Evidence Accumulation and Change Rate Inference in Dynamic Environments.

In a constantly changing world, animals must account for environmental volatility when making decisions. To appropriately discount older, irrelevant information, they need to learn the rate at which the environment changes. We develop an ideal observer model capable of inferring the present state of the environment along with its rate of change. Key to this computation is an update of the posterior probability of all possible change point counts. This computation can be challenging, as the number of possibilities grows rapidly with time. However, we show how the computations can be simplified in the continuum limit by a moment closure approximation. The resulting low-dimensional system can be used to infer the environmental state and change rate with accuracy comparable to the ideal observer. The approximate computations can be performed by a neural network model via a rate-correlation-based plasticity rule. We thus show how optimal observers accumulate evidence in changing environments and map this computation to reduced models that perform inference using plausible neural mechanisms.

Effects of cell cycle noise on excitable gene circuits.

We assess the impact of cell cycle noise on gene circuit dynamics. For bistable genetic switches and excitable circuits, we find that transitions between metastable states most likely occur just after cell division and that this concentration effect intensifies in the presence of transcriptional delay. We explain this concentration effect with a three-states stochastic model. For genetic oscillators, we quantify the temporal correlations between daughter cells induced by cell division. Temporal correlations must be captured properly in order to accurately quantify noise sources within gene networks.

Sources of Variability in a Synthetic Gene Oscillator.

Synthetic gene oscillators are small, engineered genetic circuits that produce periodic variations in target protein expression. Like other gene circuits, synthetic gene oscillators are noisy and exhibit fluctuations in amplitude and period. Understanding the origins of such variability is key to building predictive models that can guide the rational design of synthetic circuits. Here, we developed a method for determining the impact of different sources of noise in genetic oscillators by measuring the variability in oscillation amplitude and correlations between sister cells. We first used a combination of microfluidic devices and time-lapse fluorescence microscopy to track oscillations in cell lineages across many generations. We found that oscillation amplitude exhibited high cell-to-cell variability, while sister cells remained strongly correlated for many minutes after cell division. To understand how such variability arises, we constructed a computational model that identified the impact of various noise sources across the lineage of an initial cell. When each source of noise was appropriately tuned the model reproduced the experimentally observed amplitude variability and correlations, and accurately predicted outcomes under novel experimental conditions. Our combination of computational modeling and time-lapse data analysis provides a general way to examine the sources of variability in dynamic gene circuits.

Networks that learn the precise timing of event sequences.

Neuronal circuits can learn and replay firing patterns evoked by sequences of sensory stimuli. After training, a brief cue can trigger a spatiotemporal pattern of neural activity similar to that evoked by a learned stimulus sequence. Network models show that such sequence learning can occur through the shaping of feedforward excitatory connectivity via long term plasticity. Previous models describe how event order can be learned, but they typically do not explain how precise timing can be recalled. We propose a mechanism for learning both the order and precise timing of event sequences. In our recurrent network model, long term plasticity leads to the learning of the sequence, while short term facilitation enables temporally precise replay of events. Learned synaptic weights between populations determine the time necessary for one population to activate another. Long term plasticity adjusts these weights so that the trained event times are matched during playback. While we chose short term facilitation as a time-tracking process, we also demonstrate that other mechanisms, such as spike rate adaptation, can fulfill this role. We also analyze the impact of trial-to-trial variability, showing how observational errors as well as neuronal noise result in variability in learned event times. The dynamics of the playback process determines how stochasticity is inherited in learned sequence timings. Future experiments that characterize such variability can therefore shed light on the neural mechanisms of sequence learning.

Steady state analysis of Boolean molecular network models via model reduction and computational algebra.

BACKGROUND: A key problem in the analysis of mathematical models of molecular networks is the determination of their steady states. The present paper addresses this problem for Boolean network models, an increasingly popular modeling paradigm for networks lacking detailed kinetic information. For small models, the problem can be solved by exhaustive enumeration of all state transitions. But for larger models this is not feasible, since the size of the phase space grows exponentially with the dimension of the network. The dimension of published models is growing to over 100, so that efficient methods for steady state determination are essential. Several methods have been proposed for large networks, some of them heuristic. While these methods represent a substantial improvement in scalability over exhaustive enumeration, the problem for large networks is still unsolved in general. RESULTS: This paper presents an algorithm that consists of two main parts. The first is a graph theoretic reduction of the wiring diagram of the network, while preserving all information about steady states. The second part formulates the determination of all steady states of a Boolean network as a problem of finding all solutions to a system of polynomial equations over the finite number system with two elements. This problem can be solved with existing computer algebra software. This algorithm compares favorably with several existing algorithms for steady state determination. One advantage is that it is not heuristic or reliant on sampling, but rather determines algorithmically and exactly all steady states of a Boolean network. The code for the algorithm, as well as the test suite of benchmark networks, is available upon request from the corresponding author. CONCLUSIONS: The algorithm presented in this paper reliably determines all steady states of sparse Boolean networks with up to 1000 nodes. The algorithm is effective at analyzing virtually all published models even those of moderate connectivity. The problem for large Boolean networks with high average connectivity remains an open problem.

Piecewise linear and Boolean models of chemical reaction networks.

Models of biochemical networks are frequently complex and high-dimensional. Reduction methods that preserve important dynamical properties are therefore essential for their study. Interactions in biochemical networks are frequently modeled using Hill functions ([Formula: see text]). Reduced ODEs and Boolean approximations of such model networks have been studied extensively when the exponent [Formula: see text] is large. However, while the case of small constant [Formula: see text] appears in practice, it is not well understood. We provide a mathematical analysis of this limit and show that a reduction to a set of piecewise linear ODEs and Boolean networks can be mathematically justified. The piecewise linear systems have closed-form solutions that closely track those of the fully nonlinear model. The simpler, Boolean network can be used to study the qualitative behavior of the original system. We justify the reduction using geometric singular perturbation theory and compact convergence, and illustrate the results in network models of a toggle switch and an oscillator.

Modular Control of Biological Networks.

The concept of control is central to understanding and applications of biological network models. Some of their key structural features relate to control functions, through gene regulation, signaling, or metabolic mechanisms, and computational models need to encode these. Applications of models often focus on model-based control, such as in biomedicine or metabolic engineering. This paper presents an approach to model-based control that exploits two common features of biological networks, namely their modular structure and canalizing features of their regulatory mechanisms. The paper focuses on intracellular regulatory networks, represented by Boolean network models. A main result of this paper is that control strategies can be identified by focusing on one module at a time. This paper also presents a criterion based on canalizing features of the regulatory rules to identify modules that do not contribute to network control and can be excluded. For even moderately sized networks, finding global control inputs is computationally very challenging. The modular approach presented here leads to a highly efficient approach to solving this problem. This approach is applied to a published Boolean network model of blood cancer large granular lymphocyte (T-LGL) leukemia to identify a minimal control set that achieves a desired control objective.

The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes.

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can--in principle--be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode the full combinatorial data of a neural code. Our main finding is that these objects can be expressed in a "canonical form" that directly translates to a minimal description of the receptive field structure intrinsic to the code. We also find connections to Stanley-Reisner rings, and use ideas similar to those in the theory of monomial ideals to obtain an algorithm for computing the primary decomposition of pseudo-monomial ideals. This allows us to algorithmically extract the canonical form associated to any neural code, providing the groundwork for inferring stimulus space features from neural activity alone.

Modularity of biological systems: a link between structure and function.

This paper addresses two topics in systems biology, the hypothesis that biological systems are modular and the problem of relating structure and function of biological systems. The focus here is on gene regulatory networks, represented by Boolean network models, a commonly used tool. Most of the research on gene regulatory network modularity has focused on network structure, typically represented through either directed or undirected graphs. But since gene regulation is a highly dynamic process as it determines the function of cells over time, it is natural to consider functional modularity as well. One of the main results is that the structural decomposition of a network into modules induces an analogous decomposition of the dynamic structure, exhibiting a strong relationship between network structure and function. An extensive simulation study provides evidence for the hypothesis that modularity might have evolved to increase phenotypic complexity while maintaining maximal dynamic robustness to external perturbations.

Modularity of biological systems: a link between structure and function.

This paper addresses two topics in systems biology, the hypothesis that biological systems are modular and the problem of relating structure and function of biological systems. The focus here is on gene regulatory networks, represented by Boolean network models, a commonly used tool. Most of the research on gene regulatory network modularity has focused on network structure, typically represented through either directed or undirected graphs. But since gene regulation is a highly dynamic process as it determines the function of cells over time, it is natural to consider functional modularity as well. One of the main results is that the structural decomposition of a network into modules induces an analogous decomposition of the dynamic structure, exhibiting a strong relationship between network structure and function. An extensive simulation study provides evidence for the hypothesis that modularity might have evolved to increase phenotypic complexity while maintaining maximal dynamic robustness to external perturbations.

On the relationship of steady states of continuous and discrete models arising from biology.

For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.

An information theoretic approach for the inference of Boolean networks and functions from data: BoCSE.

Building predictive models from data is an important and challenging task in many fields including biology, medicine, engineering, and economy. In this issue, Sun et al.1 present a method for the inference of Boolean networks along with practical applications.

ADAM: analysis of discrete models of biological systems using computer algebra.

BACKGROUND: Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, to gain a better understanding of them. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. There exist software tools to analyze discrete models, but they either lack the algorithmic functionality to analyze complex models deterministically or they are inaccessible to many users as they require understanding the underlying algorithm and implementation, do not have a graphical user interface, or are hard to install. Efficient analysis methods that are accessible to modelers and easy to use are needed. RESULTS: We propose a method for efficiently identifying attractors and introduce the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness and robustness. For a large set of published complex discrete models, ADAM identified the attractors in less than one second. CONCLUSIONS: Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for accessible efficient analysis tools. ADAM provides analysis methods based on mathematical algorithms as a web-based tool for several different input formats, and it makes analysis of complex models accessible to a larger community, as it is platform independent as a web-service and does not require understanding of the underlying mathematics.

Polynomial algebra of discrete models in systems biology.

MOTIVATION: An increasing number of discrete mathematical models are being published in Systems Biology, ranging from Boolean network models to logical models and Petri nets. They are used to model a variety of biochemical networks, such as metabolic networks, gene regulatory networks and signal transduction networks. There is increasing evidence that such models can capture key dynamic features of biological networks and can be used successfully for hypothesis generation. RESULTS: This article provides a unified framework that can aid the mathematical analysis of Boolean network models, logical models and Petri nets. They can be represented as polynomial dynamical systems, which allows the use of a variety of mathematical tools from computer algebra for their analysis. Algorithms are presented for the translation into polynomial dynamical systems. Examples are given of how polynomial algebra can be used for the model analysis. CONTACT: alanavc@vt.edu SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Reduction of Boolean network models.

Boolean networks have been successfully used in modelling gene regulatory networks. However, for large networks, analysis by simulation becomes unfeasible. In this paper we propose a reduction method for Boolean networks that decreases the size of the network, while preserving important dynamical properties and topological features. As a result, the reduced network can be used to infer properties about the original network and to gain a better understanding of the role of network topology on the dynamics. In particular, we use the reduction method to study steady states of Boolean networks and apply our results to models of Th-lymphocyte differentiation and the lac operon.

Evolution of Cellular Differentiation: From Hypotheses to Models.

Cellular differentiation is one of the hallmarks of complex multicellularity, allowing individual organisms to capitalize on among-cell functional diversity. The evolution of multicellularity is a major evolutionary transition that allowed for the increase of organismal complexity in multiple lineages, a process that relies on the functional integration of cell-types within an individual. Multiple hypotheses have been proposed to explain the origins of cellular differentiation, but we lack a general understanding of what makes one cell-type distinct from others, and how such differentiation arises. Here, we describe how the use of Boolean networks (BNs) can aid in placing empirical findings into a coherent conceptual framework, and we emphasize some of the standing problems when interpreting data and model behaviors.

HDAC Inhibitor Titration of Transcription and Axolotl Tail Regeneration.

New patterns of gene expression are enacted and regulated during tissue regeneration. Histone deacetylases (HDACs) regulate gene expression by removing acetylated lysine residues from histones and proteins that function directly or indirectly in transcriptional regulation. Previously we showed that romidepsin, an FDA-approved HDAC inhibitor, potently blocks axolotl embryo tail regeneration by altering initial transcriptional responses to injury. Here, we report on the concentration-dependent effect of romidepsin on transcription and regeneration outcome, introducing an experimental and conceptual framework for investigating small molecule mechanisms of action. A range of romidepsin concentrations (0-10 µM) were administered from 0 to 6 or 0 to 12 h post amputation (HPA) and distal tail tip tissue was collected for gene expression analysis. Above a threshold concentration, romidepsin potently inhibited regeneration. Sigmoidal and biphasic transcription response curve modeling identified genes with inflection points aligning to the threshold concentration defining regenerative failure verses success. Regeneration inhibitory concentrations of romidepsin increased and decreased the expression of key genes. Genes that associate with oxidative stress, negative regulation of cell signaling, negative regulation of cell cycle progression, and cellular differentiation were increased, while genes that are typically up-regulated during appendage regeneration were decreased, including genes expressed by fibroblast-like progenitor cells. Using single-nuclei RNA-Seq at 6 HPA, we found that key genes were altered by romidepin in the same direction across multiple cell types. Our results implicate HDAC activity as a transcriptional mechanism that operates across cell types to regulate the alternative expression of genes that associate with regenerative success versus failure outcomes.

The dynamics of conjunctive and disjunctive Boolean network models.

For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.

Performance of normative and approximate evidence accumulation on the dynamic clicks task.

The aim of a number of psychophysics tasks is to uncover how mammals make decisions in a world that is in flux. Here we examine the characteristics of ideal and near-ideal observers in a task of this type. We ask when and how performance depends on task parameters and design, and, in turn, what observer performance tells us about their decision-making process. In the dynamic clicks task subjects hear two streams (left and right) of Poisson clicks with different rates. Subjects are rewarded when they correctly identify the side with the higher rate, as this side switches unpredictably. We show that a reduced set of task parameters defines regions in parameter space in which optimal, but not near-optimal observers, maintain constant response accuracy. We also show that for a range of task parameters an approximate normative model must be finely tuned to reach near-optimal performance, illustrating a potential way to distinguish between normative models and their approximations. In addition, we show that using the negative log-likelihood and the 0/1-loss functions to fit these types of models is not equivalent: the 0/1-loss leads to a bias in parameter recovery that increases with sensory noise. These findings suggest ways to tease apart models that are hard to distinguish when tuned exactly, and point to general pitfalls in experimental design, model fitting, and interpretation of the resulting data.

The effect of negative feedback loops on the dynamics of boolean networks.

Feedback loops play an important role in determining the dynamics of biological networks. To study the role of negative feedback loops, this article introduces the notion of distance-to-positive-feedback which, in essence, captures the number of independent negative feedback loops in the network, a property inherent in the network topology. Through a computational study using Boolean networks, it is shown that distance-to-positive-feedback has a strong influence on network dynamics and correlates very well with the number and length of limit cycles in the phase space of the network. To be precise, it is shown that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact that certain natural biological networks exhibit generally regular behavior and have fewer negative feedback loops than randomized networks with the same number of nodes and same connectivity.