TY - JOUR
AB - We study the Fokker-Planck equation derived in the large system limit of the Markovian process describing the dynamics of quantitative traits. The Fokker-Planck equation is posed on a bounded domain and its transport and diffusion coefficients vanish on the domain's boundary. We first argue that, despite this degeneracy, the standard no-flux boundary condition is valid. We derive the weak formulation of the problem and prove the existence and uniqueness of its solutions by constructing the corresponding contraction semigroup on a suitable function space. Then, we prove that for the parameter regime with high enough mutation rate the problem exhibits a positive spectral gap, which implies exponential convergence to equilibrium.Next, we provide a simple derivation of the so-called Dynamic Maximum Entropy (DynMaxEnt) method for approximation of observables (moments) of the Fokker-Planck solution, which can be interpreted as a nonlinear Galerkin approximation. The limited applicability of the DynMaxEnt method inspires us to introduce its modified version that is valid for the whole range of admissible parameters. Finally, we present several numerical experiments to demonstrate the performance of both the original and modified DynMaxEnt methods. We observe that in the parameter regimes where both methods are valid, the modified one exhibits slightly better approximation properties compared to the original one.
AU - Bodova, Katarina
AU - Haskovec, Jan
AU - Markowich, Peter
ID - 607
JF - Physica D: Nonlinear Phenomena
TI - Well posedness and maximum entropy approximation for the dynamics of quantitative traits
VL - 376-377
ER -
TY - JOUR
AB - Which properties of metabolic networks can be derived solely from stoichiometry? Predictive results have been obtained by flux balance analysis (FBA), by postulating that cells set metabolic fluxes to maximize growth rate. Here we consider a generalization of FBA to single-cell level using maximum entropy modeling, which we extend and test experimentally. Specifically, we define for Escherichia coli metabolism a flux distribution that yields the experimental growth rate: the model, containing FBA as a limit, provides a better match to measured fluxes and it makes a wide range of predictions: on flux variability, regulation, and correlations; on the relative importance of stoichiometry vs. optimization; on scaling relations for growth rate distributions. We validate the latter here with single-cell data at different sub-inhibitory antibiotic concentrations. The model quantifies growth optimization as emerging from the interplay of competitive dynamics in the population and regulation of metabolism at the level of single cells.
AU - De Martino, Daniele
AU - Mc, Andersson Anna
AU - Bergmiller, Tobias
AU - Guet, Calin C
AU - Tkacik, Gasper
ID - 161
IS - 1
JF - Nature Communications
TI - Statistical mechanics for metabolic networks during steady state growth
VL - 9
ER -
TY - JOUR
AB - Recent developments in automated tracking allow uninterrupted, high-resolution recording of animal trajectories, sometimes coupled with the identification of stereotyped changes of body pose or other behaviors of interest. Analysis and interpretation of such data represents a challenge: the timing of animal behaviors may be stochastic and modulated by kinematic variables, by the interaction with the environment or with the conspecifics within the animal group, and dependent on internal cognitive or behavioral state of the individual. Existing models for collective motion typically fail to incorporate the discrete, stochastic, and internal-state-dependent aspects of behavior, while models focusing on individual animal behavior typically ignore the spatial aspects of the problem. Here we propose a probabilistic modeling framework to address this gap. Each animal can switch stochastically between different behavioral states, with each state resulting in a possibly different law of motion through space. Switching rates for behavioral transitions can depend in a very general way, which we seek to identify from data, on the effects of the environment as well as the interaction between the animals. We represent the switching dynamics as a Generalized Linear Model and show that: (i) forward simulation of multiple interacting animals is possible using a variant of the Gillespie’s Stochastic Simulation Algorithm; (ii) formulated properly, the maximum likelihood inference of switching rate functions is tractably solvable by gradient descent; (iii) model selection can be used to identify factors that modulate behavioral state switching and to appropriately adjust model complexity to data. To illustrate our framework, we apply it to two synthetic models of animal motion and to real zebrafish tracking data.
AU - Bod’Ová, Katarína
AU - Mitchell, Gabriel
AU - Harpaz, Roy
AU - Schneidman, Elad
AU - Tkacik, Gasper
ID - 406
IS - 3
JF - PLoS One
TI - Probabilistic models of individual and collective animal behavior
VL - 13
ER -
TY - JOUR
AB - Temperate bacteriophages integrate in bacterial genomes as prophages and represent an important source of genetic variation for bacterial evolution, frequently transmitting fitness-augmenting genes such as toxins responsible for virulence of major pathogens. However, only a fraction of bacteriophage infections are lysogenic and lead to prophage acquisition, whereas the majority are lytic and kill the infected bacteria. Unless able to discriminate lytic from lysogenic infections, mechanisms of immunity to bacteriophages are expected to act as a double-edged sword and increase the odds of survival at the cost of depriving bacteria of potentially beneficial prophages. We show that although restriction-modification systems as mechanisms of innate immunity prevent both lytic and lysogenic infections indiscriminately in individual bacteria, they increase the number of prophage-acquiring individuals at the population level. We find that this counterintuitive result is a consequence of phage-host population dynamics, in which restriction-modification systems delay infection onset until bacteria reach densities at which the probability of lysogeny increases. These results underscore the importance of population-level dynamics as a key factor modulating costs and benefits of immunity to temperate bacteriophages
AU - Pleska, Maros
AU - Lang, Moritz
AU - Refardt, Dominik
AU - Levin, Bruce
AU - Guet, Calin C
ID - 457
IS - 2
JF - Nature Ecology and Evolution
TI - Phage-host population dynamics promotes prophage acquisition in bacteria with innate immunity
VL - 2
ER -
TY - JOUR
AB - The hanging-drop network (HDN) is a technology platform based on a completely open microfluidic network at the bottom of an inverted, surface-patterned substrate. The platform is predominantly used for the formation, culturing, and interaction of self-assembled spherical microtissues (spheroids) under precisely controlled flow conditions. Here, we describe design, fabrication, and operation of microfluidic hanging-drop networks.
AU - Misun, Patrick
AU - Birchler, Axel
AU - Lang, Moritz
AU - Hierlemann, Andreas
AU - Frey, Olivier
ID - 305
JF - Methods in Molecular Biology
TI - Fabrication and operation of microfluidic hanging drop networks
VL - 1771
ER -
TY - JOUR
AB - A cornerstone of statistical inference, the maximum entropy framework is being increasingly applied to construct descriptive and predictive models of biological systems, especially complex biological networks, from large experimental data sets. Both its broad applicability and the success it obtained in different contexts hinge upon its conceptual simplicity and mathematical soundness. Here we try to concisely review the basic elements of the maximum entropy principle, starting from the notion of ‘entropy’, and describe its usefulness for the analysis of biological systems. As examples, we focus specifically on the problem of reconstructing gene interaction networks from expression data and on recent work attempting to expand our system-level understanding of bacterial metabolism. Finally, we highlight some extensions and potential limitations of the maximum entropy approach, and point to more recent developments that are likely to play a key role in the upcoming challenges of extracting structures and information from increasingly rich, high-throughput biological data.
AU - De Martino, Andrea
AU - De Martino, Daniele
ID - 306
IS - 4
JF - Heliyon
TI - An introduction to the maximum entropy approach and its application to inference problems in biology
VL - 4
ER -
TY - JOUR
AB - Correlations in sensory neural networks have both extrinsic and intrinsic origins. Extrinsic or stimulus correlations arise from shared inputs to the network and, thus, depend strongly on the stimulus ensemble. Intrinsic or noise correlations reflect biophysical mechanisms of interactions between neurons, which are expected to be robust to changes in the stimulus ensemble. Despite the importance of this distinction for understanding how sensory networks encode information collectively, no method exists to reliably separate intrinsic interactions from extrinsic correlations in neural activity data, limiting our ability to build predictive models of the network response. In this paper we introduce a general strategy to infer population models of interacting neurons that collectively encode stimulus information. The key to disentangling intrinsic from extrinsic correlations is to infer the couplings between neurons separately from the encoding model and to combine the two using corrections calculated in a mean-field approximation. We demonstrate the effectiveness of this approach in retinal recordings. The same coupling network is inferred from responses to radically different stimulus ensembles, showing that these couplings indeed reflect stimulus-independent interactions between neurons. The inferred model predicts accurately the collective response of retinal ganglion cell populations as a function of the stimulus.
AU - Ferrari, Ulisse
AU - Deny, Stephane
AU - Chalk, Matthew J
AU - Tkacik, Gasper
AU - Marre, Olivier
AU - Mora, Thierry
ID - 31
IS - 4
JF - Physical Review E
SN - 24700045
TI - Separating intrinsic interactions from extrinsic correlations in a network of sensory neurons
VL - 98
ER -
TY - JOUR
AB - Self-incompatibility (SI) is a genetically based recognition system that functions to prevent self-fertilization and mating among related plants. An enduring puzzle in SI is how the high diversity observed in nature arises and is maintained. Based on the underlying recognition mechanism, SI can be classified into two main groups: self- and non-self recognition. Most work has focused on diversification within self-recognition systems despite expected differences between the two groups in the evolutionary pathways and outcomes of diversification. Here, we use a deterministic population genetic model and stochastic simulations to investigate how novel S-haplotypes evolve in a gametophytic non-self recognition (SRNase/S Locus F-box (SLF)) SI system. For this model the pathways for diversification involve either the maintenance or breakdown of SI and can vary in the order of mutations of the female (SRNase) and male (SLF) components. We show analytically that diversification can occur with high inbreeding depression and self-pollination, but this varies with evolutionary pathway and level of completeness (which determines the number of potential mating partners in the population), and in general is more likely for lower haplotype number. The conditions for diversification are broader in stochastic simulations of finite population size. However, the number of haplotypes observed under high inbreeding and moderate to high self-pollination is less than that commonly observed in nature. Diversification was observed through pathways that maintain SI as well as through self-compatible intermediates. Yet the lifespan of diversified haplotypes was sensitive to their level of completeness. By examining diversification in a non-self recognition SI system, this model extends our understanding of the evolution and maintenance of haplotype diversity observed in a self recognition system common in flowering plants.
AU - Bodova, Katarina
AU - Priklopil, Tadeas
AU - Field, David
AU - Barton, Nicholas H
AU - Pickup, Melinda
ID - 316
IS - 3
JF - Genetics
TI - Evolutionary pathways for the generation of new self-incompatibility haplotypes in a non-self recognition system
VL - 209
ER -
TY - JOUR
AB - Gene regulatory networks evolve through rewiring of individual components—that is, through changes in regulatory connections. However, the mechanistic basis of regulatory rewiring is poorly understood. Using a canonical gene regulatory system, we quantify the properties of transcription factors that determine the evolutionary potential for rewiring of regulatory connections: robustness, tunability and evolvability. In vivo repression measurements of two repressors at mutated operator sites reveal their contrasting evolutionary potential: while robustness and evolvability were positively correlated, both were in trade-off with tunability. Epistatic interactions between adjacent operators alleviated this trade-off. A thermodynamic model explains how the differences in robustness, tunability and evolvability arise from biophysical characteristics of repressor–DNA binding. The model also uncovers that the energy matrix, which describes how mutations affect repressor–DNA binding, encodes crucial information about the evolutionary potential of a repressor. The biophysical determinants of evolutionary potential for regulatory rewiring constitute a mechanistic framework for understanding network evolution.
AU - Igler, Claudia
AU - Lagator, Mato
AU - Tkacik, Gasper
AU - Bollback, Jonathan P
AU - Guet, Calin C
ID - 67
IS - 10
JF - Nature Ecology and Evolution
TI - Evolutionary potential of transcription factors for gene regulatory rewiring
VL - 2
ER -
TY - DATA
AB - Mean repression values and standard error of the mean are given for all operator mutant libraries.
AU - Igler, Claudia
AU - Lagator, Mato
AU - Tkacik, Gasper
AU - Bollback, Jonathan P
AU - Guet, Calin C
ID - 5585
TI - Data for the paper Evolutionary potential of transcription factors for gene regulatory rewiring
ER -
TY - JOUR
AB - The resolution of a linear system with positive integer variables is a basic yet difficult computational problem with many applications. We consider sparse uncorrelated random systems parametrised by the density c and the ratio α=N/M between number of variables N and number of constraints M. By means of ensemble calculations we show that the space of feasible solutions endows a Van-Der-Waals phase diagram in the plane (c, α). We give numerical evidence that the associated computational problems become more difficult across the critical point and in particular in the coexistence region.
AU - Colabrese, Simona
AU - De Martino, Daniele
AU - Leuzzi, Luca
AU - Marinari, Enzo
ID - 823
IS - 9
JF - Journal of Statistical Mechanics: Theory and Experiment
SN - 17425468
TI - Phase transitions in integer linear problems
VL - 2017
ER -
TY - JOUR
AB - The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness-of-fit. Here, we propose an exact goodness-of-fit test for the finite-lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness-of-fit testing using a Monte Carlo approach. However, this beautiful theory has fallen short of its promise for applications, because finding a Markov basis is usually computationally intractable. We develop a Monte Carlo method for exact goodness-of-fit testing for the Ising model which avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.
AU - Martin Del Campo Sanchez, Abraham
AU - Cepeda Humerez, Sarah A
AU - Uhler, Caroline
ID - 2016
IS - 2
JF - Scandinavian Journal of Statistics
SN - 03036898
TI - Exact goodness-of-fit testing for the Ising model
VL - 44
ER -
TY - JOUR
AB - Advances in multi-unit recordings pave the way for statistical modeling of activity patterns in large neural populations. Recent studies have shown that the summed activity of all neurons strongly shapes the population response. A separate recent finding has been that neural populations also exhibit criticality, an anomalously large dynamic range for the probabilities of different population activity patterns. Motivated by these two observations, we introduce a class of probabilistic models which takes into account the prior knowledge that the neural population could be globally coupled and close to critical. These models consist of an energy function which parametrizes interactions between small groups of neurons, and an arbitrary positive, strictly increasing, and twice differentiable function which maps the energy of a population pattern to its probability. We show that: 1) augmenting a pairwise Ising model with a nonlinearity yields an accurate description of the activity of retinal ganglion cells which outperforms previous models based on the summed activity of neurons; 2) prior knowledge that the population is critical translates to prior expectations about the shape of the nonlinearity; 3) the nonlinearity admits an interpretation in terms of a continuous latent variable globally coupling the system whose distribution we can infer from data. Our method is independent of the underlying system’s state space; hence, it can be applied to other systems such as natural scenes or amino acid sequences of proteins which are also known to exhibit criticality.
AU - Humplik, Jan
AU - Tkacik, Gasper
ID - 720
IS - 9
JF - PLoS Computational Biology
SN - 1553734X
TI - Probabilistic models for neural populations that naturally capture global coupling and criticality
VL - 13
ER -
TY - JOUR
AB - Individual computations and social interactions underlying collective behavior in groups of animals are of great ethological, behavioral, and theoretical interest. While complex individual behaviors have successfully been parsed into small dictionaries of stereotyped behavioral modes, studies of collective behavior largely ignored these findings; instead, their focus was on inferring single, mode-independent social interaction rules that reproduced macroscopic and often qualitative features of group behavior. Here, we bring these two approaches together to predict individual swimming patterns of adult zebrafish in a group. We show that fish alternate between an “active” mode, in which they are sensitive to the swimming patterns of conspecifics, and a “passive” mode, where they ignore them. Using a model that accounts for these two modes explicitly, we predict behaviors of individual fish with high accuracy, outperforming previous approaches that assumed a single continuous computation by individuals and simple metric or topological weighing of neighbors’ behavior. At the group level, switching between active and passive modes is uncorrelated among fish, but correlated directional swimming behavior still emerges. Our quantitative approach for studying complex, multi-modal individual behavior jointly with emergent group behavior is readily extensible to additional behavioral modes and their neural correlates as well as to other species.
AU - Harpaz, Roy
AU - Tkacik, Gasper
AU - Schneidman, Elad
ID - 725
IS - 38
JF - PNAS
SN - 00278424
TI - Discrete modes of social information processing predict individual behavior of fish in a group
VL - 114
ER -
TY - JOUR
AB - Neural responses are highly structured, with population activity restricted to a small subset of the astronomical range of possible activity patterns. Characterizing these statistical regularities is important for understanding circuit computation, but challenging in practice. Here we review recent approaches based on the maximum entropy principle used for quantifying collective behavior in neural activity. We highlight recent models that capture population-level statistics of neural data, yielding insights into the organization of the neural code and its biological substrate. Furthermore, the MaxEnt framework provides a general recipe for constructing surrogate ensembles that preserve aspects of the data, but are otherwise maximally unstructured. This idea can be used to generate a hierarchy of controls against which rigorous statistical tests are possible.
AU - Savin, Cristina
AU - Tkacik, Gasper
ID - 730
JF - Current Opinion in Neurobiology
SN - 09594388
TI - Maximum entropy models as a tool for building precise neural controls
VL - 46
ER -
TY - JOUR
AB - Cell-cell contact formation constitutes an essential step in evolution, leading to the differentiation of specialized cell types. However, remarkably little is known about whether and how the interplay between contact formation and fate specification affects development. Here, we identify a positive feedback loop between cell-cell contact duration, morphogen signaling, and mesendoderm cell-fate specification during zebrafish gastrulation. We show that long-lasting cell-cell contacts enhance the competence of prechordal plate (ppl) progenitor cells to respond to Nodal signaling, required for ppl cell-fate specification. We further show that Nodal signaling promotes ppl cell-cell contact duration, generating a positive feedback loop between ppl cell-cell contact duration and cell-fate specification. Finally, by combining mathematical modeling and experimentation, we show that this feedback determines whether anterior axial mesendoderm cells become ppl or, instead, turn into endoderm. Thus, the interdependent activities of cell-cell signaling and contact formation control fate diversification within the developing embryo.
AU - Barone, Vanessa
AU - Lang, Moritz
AU - Krens, Gabriel
AU - Pradhan, Saurabh
AU - Shamipour, Shayan
AU - Sako, Keisuke
AU - Sikora, Mateusz K
AU - Guet, Calin C
AU - Heisenberg, Carl-Philipp J
ID - 735
IS - 2
JF - Developmental Cell
SN - 15345807
TI - An effective feedback loop between cell-cell contact duration and morphogen signaling determines cell fate
VL - 43
ER -
TY - JOUR
AB - In this work maximum entropy distributions in the space of steady states of metabolic networks are considered upon constraining the first and second moments of the growth rate. Coexistence of fast and slow phenotypes, with bimodal flux distributions, emerges upon considering control on the average growth (optimization) and its fluctuations (heterogeneity). This is applied to the carbon catabolic core of Escherichia coli where it quantifies the metabolic activity of slow growing phenotypes and it provides a quantitative map with metabolic fluxes, opening the possibility to detect coexistence from flux data. A preliminary analysis on data for E. coli cultures in standard conditions shows degeneracy for the inferred parameters that extend in the coexistence region.
AU - De Martino, Daniele
ID - 548
IS - 6
JF - Physical Review E
SN - 24700045
TI - Maximum entropy modeling of metabolic networks by constraining growth-rate moments predicts coexistence of phenotypes
VL - 96
ER -
TY - DATA
AB - This repository contains the data collected for the manuscript "Biased partitioning of the multi-drug efflux pump AcrAB-TolC underlies long-lived phenotypic heterogeneity".
The data is compressed into a single archive. Within the archive, different folders correspond to figures of the main text and the SI of the related publication.
Data is saved as plain text, with each folder containing a separate readme file describing the format. Typically, the data is from fluorescence microscopy measurements of single cells growing in a microfluidic "mother machine" device, and consists of relevant values (primarily arbitrary unit or normalized fluorescence measurements, and division times / growth rates) after raw microscopy images have been processed, segmented, and their features extracted, as described in the methods section of the related publication.
AU - Bergmiller, Tobias
AU - Andersson, Anna M
AU - Tomasek, Kathrin
AU - Balleza, Enrique
AU - Kiviet, Daniel
AU - Hauschild, Robert
AU - Tkacik, Gasper
AU - Guet, Calin C
ID - 5560
KW - single cell microscopy
KW - mother machine microfluidic device
KW - AcrAB-TolC pump
KW - multi-drug efflux
KW - Escherichia coli
TI - Biased partitioning of the multi-drug efflux pump AcrAB-TolC underlies long-lived phenotypic heterogeneity
ER -
TY - DATA
AB - This data was collected as part of the study [1]. It consists of preprocessed multi-electrode array recording from 160 salamander retinal ganglion cells responding to 297 repeats of a 19 s natural movie. The data is available in two formats: (1) a .mat file containing an array with dimensions “number of repeats” x “number of neurons” x “time in a repeat”; (2) a zipped .txt file containing the same data represented as an array with dimensions “number of neurons” x “number of samples”, where the number of samples is equal to the product of the number of repeats and timebins within a repeat. The time dimension is divided into 20 ms time windows, and the array is binary indicating whether a given cell elicited at least one spike in a given time window during a particular repeat. See the reference below for details regarding collection and preprocessing:
[1] Tkačik G, Marre O, Amodei D, Schneidman E, Bialek W, Berry MJ II. Searching for Collective Behavior in a Large Network of Sensory Neurons. PLoS Comput Biol. 2014;10(1):e1003408.
AU - Marre, Olivier
AU - Tkacik, Gasper
AU - Amodei, Dario
AU - Schneidman, Elad
AU - Bialek, William
AU - Berry, Michael
ID - 5562
KW - multi-electrode recording
KW - retinal ganglion cells
TI - Multi-electrode array recording from salamander retinal ganglion cells
ER -
TY - JOUR
AB - Bacteria in groups vary individually, and interact with other bacteria and the environment to produce population-level patterns of gene expression. Investigating such behavior in detail requires measuring and controlling populations at the single-cell level alongside precisely specified interactions and environmental characteristics. Here we present an automated, programmable platform that combines image-based gene expression and growth measurements with on-line optogenetic expression control for hundreds of individual Escherichia coli cells over days, in a dynamically adjustable environment. This integrated platform broadly enables experiments that bridge individual and population behaviors. We demonstrate: (i) population structuring by independent closed-loop control of gene expression in many individual cells, (ii) cell-cell variation control during antibiotic perturbation, (iii) hybrid bio-digital circuits in single cells, and freely specifiable digital communication between individual bacteria. These examples showcase the potential for real-time integration of theoretical models with measurement and control of many individual cells to investigate and engineer microbial population behavior.
AU - Chait, Remy P
AU - Ruess, Jakob
AU - Bergmiller, Tobias
AU - Tkacik, Gasper
AU - Guet, Calin C
ID - 613
IS - 1
JF - Nature Communications
SN - 20411723
TI - Shaping bacterial population behavior through computer interfaced control of individual cells
VL - 8
ER -