transition back to an active growing state and give rise to resistant tumors. Here we introduce a quantitative genetics approach to drug-exposed populations of cancer cells in order to interpret recent experimental data regarding inheritance of persister probability. Our results indicate that alternating periods of drug treatment and drug removal may not be an effective strategy for eliminating persisters.

Discrete Sampling of Extreme Events Modifies Their Statistics.

Extreme value (EV) statistics of correlated systems are widely investigated in many fields, spanning the spectrum from weather forecasting to earthquake prediction. Does the unavoidable discrete sampling of a continuous correlated stochastic process change its EV distribution? We explore this question for correlated random variables modeled via Langevin dynamics for a particle in a potential field. For potentials growing at infinity faster than linearly and for long measurement times, we find that the EV distribution of the discretely sampled process diverges from that of the full continuous dataset and converges to that of independent and identically distributed random variables drawn from the process's equilibrium measure. However, for processes with sublinear potentials, the long-time limit is the EV statistics of the continuously sampled data. We treat processes whose equilibrium measures belong to the three EV attractors: Gumbel, Fréchet, and Weibull. Our Letter shows that the EV statistics can be extremely sensitive to the sampling rate of the data.

Ordered hexagonal patterns via notch-delta signaling.

Many developmental processes in biology utilize notch-delta signaling to construct an ordered pattern of cellular differentiation. This signaling modality is based on nearest-neighbor contact, as opposed to the more familiar mechanism driven by the release of diffusible ligands. Here, exploiting this 'juxtacrine' property, we present an exact treatment of the pattern formation problem via a system of nine coupled ordinary differential equations. The possible patterns that are realized for realistic parameters can be analyzed by considering a co-dimension 2 pitchfork bifurcation of this system. This analysis explains the observed prevalence of hexagonal patterns with high delta at their center, as opposed to those with central high notch levels (referred to as anti-hexagons). We show that outside this range of parameters, in particular for lowcis-coupling, a novel kind of pattern is produced, where high delta cells have high notch as well. It also suggests that the biological system is only weakly first order, so that an additional mechanism is required to generate the observed defect-free patterns. We construct a simple strategy for producing such defect-free patterns.

Non-Normalizable Quasi-Equilibrium Solution of the Fokker-Planck Equation for Nonconfining Fields.

We investigate the overdamped Langevin motion for particles in a potential well that is asymptotically flat. When the potential well is deep as compared to the temperature, physical observables, like the mean square displacement, are essentially time-independent over a long time interval, the stagnation epoch. However, the standard Boltzmann-Gibbs (BG) distribution is non-normalizable, given that the usual partition function is divergent. For this regime, we have previously shown that a regularization of BG statistics allows for the prediction of the values of dynamical and thermodynamical observables in the non-normalizable quasi-equilibrium state. In this work, based on the eigenfunction expansion of the time-dependent solution of the associated Fokker-Planck equation with free boundary conditions, we obtain an approximate time-independent solution of the BG form, being valid for times that are long, but still short as compared to the exponentially large escape time. The escaped particles follow a general free-particle statistics, where the solution is an error function, which is shifted due to the initial struggle to overcome the potential well. With the eigenfunction solution of the Fokker-Planck equation in hand, we show the validity of the regularized BG statistics and how it perfectly describes the time-independent regime though the quasi-stationary state is non-normalizable.

Biological Networks Regulating Cell Fate Choice Are Minimally Frustrated.

Characterization of the differences between biological and random networks can reveal the design principles that enable the robust realization of crucial biological functions including the establishment of different cell types. Previous studies, focusing on identifying topological features that are present in biological networks but not in random networks, have, however, provided few functional insights. We use a Boolean modeling framework and ideas from the spin glass literature to identify functional differences between five real biological networks and random networks with similar topological features. We show that minimal frustration is a fundamental property that allows biological networks to robustly establish cell types and regulate cell fate choice, and that this property can emerge in complex networks via Darwinian evolution. The study also provides clues regarding how the regulation of cell fate choice can go awry in a disease like cancer and lead to the emergence of aberrant cell types.

Effects of thymic selection on T cell recognition of foreign and tumor antigenic peptides.

The advent of cancer immunotherapy has generated renewed hope for the treatment of many malignancies by introducing a number of novel strategies that exploit various properties of the immune system. These therapies are based on the idea that cytotoxic T lymphocytes (CTLs) directly recognize and respond to tumor-associated neoantigens (TANs) in much the same way as they would to foreign peptides presented on cell surfaces. To date, however, nearly all attempts to optimize immunotherapeutic strategies have been empirical. Here, we develop a model of T cell selection based on the assumption of random interaction strengths between a self-peptide and the various T cell receptors. The model enables the analytical study of the effects of selection on the CTL recognition of TANs and completely foreign peptides and can estimate the number of CTLs that can detect donor-matched transplants. We show that negative selection thresholds chosen to reflect experimentally observed thymic survival rates result in near-optimal production of T cells that are capable of surviving selection and recognizing foreign antigen. These analytical results are confirmed by simulation.

From Non-Normalizable Boltzmann-Gibbs Statistics to Infinite-Ergodic Theory.

We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as 1/x, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a mean-squared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.

Mechanical bounds to transcriptional noise.

Over the past several decades it has been increasingly recognized that stochastic processes play a central role in transcription. Although many stochastic effects have been explained, the source of transcriptional bursting (one of the most well-known sources of stochasticity) has continued to evade understanding. Recent results have pointed to mechanical feedback as the source of transcriptional bursting, but a reconciliation of this perspective with preexisting views of transcriptional regulation is lacking. In this article, we present a simple phenomenological model that is able to incorporate the traditional view of gene expression within a framework with mechanical limits to transcription. By introducing a simple competition between mechanical arrest and relaxation copy number probability distributions collapse onto a shared universal curve under shifting and rescaling and a lower limit of intrinsic noise for any mean expression level is found.

First Detected Arrival of a Quantum Walker on an Infinite Line.

The first detection of a quantum particle on a graph is shown to depend sensitively on the distance ξ between the detector and initial location of the particle, and on the sampling time τ. Here, we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest-neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the large ξ limit, the small τ limit, and the power law decay with the attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width nonanalytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value that is distance independent.

Stability of two-species communities: Drift, environmental stochasticity, storage effect and selection.

The dynamics of two competing species in a finite size community is one of the most studied problems in population genetics and community ecology. Stochastic fluctuations lead, inevitably, to the extinction of one of the species, but the relevant timescale depends on the underlying dynamics. The persistence time of the community has been calculated both for neutral models, where the only driving force of the system is drift (demographic stochasticity), and for models with strong selection. Following recent analyses that stress the importance of environmental stochasticity in empirical systems, we present here a general theory of the persistence time of a two-species community where drift, environmental variations and time independent selective advantage are all taken into account.

Large Fluctuations for Spatial Diffusion of Cold Atoms.

We use a new approach to study the large fluctuations of a heavy-tailed system, where the standard large-deviations principle does not apply. Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example, spreading packets of particles. Mathematically, it concerns the exponential falloff of the density of thin-tailed systems. Here we investigate the spatial density P_{t}(x) of laser-cooled atoms, where at intermediate length scales the shape is fat tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed non-normalized infinite-covariant density approach. The small and large fluctuations give rise to a bifractal nature of the spreading packet. We derive general relations which extend our theory to a class of systems with multifractal moments.

Growth feedback as a basis for persister bistability.

A small fraction of cells in many bacterial populations, called persisters, are much less sensitive to antibiotic treatment than the majority. Persisters are in a dormant metabolic state, even while remaining genetically identical to the actively growing cells. Toxin and antitoxin modules in bacteria are believed to be one possible cause of persistence. A two-gene operon, HipBA, is one of many chromosomally encoded toxin and antitoxin modules in Escherichia coli and the HipA7 allelic variant was the first validated high-persistence mutant. Here, we present a stochastic model that can generate bistability of the HipBA system, via the reciprocal coupling of free HipA to the cellular growth rate. The actively growing state and the dormant state each correspond to a stable state of this model. Fluctuations enable transitions from one to the other. This model is fully in agreement with experimental data obtained with synthetic promoter constructs.

The effect of environmental stochasticity on species richness in neutral communities.

Environmental stochasticity is known to be a destabilizing factor, increasing abundance fluctuations and extinction rates of populations. However, the stability of a community may benefit from the differential response of species to environmental variations due to the storage effect. This paper provides a systematic and comprehensive discussion of these two contradicting tendencies, using the metacommunity version of the recently proposed time-average neutral model of biodiversity which incorporates environmental stochasticity and demographic noise and allows for extinction and speciation. We show that the incorporation of demographic noise into the model is essential to its applicability, yielding realistic behavior of the system when fitness variations are relatively weak. The dependence of species richness on the strength of environmental stochasticity changes sign when the correlation time of the environmental variations increases. This transition marks the point at which the storage effect no longer succeeds in stabilizing the community.

Large population solution of the stochastic Luria-Delbruck evolution model.

Luria and Delbrück introduced a very useful and subsequently widely adopted framework for quantitatively understanding the emergence of new cellular lineages. Here, we provide an analytical treatment of the fully stochastic version of the model, enabled by the fact that population sizes at the time of measurement are invariably very large and mutation rates are low. We show that the Lea-Coulson generating function describes the "inner solution," where the number of mutants is much smaller than the total population. We find that the corresponding distribution function interpolates between a monotonic decrease at relatively small populations, (compared with the inverse of the mutation probability), whereas it goes over to a Lévy α-stable distribution in the very large population limit. The moments are completely determined by the outer solution, and so are devoid of practical significance. The key to our solution is focusing on the fixed population size ensemble, which we show is very different from the fixed time ensemble due to the extreme variability in the evolutionary process.

Model for macroevolutionary dynamics.

The highly skewed distribution of species among genera, although challenging to macroevolutionists, provides an opportunity to understand the dynamics of diversification, including species formation, extinction, and morphological evolution. Early models were based on either the work by Yule [Yule GU (1925) Philos Trans R Soc Lond B Biol Sci 213:21-87], which neglects extinction, or a simple birth-death (speciation-extinction) process. Here, we extend the more recent development of a generic, neutral speciation-extinction (of species)-origination (of genera; SEO) model for macroevolutionary dynamics of taxon diversification. Simulations show that deviations from the homogeneity assumptions in the model can be detected in species-per-genus distributions. The SEO model fits observed species-per-genus distributions well for class-to-kingdom-sized taxonomic groups. The model's predictions for the appearance times (the time of the first existing species) of the taxonomic groups also approximately match estimates based on molecular inference and fossil records. Unlike estimates based on analyses of phylogenetic reconstruction, fitted extinction rates for large clades are close to speciation rates, consistent with high rates of species turnover and the relatively slow change in diversity observed in the fossil record. Finally, the SEO model generally supports the consistency of generic boundaries based on morphological differences between species and provides a comparator for rates of lineage splitting and morphological evolution.

Singularity screening in generic optical fields.

We revisit the widely studied subject of screening in optical fields by topological charges and show that screening does not depend on charge ordering. Instead, for an array of N charges, screening requires that the variance of the charge fluctuations be small compared to N. We show by means of explicit examples that, when this requirement is met, screening can be complete, even for a spatially random arrangement of charges. We derive a minimal screening constraint on the charge correlation function and show that it is this constraint that is met in practice, rather than the more stringent constraints previously assumed.

Deviations from Boltzmann-Gibbs Statistics in Confined Optical Lattices.

We investigate the semiclassical phase-space probability distribution P(x,p) of cold atoms in a Sisyphus cooling lattice with an additional harmonic confinement. We pose the question of whether this nonequilibrium steady state satisfies the equivalence of energy and probability. This equivalence is the foundation of Boltzmann-Gibbs and generalized thermostatic statistics, and a prerequisite for the description in terms of a temperature. At large energies, P(x,p) depends only on the Hamiltonian H(x,p) and the answer to the question is yes. In distinction to the Boltzmann-Gibbs state, the large-energy tails are power laws P(x,p)∝H(x,p)(-1/D), where D is related to the depth of the optical lattice. At intermediate energies, however, P(x,p) cannot be expressed as a function of the Hamiltonian and the equivalence between energy and probability breaks down. As a consequence the average potential and kinetic energy differ and no well-defined temperature can be assigned. The Boltzmann-Gibbs state is regained only in the limit of deep optical lattices. For strong confinement relative to the damping, we derive an explicit expression for the stationary phase-space distribution.

Emergence of structured communities through evolutionary dynamics.

Species-rich communities, in which many competing species coexist in a single trophic level, are quite frequent in nature, but pose a formidable theoretical challenge. In particular, it is known that complex competitive systems become unstable and unfeasible when the number of species is large. Recently, many studies have attributed the stability of natural communities to the structure of the interspecific interaction network, yet the nature of such structures and the underlying mechanisms responsible for them remain open questions. Here we introduce an evolutionary model, based on the generic Lotka-Volterra competitive framework, from which a stable, structured, diverse community emerges spontaneously. The modular structure of the competition matrix reflects the phylogeny of the community, in agreement with the hierarchial taxonomic classification. Closely related species tend to have stronger niche overlap and weaker fitness differences, as opposed to pairs of species from different modules. The competitive-relatedness hypothesis and the idea of emergent neutrality are discussed in the context of this evolutionary model.

Temporal fluctuation scaling in populations and communities.

Taylor's law, one of the most widely accepted generalizations in ecology, states that the variance of a population abundance time series scales as a power law of its mean. Here we reexamine this law and the empirical evidence presented in support of it. Specifically, we show that the exponent generally depends on the length of the time series, and its value reflects the combined effect of many underlying mechanisms. Moreover, sampling errors alone, when presented on a double logarithmic scale, are sufficient to produce an apparent power law. This raises questions regarding the usefulness of Taylor's law for understanding ecological processes. As an alternative approach, we focus on short-term fluctuations and derive a generic null model for the variance-to-mean ratio in population time series from a demographic model that incorporates the combined effects of demographic and environmental stochasticity. After comparing the predictions of the proposed null model with the fluctuations observed in empirical data sets, we suggest an alternative expression for fluctuation scaling in population time series. Analyzing population fluctuations as we have proposed here may provide new applied (e.g., estimation of species persistence times) and theoretical (e.g., the neutral theory of biodiversity) insights that can be derived from more generally available short-term monitoring data.