Publications by Eduardo D. Sontag in year 2017 |
Articles in journal or book chapters |
This is an expository paper, which compares in detail various alternative weak contraction ideas for nonlinear system stability. |
In biological processes such as embryonic development, hematopoietic cell differentiation, and the arising of tumor heterogeneity and consequent resistance to therapy, mechanisms of gene activation and deactivation may play a role in the emergence of phenotypically heterogeneous yet genetically identical (clonal) cellular populations. Mathematically, the variability in phenotypes in the absence of genetic variation can be modeled through the existence of multiple metastable attractors in nonlinear systems subject with stochastic switching, each one of them associated to an alternative epigenetic state. An important theoretical and practical question is that of estimating the number and location of these states, as well as their relative probabilities of occurrence. This paper focuses on a rigorous analytic characterization of multiple modes under slow promoter kinetics, which is a feature of epigenetic regulation. It characterizes the stationary distributions of Chemical Master Equations for gene regulatory networks as a mixture of Poisson distributions. As illustrations, the theory is used to tease out the role of cooperative binding in stochastic models in comparison to deterministic models, and applications are given to various model systems, such as toggle switches in isolation or in communicating populations and a trans-differentiation network. |
This paper proposes a technique that combines experimental data, mathematical modeling, and statistical analyses for identifying optimal treatment protocols that are robust with respect to individual variability. Experimental data from a small sample population is amplified using bootstrapping to obtain a large number of virtual populations that statistically match the expected heterogeneity. Alternative therapies chosen from among a set of clinically-realizable protocols are then compared and scored according to coverage. As proof of concept, the method is used to evaluate a treatment with oncolytic viruses and dendritic cell vaccines in a mouse model of melanoma. The analysis shows that while every scheduling variant of an experimentally-utilized treatment protocol is fragile (non-robust), there is an alternative region of dosing space (lower oncolytic virus dose, higher dendritic cell dose) for which a robust optimal protocol exists. |
Resistance to chemotherapy is a major impediment to the successful treatment of cancer. Classically, resistance has been thought to arise primarily through random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that the progression to resistance need not occur randomly, but instead may be induced by the therapeutic agent itself.This process of resistance induction can be a result of genetic changes, or can occur through epigenetic alterations that cause otherwise drug-sensitive cancer cells to undergo ``phenotype switching''. This relatively novel notion of resistance further complicates the already challenging task of designing treatment protocols that minimize the risk of evolving resistance. In an effort to better understand treatment resistance, we have developed a mathematical modeling framework that incorporates both random and drug-induced resistance. Our model demonstrates that the ability (or lack thereof) of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. The importance of induced resistance in treatment response led us to ask if, in our model, one can determine the resistance induction rate of a drug for a given treatment protocol. Not only could we prove that the induction parameter in our model is theoretically identifiable, we have also proposed a possible in vitro experiment which could practically be used to determine a treatment's propensity to induce resistance. |
Biochemical reaction networks in cells frequently consist of reactions with disparate timescales. Stochastic simulations of such multiscale BRNs are prohibitively slow due to the high computational cost incurred in the simulations of fast reactions. One way to resolve this problem is to replace fast species by their stationary conditional expectation values conditioned on slow species. While various approximations schemes for this quasi-steady state approximation have been developed, they often lead to considerable errors. This paper considers two classes of multiscale BRNs which can be reduced by through an exact QSS rather than approximations. Specifically, we assume that fast species constitute either a feedforward network or a complex balanced network. Exact reductions for various examples are derived, and the computational advantages of this approach are illustrated through simulations. |
This paper introduces two generalizations of systems invariant with respect to continuous sets of input transformations, that is, systems whose output dynamics remain invariant when applying a transformation to the input and simultaneously adjusting the initial conditions. These generalizations concern systems invariant with respect to time-dependent input transformations with exponentially increasing or decreasing ``strength'', and systems invariant with respect to transformations of the "nonlinear derivatives" of the input. Interestingly, these two generalizations of invariant systems encompass linear time-invariant (LTI) systems with real transfer function zeros of arbitrary multiplicity. Furthermore, the zero-dynamics of systems possessing our generalized invariances show properties analogous to those of LTI systems with transfer function zeros, generalizing concepts like pole-zero cancellation, the rejection of ramps by Hurwitz LTI systems with a zero at the origin with multiplicity two, and (to a certain extend) the superposition principle with respect to inputs zeroing the output. |
Aerotaxis, the directed migration along oxygen gradients, allows many microorganisms to locate favorable oxygen concentrations. Despite oxygen's fundamental role for life, even key aspects of aerotaxis remain poorly understood. In Bacillus subtilis, for example, there is conflicting evidence of whether migration occurs to the maximal oxygen concentration available or to an optimal intermediate one, and how aerotaxis can be maintained over a broad range of conditions. Using precisely controlled oxygen gradients in a microfluidic device, spanning the full spectrum of conditions from quasi-anoxic to oxic (60nM-1mM), we resolved B. subtilis' ``oxygen preference conundrum'' by demonstrating consistent migration towards maximum oxygen concentrations. Surprisingly, the strength of aerotaxis was largely unchanged over three decades in oxygen concentration (131nM-196mM). We discovered that in this range B. subtilis responds to the logarithm of the oxygen concentration gradient, a log-sensing strategy that affords organisms high sensitivity over a wide range of conditions. |
Utilizing the synthetic transcription-translation (TX-TL) system, this paper studies the impact of nucleotide triphosphates (NTPs) and magnesium (Mg2+), on gene expression, in the context of the counterintuitive phenomenon of suppression of gene expression at high NTP concentration. Measuring translation rates for different Mg2+ and NTP concentrations, we observe a complex resource dependence. We demonstrate that translation is the rate-limiting process that is directly inhibited by high NTP concentrations. Additional Mg2+ can partially reverse this inhibition. In several experiments, we observe two maxima of the translation rate viewed as a function of both Mg2+ and NTP concentration, which can be explained in terms of an NTP-independent effect on the ribosome complex and an NTP- Mg2+ titration effect. The non-trivial compensatory effects of abundance of different vital resources signals the presence of complex regulatory mechanisms to achieve optimal gene expression. |
What complicated dynamics can arise in the simplest biochemical systems, in response to a periodic input? This paper discusses two models that commonly appear as components of larger sensing and signal transduction pathways in systems biology: a simple two-species negative feedback loop, and a prototype nonlinear integral feedback. These systems have globally attracting steady states when unforced, yet, when subject to a periodic excitation, subharmonic responses and strange attractors can arise via period-doubling cascades. These behaviors are similar to those exhibited by classical forced nonlinear oscillators such as those described by van der Pol or Duffing equations. The lack of entrainment to external oscillations, in even the simplest biochemical networks, represents a level of additional complexity in molecular biology. |
Elucidating the structure of biological intracellular networks from experimental data remains a major challenge. This paper studies two types of ``response signatures'' to identify specific circuit motifs, from the observed response to periodic inputs. In particular, the objective is to distinguish negative feedback loops (NFLs) from incoherent feedforward loops (IFFLs), which are two types of circuits capable of producing exact adaptation. The theory of monotone systems with inputs is used to show that ``period skipping'' (non-harmonic responses) is ruled out in IFFL's, and a notion called ``refractory period stabilization'' is also analyzed. The approach is then applied to identify a circuit dominating cell cycle timing in yeast, and to uncover a calcium-mediated NFL circuit in \emph{C.elegans} olfactory sensory neurons. |
This paper analizes a model for the initial stage of T cell activation. The state variables in the model are the concentrations of phosphorylation states of the T cell receptor complex and the phosphatase SHP-1 in the cell. It is shown that these quantities cannot approach zero, and that there is more than one positive steady state for certain values of the parameters; in addition, damped oscillations are possible. It is also shown that the chemical concentration which represents the degree of activation of the cell, represented by the maximally phosphorylated form of the T cell receptor complex, is in general a non-monotone function of the activating signal. In particular there are cases where there is a value of the dissociation constant of the ligand from the receptor which produces an optimal activation of the T cell. In this way the results of certain simulations in the literature have been confirmed rigorously and new features are discovered. |
This paper deals with the design of promoters that maintain constant levels of expression, whether they are carried at single copy in the genome or on high-copy plasmids. The design is based on an incoherent feedforward loop (iFFL) with a perfectly non-cooperative repression. The circuits are implemented in E. coli using Transcription Activator Like Effectors (TALEs). The resulting stabilized promoters generate near identical expression across different genome locations and plasmid backbones (pSC101, p15a, ColE1, pUC), and also provide robustness to strain mutations and growth media. Further, their strength is tunable and can be used to maintain constant ratios between proteins. |
This paper describes a novel approach for characterization of chemosensitivity and prediction of clinical response in multiple myeloma. It relies upon a patient-specific computational model of clinical response, parameterized by a high-throughput ex vivo assay that quantifies sensitivity of primary MM cells to 31 agents or combinations, in a reconstruction of the tumor microenvironment. The mathematical model, which inherently accounts for intra-tumoral heterogeneity of drug sensitivity, combined with drug- and regimen-specific pharmacokinetics, produces patient-specific predictions of clinical response 5 days post-biopsy. |
Since the early 1990s, many authors have independently suggested that self/nonself recognition by the immune system might be modulated by the rates of change of antigen challenges. This paper introduces an extremely simple and purely conceptual mathematical model that allows dynamic discrimination of immune challenges. The main component of the model is a motif which is ubiquitous in systems biology, the incoherent feedforward loop, which endows the system with the capability to estimate exponential growth exponents, a prediction which is consistent with experimental work showing that exponentially increasing antigen stimulation is a determinant of immune reactivity. Combined with a bistable system and a simple feedback repression mechanism, an interesting phenomenon emerges as a tumor growth rate increases: elimination, tolerance (tumor growth), again elimination, and finally a second zone of tolerance (tumor escape). This prediction from our model is analogous to the ``two-zone tumor tolerance'' phenomenon experimentally validated since the mid 1970s. Moreover, we provide a plausible biological instantiation of our circuit using combinations of regulatory and effector T cells. |
A recent paper by Karin et al. introduced a mathematical notion called dynamical compensation (DC) of biological circuits. DC was shown to play an important role in glucose homeostasis as well as other key physiological regulatory mechanisms. Karin et al.\ went on to provide a sufficient condition to test whether a given system has the DC property. Here, we show how DC is a reformulation of a well-known concept in systems biology, statistics, and control theory -- that of parameter structural non-identifiability. Viewing DC as a parameter identification problem enables one to take advantage of powerful theoretical and computational tools to test a system for DC. We obtain as a special case the sufficient criterion discussed by Karin et al. We also draw connections to system equivalence and to the fold-change detection property. |
Emergent responses of the immune system result from the integration of molecular and cellular networks over time and across multiple organs. High-content and high-throughput analysis technologies, concomitantly with data-driven and mechanistic modeling, hold promise for the systematic interrogation of these complex pathways. However, connecting genetic variation and molecular mechanisms to individual phenotypes and health outcomes has proven elusive. Gaps remain in data, and disagreements persist about the value of mechanistic modeling for immunology. This paper presents perspectives that emerged from the National Institute of Allergy and Infectious Disease (NIAID) workshop `Complex Systems Science, Modeling and Immunity' and subsequent discussions regarding the potential synergy of high-throughput data acquisition, data-driven modeling, and mechanistic modeling to define new mechanisms of immunological disease and to accelerate the translation of these insights into therapies. |
This paper reports on the construction of a phosphorylation- and optically-responsive supramolecular complex of metabolic pathway enzymes for the biodegradation of an environmental pollutant. Fusing of enzymes led to an increase in pathway efficiency, and illustrates the possibility of spatio-temporal control over formation and functioning of a wide variety of synthetic biotransformations. |
The ribosomal density along the coding region of the mRNA molecule affects various fundamental intracellular phenomena including: protein production rates, organismal fitness, ribosomal drop off, and co-translational protein folding. Thus, regulating translation in order to obtain a desired ribosomal profile along the mRNA molecule is an important biological problem. This paper studies this problem formulated in the context of the ribosome flow model (RFM) in which one views the transition rates between site as controls. |
Conference articles |
Deterministic models remain the most common option for modeling gene regulatory networks even when the underlying assumptions of high copy numbers and fast promoter kinetics are unsatisfied. Here, we analyze a widely studied differentiation network motif known as the PU.1-GATA-1 circuit and we show that an ODE model of the biomolecular reactions consistent with known biology is incapable of exhibiting multistability, a defining behaviour for such a network. Thus, we consider the chemical master equation model of the same biomolecular reactions and using results recently developed by the authors, we analytically construct the stationary distribution. We show that this distribution is indeed capable of admitting a multitude of modes. We illustrate the results with a numerical example. |
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