Aleksandra M. Walczak

Statistical mechanics for natural flocks of birds

Flocking is a typical example of emergent collective behavior, where interactions between individuals produce collective patterns on the large scale. Here we show how a quantitative microscopic theory for directional ordering in a flock can be derived directly from field data. We construct the minimally structured (maximum entropy) model consistent with experimental correlations in large flocks of starlings. The maximum entropy model shows that local, pairwise interactions between birds are sufficient to correctly predict the propagation of order throughout entire flocks of starlings, with no free parameters. We also find that the number of interacting neighbors is independent of flock density, confirming that interactions are ruled by topological rather than metric distance. Finally, by comparing flocks of different sizes, the model correctly accounts for the observed scale invariance of long-range correlations among the fluctuations in flight direction.

Maximum entropy models for antibody diversity

Recognition of pathogens relies on families of proteins showing great diversity. Here we construct maximum entropy models of the sequence repertoire, building on recent experiments that provide a nearly exhaustive sampling of the IgM sequences in zebrafish. These models are based solely on pairwise correlations between residue positions but correctly capture the higher order statistical properties of the repertoire. By exploiting the interpretation of these models as statistical physics problems, we make several predictions for the collective properties of the sequence ensemble: The distribution of sequences obeys Zipf’s law, the repertoire decomposes into several clusters, and there is a massive restriction of diversity because of the correlations. These predictions are completely inconsistent with models in which amino acid substitutions are made independently at each site and are in good agreement with the data. Our results suggest that antibody diversity is not limited by the sequences encoded in the genome and may reflect rapid adaptation to antigenic challenges. This approach should be applicable to the study of the global properties of other protein families.

Statistical inference of the generation probability of T-cell receptors from sequence repertoires

Stochastic rearrangement of germline V-, D-, and J-genes to create variable coding sequence for certain cell surface receptors is at the origin of immune system diversity. This process, known as “VDJ recombination”, is implemented via a series of stochastic molecular events involving gene choices and random nucleotide insertions between, and deletions from, genes. We use large sequence repertoires of the variable CDR3 region of human CD4+ T-cell receptor beta chains to infer the statistical properties of these basic biochemical events. Because any given CDR3 sequence can be produced in multiple ways, the probability distribution of hidden recombination events cannot be inferred directly from the observed sequences; we therefore develop a maximum likelihood inference method to achieve this end. To separate the properties of the molecular rearrangement mechanism from the effects of selection, we focus on nonproductive CDR3 sequences in T-cell DNA. We infer the joint distribution of the various generative events that occur when a new T-cell receptor gene is created. We find a rich picture of correlation (and absence thereof), providing insight into the molecular mechanisms involved. The generative event statistics are consistent between individuals, suggesting a universal biochemical process. Our probabilistic model predicts the generation probability of any specific CDR3 sequence by the primitive recombination process, allowing us to quantify the potential diversity of the T-cell repertoire and to understand why some sequences are shared between individuals. We argue that the use of formal statistical inference methods, of the kind presented in this paper, will be essential for quantitative understanding of the generation and evolution of diversity in the adaptive immune system.

Absolute rate theories of epigenetic stability

Spontaneous switching events in most characterized genetic switches are rare, resulting in extremely stable epigenetic properties. We show how simple arguments lead to theories of the rate of such events much like the absolute rate theory of chemical reactions corrected by a transmission factor. Both the probability of the rare cellular states that allow epigenetic escape and the transmission factor depend on the rates of DNA binding and unbinding events and on the rates of protein synthesis and degradation. Different mechanisms of escape from the stable attractors occur in the nonadiabatic, weakly adiabatic, and strictly adiabatic regimes, characterized by the relative values of those input rates.

Self-Consistent Proteomic Field Theory of Stochastic Gene Switches

We present a self-consistent field approximation approach to the problem of the genetic switch composed of two mutually repressing/activating genes. The protein and DNA state dynamics are treated stochastically and on an equal footing. In this approach the mean influence of the proteomic cloud created by one gene on the action of another is self-consistently computed. Within this approximation a broad range of stochastic genetic switches may be solved exactly in terms of finding the probability distribution and its moments. A much larger class of problems, such as genetic networks and cascades, also remain exactly solvable with this approximation. We discuss, in depth, certain specific types of basic switches used by biological systems and compare their behavior to the expectation for a deterministic switch.

The past, present, and future of immune repertoire biology - the rise of next-generation repertoire analysis

T and B cell repertoires are collections of lymphocytes, each characterized by its antigen-specific receptor. We review here classical technologies and analysis strategies developed to assess immunoglobulin (IG) and T cell receptor (TR) repertoire diversity, and describe recent advances in the field. First, we describe the broad range of available methodological tools developed in the past decades, each of which answering different questions and showing complementarity for progressive identification of the level of repertoire alterations: global overview of the diversity by flow cytometry, IG repertoire descriptions at the protein level for the identification of IG reactivities, IG/TR CDR3 spectratyping strategies, and related molecular quantification or dynamics of T/B cell differentiation. Additionally, we introduce the recent technological advances in molecular biology tools allowing deeper analysis of IG/TR diversity by next-generation sequencing (NGS), offering systematic and comprehensive sequencing of IG/TR transcripts in a short amount of time. NGS provides several angles of analysis such as clonotype frequency, CDR3 diversity, CDR3 sequence analysis, V allele identification with a quantitative dimension, therefore requiring high-throughput analysis tools development. In this line, we discuss the recent efforts made for nomenclature standardization and ontology development. We then present the variety of available statistical analysis and modeling approaches developed with regards to the various levels of diversity analysis, and reveal the increasing sophistication of those modeling approaches. To conclude, we provide some examples of recent mathematical modeling strategies and perspectives that illustrate the active rise of a “next-generation” of repertoire analysis.

Social interactions dominate speed control in poising natural flocks near criticality.

Significance The coherent flight of bird flocks is one of nature’s most impressive aerial displays. Beyond the fact that thousands of birds fly, on average, with the same velocity, quantitative observations show that small deviations of individual birds from this average are correlated across the entire flock. By learning minimally structured models from field data, we show that these long-ranged correlations are consistent with local interactions among neighboring birds, but only because the parameters of the flock are tuned to special values, mathematically equivalent to a critical point in statistical mechanics. Being in this critical regime allows information to propagate almost without loss throughout the flock, while keeping the variance of individual velocities small. Flocks of birds exhibit a remarkable degree of coordination and collective response. It is not just that thousands of individuals fly, on average, in the same direction and at the same speed, but that even the fluctuations around the mean velocity are correlated over long distances. Quantitative measurements on flocks of starlings, in particular, show that these fluctuations are scale-free, with effective correlation lengths proportional to the linear size of the flock. Here we construct models for the joint distribution of velocities in the flock that reproduce the observed local correlations between individuals and their neighbors, as well as the variance of flight speeds across individuals, but otherwise have as little structure as possible. These minimally structured or maximum entropy models provide quantitative, parameter-free predictions for the spread of correlations throughout the flock, and these are in excellent agreement with the data. These models are mathematically equivalent to statistical physics models for ordering in magnets, and the correct prediction of scale-free correlations arises because the parameters—completely determined by the data—are in the critical regime. In biological terms, criticality allows the flock to achieve maximal correlation across long distances with limited speed fluctuations.

Live Imaging of Bicoid-Dependent Transcription in Drosophila Embryos

The early Drosophila embryo is an ideal model to understand the transcriptional regulation of well-defined patterns of gene expression in a developing organism. In this system, snapshots of transcription measurements obtained by RNA FISH on fixed samples cannot provide the temporal resolution needed to distinguish spatial heterogeneity from inherent noise. Here, we used the MS2-MCP system to visualize in living embryos nascent transcripts expressed from the canonical hunchback (hb) promoter under the control of Bicoid (Bcd). The hb-MS2 reporter is expressed as synchronously as endogenous hb in the anterior half of the embryo, but unlike hb it is also active in the posterior, though more heterogeneously and more transiently than in the anterior. The length and intensity of active transcription periods in the anterior are strongly reduced in absence of Bcd, whereas posterior ones are mostly Bcd independent. This posterior noisy signal decreases progressively through nuclear divisions, so that the MS2 reporter expression mimics the known anterior hb pattern at cellularization. We propose that the establishment of the hb pattern relies on Bcd-dependent lengthening of transcriptional activity periods in the anterior and may require two distinct repression mechanisms in the posterior.

Quantifying selection in immune receptor repertoires

Significance The immune system defends against pathogens through a diverse population of T cells that display different antigen recognition surface receptor proteins. Receptor diversity is produced by an initial random gene recombination process followed by selection for a desirable range of peptide binding. Although recombination is well-understood, selection has not been quantitatively characterized. By combining high-throughput sequencing data with modeling, we quantify the selection pressure that shapes functional repertoires. Selection is found to vary little between individuals or between naive and memory repertoires. It reinforces the biases of the recombination process, meaning that sequences more likely to be produced are also more likely to pass selection. The model accounts for public sequences shared between individuals as resulting from pure chance. The efficient recognition of pathogens by the adaptive immune system relies on the diversity of receptors displayed at the surface of immune cells. T-cell receptor diversity results from an initial random DNA editing process, called VDJ recombination, followed by functional selection of cells according to the interaction of their surface receptors with self and foreign antigenic peptides. Using high-throughput sequence data from the β-chain of human T-cell receptors, we infer factors that quantify the overall effect of selection on the elements of receptor sequence composition: the V and J gene choice and the length and amino acid composition of the variable region. We find a significant correlation between biases induced by VDJ recombination and our inferred selection factors together with a reduction of diversity during selection. Both effects suggest that natural selection acting on the recombination process has anticipated the selection pressures experienced during somatic evolution. The inferred selection factors differ little between donors or between naive and memory repertoires. The number of sequences shared between donors is well-predicted by our model, indicating a stochastic origin of such public sequences. Our approach is based on a probabilistic maximum likelihood method, which is necessary to disentangle the effects of selection from biases inherent in the recombination process.

Inferring processes underlying B-cell repertoire diversity

We quantify the VDJ recombination and somatic hypermutation processes in human B cells using probabilistic inference methods on high-throughput DNA sequence repertoires of human B-cell receptor heavy chains. Our analysis captures the statistical properties of the naive repertoire, first after its initial generation via VDJ recombination and then after selection for functionality. We also infer statistical properties of the somatic hypermutation machinery (exclusive of subsequent effects of selection). Our main results are the following: the B-cell repertoire is substantially more diverse than T-cell repertoires, owing to longer junctional insertions; sequences that pass initial selection are distinguished by having a higher probability of being generated in a VDJ recombination event; somatic hypermutations have a non-uniform distribution along the V gene that is well explained by an independent site model for the sequence context around the hypermutation site.