Publications about 'robustness' |
Articles in journal or book chapters |
This paper studies the effect of perturbations on the gradient flow of a general constrained nonlinear programming problem, where the perturbation may arise from inaccurate gradient estimation in the setting of data-driven optimization. Under suitable conditions on the objective function, the perturbed gradient flow is shown to be small-disturbance input-to-state stable (ISS), which implies that, in the presence of a small-enough perturbation, the trajectory of the perturbed gradient flow must eventually enter a small neighborhood of the optimum. This work was motivated by the question of robustness of direct methods for the linear quadratic regulator problem, and specifically the analysis of the effect of perturbations caused by gradient estimation or round-off errors in policy optimization. Interestingly, we show small-disturbance ISS for three of the most common optimization algorithms: standard gradient flow, natural gradient flow, and Newton gradient flow. |
We develop some basic principles for the design and robustness analysis of a continuous-time bilinear dynamical network, where an attacker can manipulate the strength of the interconnections/edges between some of the agents/nodes. We formulate the edge protection optimization problem of picking a limited number of attack-free edges and minimizing the impact of the attack over the bilinear dynamical network. In particular, the H2-norm of bilinear systems is known to capture robustness and performance properties analogous to its linear counterpart and provides valuable insights for identifying which edges arem ost sensitive to attacks. The exact optimization problem is combinatorial in the number of edges, and brute-force approaches show poor scalability. However, we show that the H2-norm as a cost function is supermodular and, therefore, allows for efficient greedy approximations of the optimal solution. We illustrate and compare the effectiveness of our theoretical findings via numerical simulation |
Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. Specifically, we merge nodes and edges of BRNs to match in- and out-flows of each node. This allows us to derive the stationary distributions of a large class of BRNs, including autophosphorylation networks of EGFR, PAK1, and Aurora B kinase and a genetic toggle switch. This reveals the unique properties of their stochastic dynamics such as robustness, sensitivity, and multimodality. Importantly, we provide a user-friendly computational package, CASTANET, that automatically derives symbolic expressions of the stationary distributions of BRNs to understand their long-term stochasticity. |
Cells respond to biochemical and physical internal as well as external signals. These signals can be broadly classified into two categories: (a) ``actionable'' or ``reference'' inputs that should elicit appropriate biological or physical responses such as gene expression or motility, and (b) ``disturbances'' or ``perturbations'' that should be ignored or actively filtered-out. These disturbances might be exogenous, such as binding of nonspecific ligands, or endogenous, such as variations in enzyme concentrations or gene copy numbers. In this context, the term robustness describes the capability to produce appropriate responses to reference inputs while at the same time being insensitive to disturbances. These two objectives often conflict with each other and require delicate design trade-offs. Indeed, natural biological systems use complicated and still poorly understood control strategies in order to finely balance the goals of responsiveness and robustness. A better understanding of such natural strategies remains an important scientific goal in itself and will play a role in the construction of synthetic circuits for therapeutic and biosensing applications. A prototype problem in robustly responding to inputs is that of ``robust tracking'', defined by the requirement that some designated internal quantity (for example, the level of expression of a reporter protein) should faithfully follow an input signal while being insensitive to an appropriate class of perturbations. Control theory predicts that a certain type of motif, called integral feedback, will help achieve this goal, and this motif is, in fact, a necessary feature of any system that exhibits robust tracking. Indeed, integral feedback has always been a key component of electrical and mechanical control systems, at least since the 18th century when James Watt employed the centrifugal governor to regulate steam engines. Motivated by this knowledge, biological engineers have proposed various designs for biomolecular integral feedback control mechanisms. However, practical and quantitatively predictable implementations have proved challenging, in part due to the difficulty in obtaining accurate models of transcription, translation, and resource competition in living cells, and the stochasticity inherent in cellular reactions. These challenges prevent first-principles rational design and parameter optimization. In this work, we exploit the versatility of an Escherichia coli cell-free transcription-translation (TXTL) to accurately design, model and then build, a synthetic biomolecular integral controller that precisely controls the expression of a target gene. To our knowledge, this is the first design of a functioning gene network that achieves the goal of making gene expression track an externally imposed reference level, achieves this goal even in the presence of disturbances, and whose performance quantitatively agrees with mathematical predictions. |
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 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. |
Synthetic constructs in biotechnology, bio-computing, and proposed gene therapy interventions are often based on plasmids or transfected circuits which implement some form of on-off (toggle or flip-flop) switch. For example, the expression of a protein used for therapeutic purposes might be triggered by the recognition of a specific combination of inducers (e.g., antigens), and memory of this event should be maintained across a cell population until a specific stimulus commands a coordinated shut-off. The robustness of such a design is hampered by molecular (intrinsic) or environmental (extrinsic) noise, which may lead to spontaneous changes of state in a subset of the population and is reflected in the bimodality of protein expression, as measured for example using flow cytometry. In this context, a majority-vote correction circuit, which brings deviant cells back into the required state, is highly desirable. To address this concrete challenge, we have developed a new theoretical design for quorum-sensing (QS) synthetic toggles. QS provides a way for cells to broadcast their states to the population as a whole so as to facilitate consensus. Our design is endowed with strong theoretical guarantees, based on monotone dynamical systems theory, of global stability and no oscillations, and which leads to robust consensus states. |
Synthetic biology efforts have largely focused on small engineered gene networks, yet understanding how to integrate multiple synthetic modules and interface them with endogenous pathways remains a challenge. Here we present the design, system integration, and analysis of several large scale synthetic gene circuits for artificial tissue homeostasis. Diabetes therapy represents a possible application for engineered homeostasis, where genetically programmed stem cells maintain a steady population of beta-cells despite continuous turnover. We develop a new iterative process that incorporates modular design principles with hierarchical performance optimization targeted for environments with uncertainty and incomplete information. We employ theoretical analysis and computational simulations of multicellular reaction/diffusion models to design and understand system behavior, and find that certain features often associated with robustness (e.g., multicellular synchronization and noise attenuation) are actually detrimental for tissue homeostasis. We overcome these problems by engineering a new class of genetic modules for 'synthetic cellular heterogeneity' that function to generate beneficial population diversity. We design two such modules (an asynchronous genetic oscillator and a signaling throttle mechanism), demonstrate their capacity for enhancing robust control, and provide guidance for experimental implementation with various computational techniques. We found that designing modules for synthetic heterogeneity can be complex, and in general requires a framework for non-linear and multifactorial analysis. Consequently, we adapt a 'phenotypic sensitivity analysis' method to determine how functional module behaviors combine to achieve optimal system performance. We ultimately combine this analysis with Bayesian network inference to extract critical, causal relationships between a module's biochemical rate-constants, its high level functional behavior in isolation, and its impact on overall system performance once integrated. |
Natural and synthetic biological networks must function reliably in the face of fluctuating stoichiometry of their molecular components. These fluctuations are caused in part by changes in relative expression efficiency and the DNA template amount of the network-coding genes. Gene product levels could potentially be decoupled from these changes via built-in adaptation mechanisms, thereby boosting network reliability. Here we show that a mechanism based on an incoherent feed-forward motif enables adaptive gene expression in mammalian cells. We modeled, synthesized, and tested transcriptional and post-transcriptional incoherent loops and found that in all cases the gene product adapts to changes in DNA template abundance. We also observed that the post-transcriptional form results in superior adaptation behavior, higher absolute expression levels, and lower intrinsic fluctuations. Our results support a previously-hypothesized endogenous role in gene dosage compensation for such motifs and suggest that their incorporation in synthetic networks will improve their robustness and reliability. |
The concept of robustness of regulatory networks has been closely related to the nature of the interactions among genes, and the capability of pattern maintenance or reproducibility. Defining this robustness property is a challenging task, but mathematical models have often associated it to the volume of the space of admissible parameters. Not only the volume of the space but also its topology and geometry contain information on essential aspects of the network, including feasible pathways, switching between two parallel pathways or distinct/disconnected active regions of parameters. A method is presented here to characterize the space of admissible parameters, by writing it as a semi-algebraic set, and then theoretically analyzing its topology and geometry, as well as volume. This method provides a more objective and complete measure of the robustness of a developmental module. As a detailed case study, the segment polarity gene network is analyzed. |
The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In recent work, we emphasized that topology and geometry matter, as well as volume. In this paper, and using the segment polarity gene network to illustrate our approach, we show that random walks in parameter space and how they exit the feasible region provide a rich perspective on the different modes of failure of a model. In particular, for the segment polarity network, we found that, between two alternative ways of activating Wingless, one is more robust. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks. |
This paper presents a stability test for a class of interconnected nonlinear systems motivated by biochemical reaction networks. One of the main results determines global asymptotic stability of the network from the diagonal stability of a "dissipativity matrix" which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the interconnection terms. This stability test encompasses the "secant criterion" for cyclic networks presented in our previous paper, and extends it to a general interconnection structure represented by a graph. A second main result allows one to accommodate state products. This extension makes the new stability criterion applicable to a broader class of models, even in the case of cyclic systems. The new stability test is illustrated on a mitogen activated protein kinase (MAPK) cascade model, and on a branched interconnection structure motivated by metabolic networks. Finally, another result addresses the robustness of stability in the presence of diffusion terms in a compartmental system made out of identical systems. |
This expository presentation, prepared for a summer course, addresses the precise formulation of questions of robustness with respect to disturbances, using the paradigm of input to state stability. It provides an intuitive and informal presentation of the main concepts. |
As a discrete approach to genetic regulatory networks, Boolean models provide an essential qualitative description of the structure of interactions among genes and proteins. Boolean models generally assume only two possible states (expressed or not expressed) for each gene or protein in the network as well as a high level of synchronization among the various regulatory processes. In this paper, we discuss and compare two possible methods of adapting qualitative models to incorporate the continuous-time character of regulatory networks. The first method consists of introducing asynchronous updates in the Boolean model. In the second method, we adopt the approach introduced by L. Glass to obtain a set of piecewise linear differential equations which continuously describe the states of each gene or protein in the network. We apply both methods to a particular example: a Boolean model of the segment polarity gene network of Drosophila melanogaster. We analyze the dynamics of the model, and provide a theoretical characterization of the model's gene pattern prediction as a function of the timescales of the various processes. |
Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. Here we propose a framework for gene regulatory networks that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre- and post-translational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations. |
Some biological systems operate at the critical point between stability and instability and this requires a fine-tuning of parameters. We bring together two examples from the literature that illustrate this: neural integration in the nervous system and hair cell oscillations in the auditory system. In both examples the question arises as to how the required fine-tuning may be achieved and maintained in a robust and reliable way. We study this question using tools from nonlinear and adaptive control theory. We illustrate our approach on a simple model which captures some of the essential features of neural integration. As a result, we propose a large class of feedback adaptation rules that may be responsible for the experimentally observed robustness of neural integration. We mention extensions of our approach to the case of hair cell oscillations in the ear. |
This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and Jackson, of which a self-contained and streamlined exposition is included. The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan for T-cell receptor signal transduction. |
Conference articles |
Recent research in neural networks and machine learning suggests that using many more parameters than strictly required by the initial complexity of a regression problem can result in more accurate or faster-converging models -- contrary to classical statistical belief. This phenomenon, sometimes known as ``benign overfitting'', raises questions regarding in what other ways might overparameterization affect the properties of a learning problem. In this work, we investigate the effects of overfitting on the robustness of gradient-descent training when subject to uncertainty on the gradient estimation. This uncertainty arises naturally if the gradient is estimated from noisy data or directly measured. Our object of study is a linear neural network with a single, arbitrarily wide, hidden layer and an arbitrary number of inputs and outputs. In this paper we solve the problem for the case where the input and output of our neural-network are one-dimensional, deriving sufficient conditions for robustness of our system based on necessary and sufficient conditions for convergence in the undisturbed case. We then show that the general overparametrized formulation introduces a set of spurious equilibria which lay outside the set where the loss function is minimized, and discuss directions of future work that might extend our current results for more general formulations. |
In large-scale networks, agents and links are often vulnerable to attacks. This paper focuses on continuous-time bilinear networks, where additive disturbances model attacks or uncertainties on agents/states (node disturbances), and multiplicative disturbances model attacks or uncertainties on couplings between agents/states (link disturbances). It investigates network robustness notion in terms of the underlying digraph of the network, and structure of exogenous uncertainties and attacks. Specifically, it defines a robustness measure using the $\mathcal H_2$-norm of the network and calculates it in terms of the reachability Gramian of the bilinear system. The main result is that under certain conditions, the measure is supermodular over the set of all possible attacked links. The supermodular property facilitates the efficient solution finding of the optimization problem. Examples illustrate how different structures can make the system more or less vulnerable to malicious attacks on links. |
When measuring importance of nodes in a network, the interconnections and dynamics are often supposed to be perfectly known. In this paper, we consider networks of agents with both uncertain couplings and dynamics. Network uncertainty is modeled by structured additive stochastic disturbances on each agent's update dynamics and coupling weights. We then study how these uncertainties change the network's centralities. Disturbances on the couplings between agents resul in bilinear dynamics, and classical centrality indices from linear network theory need to be redefined. To do that, we first show that, similarly to its linear counterpart, the squared H2 norm of bilinear systems measures the trace of the steady-state error covariance matrix subject to stochastic disturbances. This makes the H2 norm a natural candidate for a performance metric of the system. We propose a centrality index for the agents based on the H2 norm, and show how it depends on the network topology and the noise structure. Finally, we simulate a few graphs to illustrate how uncertainties on different couplings affect the agents' centrality rankings compared to a linearized model of the same system. |
Recent experimental work has shown that transient E. coli chemotactic response is unchanged by a scaling of its ligand input signal (fold change detection, or FCD), and this is in agreement with earlier mathematical predictions. However, this prediction was based on certain particular assumptions on the structure of the chemotaxis pathway. In this work, we begin by showing that behavior similar to FCD can be obtained under weaker conditions on the system structure. Namely, we show that under relaxed conditions, a scaling of the chemotaxis system's inputs leads to a time scaling of the output response. We propose that this may be a contributing factor to the robustness of the experimentally observed FCD. We further show that FCD is a special case of this time scaling behavior for which the time scaling factor is unity. We then proceed to extend the conditions for output time scaling to more general adapting systems, and demonstrate this time scaling behavior on a published model of the chemotaxis pathway of the bacterium Rhodobacter sphaeroides. This work therefore provides examples of how robust biological behavior can arise from simple yet realistic conditions on the underlying system structure. |
This work is concerned with the study of the robustness and fragility of gene regulation networks to variability in the timescales of the distinct biological processes involved. It explores and compares two methods: introducing asynchronous updates in a Boolean model, or integrating the Boolean rules in a continuous, piecewise linear model. As an example, the segment polarity network of the fruit fly is analyzed. A theoretical characterization is given of the model's ability to predict the correct development of the segmented embryo, in terms of the specific timescales of the various regulation interactions. |
We showned in another recent paper that any asymptotically controllable system can be stabilized by means of a certain type of discontinuous feedback. The feedback laws constructed in that work are robust with respect to actuator errors as well as to perturbations of the system dynamics. A drawback, however, is that they may be highly sensitive to errors in the measurement of the state vector. This paper addresses this shortcoming, and shows how to design a dynamic hybrid stabilizing controller which, while preserving robustness to external perturbations and actuator error, is also robust with respect to measurement error. This new design relies upon a controller which incorporates an internal model of the system driven by the previously constructed feedback. |
We suggest that a very natural mathematical framework for the study of dissipation -in the sense of Willems, Moylan and Hill, and others- is that of indefinite quasimetric spaces. Several basic facts about dissipative systems are seen to be simple consequences of the properties of such spaces. Quasimetric spaces provide also one natural context for optimal control problems, and even for "gap" formulations of robustness. |
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