Perhaps the best understood biochemical networks are those of the circadian (near 24-hour) clock within cells. A mathematical model of the mammalian circadian clock is developed which incorporates a wide range of experimental data, and is by far the most detailed mathematical model of a circadian clock yet derived. Despite its complexity, there is enough experimental data to estimate the parameters of the model as an inverse problem. The model is accurate in its predictions with respect to mutations and can be used to understand key questions about clock structure and phase resetting.
We then investigate the behavior of an earlier circadian clock model in the presence of molecular noise. Despite a previous report, we find very accurate rhythms from this model, and study the physiological causes of this robustness. Unfortunately, this model is not detailed enough to specify individual molecular interactions, which has lead to conflicting results in the literature.
Based on an experimental estimate of the number of molecules of key proteins within the mammalian circadian clock, we can directly, without ambiguity, simulate our model of the mammalian circadian clock with stochastic molecular interactions. Amazingly, interactions with promoters on the time scale of seconds are required for accurate 24-hour timekeeping. The stochasticity of our model follows the central limit theorem. Finally we find that non-redundant gene-duplication can increase immunity to molecular noise by allowing for more interactions with promoters. This work was conducted with Charles Peskin.
Work done in collaboration with Justin Blau.
Regulatory processes, especially those involving reactions between species that exist at very low concentrations, are inherently stochastic in nature. It is presently not clear how the structure of the reaction network model affects the results from the (numerical) calculation of the distribution of species. As a step towards analyzing the time-dependent behavior of the concentration distribution of each species in a network, we derive analytical expressions for the mean and variance of the concentration of all species in an arbitrary network where all the interactions are zero-order production reactions or first order conversion, catalytic or degradation reactions. We find the surprising, and apparently unknown, result that the time evolution of the second moments is governed by linear combinations of the eigenvalues of the matrix for the evolution of the means.
We use this theoretical framework to analyze the effect of network topology on the evolution of the mean and variance of various species in the network. In particular, we analyze the slowest time-scales for relaxation of the mean and variance for networks that are linear, and those that have positive feedback or feedforward loops. For a stochastic analysis of diffusion-reaction processes, we derive a framework that facilitates the separation of the effects of domain geometry, diffusion, and reaction rates on the distribution of species. We discuss the use of various measures to describe the 'noise' in stochastic systems, and show that the choice of the noise measure can lead to completely different conclusions for the same system.
We have developed a systematic methods to infer regulatory structures and properties of gene networks using microarray expression data. The methods learn first-order models of regulatory influences using RNA expression profiles for a diverse set of treatments, including exogenous compounds, environmental stresses, genetic mutations, and RNA inhibition. We have successfully applied the methods in E. coli and yeast to infer networks of 10s to 1000s of genes. The resulting network models can be used to identify transcription factor interactions, critical regulatory hubs, and to predict the mode of action of compounds and metabolites. In yeast, for example, the method was applied to a microarray data set measuring 6000 RNAs in 300 treatments. The resulting network model was used correctly identify the gene target of terbinafine, itraconazole and several other drugs. This method may be similarly applied to identify the feedback interactions between metabolic compounds and regulatory genes. These regulatory models may improve the optimization of metabolic pathways for biotechnology applications and may create new opportunities for target identification and lead optimization in drug discovery.
The time evolution of a well-stirred chemically reacting system is traditionally modeled by a set of coupled ordinary differential equations called the reaction rate equation (RRE). The resulting picture of continuous deterministic evolution is, however, valid only for infinitely large systems. That condition is usually well approximated in macroscopic chemical systems. But in biological systems formed by single living cells, the small population numbers of some reactant species can result in dynamical behavior that is noticeably discrete rather than continuous, and stochastic rather than deterministic. In that case, a more accurate mathematical modeling is obtained by using the machinery of Markov process theory, specifically, the chemical master equation (CME) and the stochastic simulation algorithm (SSA). This talk will review the theoretical foundations of stochastic chemical kinetics, and then discuss some recent efforts to (1) approximate the SSA by a faster simulation procedure, and (2) establish the formal connection between the CME/SSA description and the RRE description.
Little is known about the design principles of transcriptional regulation networks that control gene expression in cells. Recent advances in data collection and analysis, however, are generating unprecedented amounts of information about gene regulation networks. To understand these complex wiring diagrams, we sought to break down such networks into basic building blocks. We generalized the notion of motifs, widely used for sequence analysis, to the level of networks. We define 'network motifs' as patterns of interconnections that recur in many different parts of a network at frequencies much higher than those found in randomized networks. We found such motifs in networks from biochemistry, neurobiology, sociology and engineering. One of the best-characterized regulation networks is that of direct transcriptional interactions in Escherichia coli. We find that much of the network is composed of repeated appearances of several highly significant motifs. Each network motif has a specific function in determining gene expression, such as generating temporal expression programs and governing the responses to fluctuating external signals. The talk will present the theoretical and experimental approaches used to detect, measure and analyze functional circuits in this genetic regulatory network.
The fruit fly Drosophila is a premier system for investigating how animal embryos self-organize their body plan. The blueprint for the fly's body is created by networks of genes operating in an ellipsoidal shell of cell nuclei called the blastoderm. We create predictive models of this process using systems of ordinary or partial differential equations fit to gene expression data by simulated annealing and/or Lagrangian methods. In this talk I will discuss the entire pattern formation project, from colorful fluorescently stained embryos to image processing, new optimization algorithms, and finally to new biological results. Also, although the notion of 'cis-regulatory modules' central to modern molecular biology, I will show that our understanding of the function and organization of these entities is fundamentally insufficient for understanding developmental biology. I will propose a solution to this problem through a new theoretical approach in concert with quantitative data from promoter-reporter constructs.
The ability to comprehensively and quantitatively monitor dynamic changes in gene expression, together with new genome-scale informatic methods, is enabling high-throughput characterization of genetic regulatory networks. In addition, methods of genetic engineering now allow synthetic regulatory circuits to be readily built. Attention is currently being turned towards manipulating genetic regulatory circuits for therapeutic and technological applications, which increases the need to understand the functional consequences of genetic manipulations and to discover principles that can guide the design process. This issue will be addressed by comparing and contrasting what has been learned about design principles for gene circuits in their complex natural setting and how these have been put to use in designing, constructing and analyzing simple synthetic gene circuits.
To understand in a quantitative manner how biological systems operate we need to achieve several things. First we need accurate and meaningful data of biologically relevant interactions. Second, we need to have experimental methodologies that allow us to dissect the behavior of the network in a context free environment. Third, we need computer algorithms to explore and simulate many different parameters, proposing new experiments to do. Finally we need to be able to modify and design the properties of the target network based on the previous analysis. In my presentation I will deal with the first two points: How to validate biologically meaningful interactions and how to analyze the properties of a network in an "in theory" context free environment.
Biological Validation of Protein interaction Networks:
Protein interaction networks are an important part of the post-genomic effort to integrate a parts-list view of the cell into system-level understanding. Using a set of 11 yeast genomes we show that combining comparative genomics and secondary structure information can greatly increase consensus based prediction of SH3 targets. Careful benchmarking of our method against positive and negative standards gives 83% accuracy with 26% coverage. We demonstrate the concept of an optimal divergence time, for effective comparative genomics studies, by proving that genomes of species that diverged very recently from S. cerevisiae (S. mikatae, S. bayanus and S. paradoxus), or a long time ago (S. pombe) contain less information for accurate prediction of SH3 targets. Our findings highlight several novel S. cerevisiae SH3 protein-interactions and the importance of selection of optimal divergence times in comparative genomics studies.
Engineering Gene Networks to Emulate Drosophila Embryonic Pattern Formation:
Pattern formation is essential in the development of higher eukaryotes. For example, in the Drosophila embryo, maternal morphogen gradients establish gap gene expression domain patterning along the anterior-posterior axis, through linkage with an elaborate gene network. To understand better the evolution and behaviour of such systems, it is important to establish the minimal determinants required for patterning. We have therefore engineered artificial transcription/translation networks, that generate simple patterns, crudely analogous to the Drosophila gap gene system. The Drosophila syncytium was modelled using DNA-coated paramagnetic beads, fixed by magnets in an artificial chamber, forming a gene expression network. Transient expression domain patterns were generated using various levels of network connectivity. Generally, adding more transcription repression interactions increased the 'sharpness' of the pattern while reducing overall expression levels. An accompanying computer model for our system allowed us to search for parameter sets compatible with patterning. While it is clear that the Drosophila embryo is far more complex than our simplified model, several features of interest emerge. For example, the model suggests that simple diffusion may be too rapid for Drosophila-scale patterning, implying that sublocalization or 'trapping' is required. Secondly, we find that for pattern formation to occur under the conditions of our in vitro reaction-diffusion system, the activator molecules must propagate faster than the inhibitors. Thirdly, adding controlled protease degradation to the system stabilizes pattern formation over time.
Work done in collaboration with Mark Isalan, Caroline Lemerle, Pedro Beltrao, and Luis Serrano.
The study of gene circuits is similar to many other areas of biology in as much as the principal aim is to understand the relationships between structure and function. This is true not only of chemical (e.g. the sequences of bases) or physical (e.g., 3-D structure of proteins) structure, but also the informational structure of genetic circuits and networks. New insights are emerging from the top-down analysis of the biomolecular networks from which complex cellular function emerges. Network motifs have been found that occur significantly more often than would be expected in random networks, providing a rational basis to search for the structure-function relationships in these systems. However, topology alone does not define function, which is sensitive to the specifics of kinetic parameters and the structure and function of individual gene circuits which comprise the higher order networks. Unfortunately, these parameters are usually very difficult to measure or infer. The problem is exacerbated by the fact that we wish to measure these parameters within the context of the fully functioning system of the cell, especially as intracellular molecular crowding generates kinetics that are vastly different than those found from in vitro measurements.
Concurrent with this emphasis on the informational architecture of intracellular molecular networks, a new appreciation of the role of stochastic processes in decision making in biological systems has emerged. Efforts in this direction have developed analysis and simulation techniques; described the noise consequences of gene circuit structure; and have explored how stochastic processes may play a pivotal role in gene circuit functionality. However, the use of inherent noise as a gene circuit probe has been largely ignored. Stochastic fluctuations are a broad-spectrum input excitation, and the frequency-domain structure of the resulting output spectra reveal details about the underlying gene circuit structure and parameter values. In this talk I will describe the frequency-domain processing of stochastic fluctuations by genetic circuits, the measurement of the output noise spectral densities, and the use of these spectra to infer gene circuit structure and reaction rate constants.
A living cell is a noisy biochemical reactor in which low reactant concentrations lead to significant statistical fluctuations, or noise, in molecule numbers and reaction rates. This noise is often perceived as being undesirable and unpredictable. However, living systems are inherently noisy and are optimized to function in the presence of stochastic fluctuations. Some organisms can exploit noise to introduce diversity into a population. In contrast, stability against fluctuations is essential in case of a gene regulatory cascade controlling cell differentiation in a developing embryo. Stability in biological systems is often obtained by feedback regulation in the underlying regulatory network. In this talk I will address how biological systems can tune the balance between stability and noise at the level of a gene, cell, and cell community.
Cell-cell communication is a pervasive activity common to both single cell and multicellular organisms, and is used in coordinating cell behavior for a variety of tasks ranging from quorum sensing in bacteria to embryogenesis in mammalian cells. Engineering synthetic multicellular communication systems to exhibit desired functions will improve our quantitative understanding of naturally occurring cell-cell communication, and will also have biotechnology applications in areas such as biosensing, biomaterial fabrication, and tissue engineering. Here we will present theoretical and experimental results from three synthetic multicellular communication systems implemented in bacteria that have been programmed to exhibit unique coordinated cell behavior. The first system is the pulse generator where sender cells communicate to nearby receiver cells, which then respond with a transient burst of gene expression whose amplitude and duration depends on the distance from the senders. In the second system, receiver cells have been engineered to respond to cell-cell communication signals only within prespecified ranges. We will demonstrate how this system can be used to generate a variety of interesting spatial patterns. In the third system, cells have been engineered to "play" Conway's Game of Life, where cells live or die based on the density of their neighbors. This system exhibits complex global emergent behavior that arises from the interaction of cells based on simple local rules. In this talk, we will correlate experimental results from observing the behavior of these systems with our quantitative spatiotemporal models.
De novo engineering of gene circuits inside cells has emerged as a powerful approach to decoding 'design principles' of biological systems. Such circuits are also of great interest for their potential applications in computation, engineering, and medicine. However, it has been challenging to realize predictable and robust circuit performance due to some major hurdles, such as noise in gene expression and cell-to-cell variation in phenotype. We address these issues by using cell-cell communication to coordinate cellular behavior across the population. To establish cell-cell communication, we take advantage of 'quorum sensing' systems that many bacteria use to detect and respond to changes in the cell density. As a prototype example, we have built and characterized a 'population control' circuit in bacterium E. coli. This circuit autonomously regulates the cell density using a negative feedback loop acting on the entire population. With the circuit, the cell density is broadcasted and detected by a quorum sensing system, which modulates the expression of a killer gene. The killer gene in turn regulates the cell density by controlling the death rate. Upon activation, the circuit will lead to a stable steady state or sustained oscillations in terms of cell density and gene expression. This circuit lays down the conceptual foundation to program interactions among multiple cell populations - essentially creating 'synthetic ecosystems' from well-characterized genetic modules.
We consider a biochemically reacting network of different species present in small quantities through several reaction channels. Due to the biochemical significance of molecular fluctuations, interactions between species in the system are considered at the molecular level. Since reactions at the molecular level are inherently stochastic, the system is modeled in terms of random processes. Understanding the time-dependent stochastic behavior of such reaction systems is necessary for analyzing numerous problems, including gene expression profiles, signal transduction and other biochemical processes.
In this work we formulate the master equation and analyze the time evolution of the number density of species that participate in the network of a general first-order reaction network. The result can be applied to numerous examples: Transcription and translation in gene network, transitions between conformational states of proteins and so on. The governing master equation is formulated in a manner that explicitly separates the effects of network topology from other species and the evolution equations for the first two moments are derived and discussed.
We discuss the methods for separation of the system into fast and slow variables and present possible analogies between biochemical networks and queueing theory, a well-established field in operations research.
Due to the small number of reactants gene expression is a stochastic phenomenon. In eucaryotic cells, in which the number of protein or mRNA molecules is relatively large, the stochastic effects originate primarily in regulation of gene activity. Transcriptional activity of a gene can be initiated by a single trans-activator molecule bound to the specific regulatory site in the target gene. The stochasticity of activator binding and dissociation is amplified by transcription and translation, since target gene activation results in a burst of mRNA molecules, and each copy of mRNA then serves as a template for numerous protein molecules. In the present paper, we briefly discuss various stochastic effects in gene expression, and then focus on regulation of gene activity in eukaryotes. We introduce a mathematical description of the stochastic effects and consider as an example regulation of a single auto-repressing gene. The ordinary differential equations with stochastic component for mRNA and protein levels in a single cell are transformed into partial differential system for probability density functions. The numerical problems in solving these equations are overcome by construction of the cellular automata.
Work done in collaboration with T. Lipniacki, P. Paszek, A. Marciniak, A. Brasier, and M. Kimmel.