Karthik Raman

Network architecture plays a crucial role in governing the dynamics of any biological network. Further, network structures have been shown to remain conserved across organisms for a given phenotype. Therefore, the mapping between network structures and the output functionality not only aids in understanding of biological systems but also finds application in synthetic biology and therapeutics. Based on the approaches involved, most of the efforts hitherto invested in this field can be classified into three broad categories, namely, computational efforts, rule-based methods and systems-theoretic approaches. The present review provides a qualitative and quantitative study of all three approaches in the light of three well-researched biological phenotypes, namely, oscillation, toggle switching, and adaptation. We also discuss the advantages, limitations, and future research scope for all three approaches along with their possible applications to other emergent properties of biological relevance.

In data-driven dynamical modeling, precise estimation of the parameters of large models from limited data has been considered a challenging task. The precision of the parameter estimates is highly dependent upon the information contained in the data; Loss of practical identifiability and sloppiness in the model structure are major challenges in estimating parameters precisely and closely related to the information contained in the data. Therefore, quantifying information is an important step in data-driven modeling. Quantifying information is a well-studied problem in the frequentist approach, where Fisher Information is one of the widely used metrics. However, Fisher Information computed via maximum likelihood estimation cannot accommodate any known prior knowledge about the parameters. Prior knowledge of the parameters along with informative experiments will improve the precision of the estimates. Bayesian estimation accommodates prior information in the form of a p.d.f. There has been very little work in the literature for quantifying information in the Bayesian framework. In this work, we introduce a new method for estimating information gain in the Bayesian framework using what is known as the Bhattacharyya coefficient. It is seen that the bounds of the coefficient have an insightful interpretation naturally in terms of information gain on the parameter of interest. We also demonstrate using case studies that the information gain of each parameter is an indication of loss of practical identifiability and sloppy parameters. It is also shown that the proposed information gain can be used as a model selection tool in black-box identification.

Designing biological networks that are capable of achieving specific functionality has been of sustained interest in the field of synthetic biology for nearly a decade. Adaptation is one such important functionality that is observed in bacterial chemotaxis, cell signalling and homoeostasis. It refers to the ability of a cell to cope with environmental perturbations. All of these adaptation networks, involve negative feedback loops or open loop control strategies. A typical enzymatic network is a circuit of enzymes whose connections are characterized by enzymatic reactions that exhibit non-linear dynamics. Previous approaches to design of enzymatic networks capable of perfect adaptation have used brute force searches encompassing the complete set of possibilities to identify suitable circuit designs. In contrast, this work presents a systematic algorithm for circuit design, using a linear systems-theoretic approach. The key idea is to set up a design-oriented problem formulation as against employing a brute force search in the space of possible circuits. To this effect, we first linearize the non-linear dynamical circuit, subsequently, we translate the requirements for adaptation to design specifications for a linear time-invariant system and imposing these design specifications on the linearized system, we obtain the minimal topologies or motifs that can perform perfect adaptation, with an optimal set of rate constants. The optimal set of rate constants is obtained by solving a structure-specific constrained optimisation problem. In effect, we demonstrate that the proposed approach identifies the key motifs of the biological network that were identified by the existing brute force approach, albeit in a systematic manner and with very little computational effort.

Constructing biological networks capable of performing specific biological functionalities has been of sustained interest in synthetic biology. Adaptation is one such ubiquitous functional property, which enables every living organism to sense a change in its surroundings and return to its operating condition prior to the disturbance. In this paper, we present a generic systems theory-driven method for designing adaptive protein networks. First, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as ‘design requirements’ for the underlying networks. We go on to prove that a protein network with different input–output nodes (proteins) needs to be at least of third-order in order to provide adaptation. Next, we show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. We argue that presence of a particular class of negative feedback or feed-forward realization is necessary for a network of any size to provide adaptation. Further, we claim that the necessary structural conditions derived in this work are the strictest among the ones hitherto existed in the literature. Finally, we prove that the capability of producing adaptation is retained for the admissible motifs even when the output node is connected with a downstream system in a feedback fashion. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality. We corroborate our theoretical results with detailed and thorough numerical simulations. Overall, our results present a generic, systematic and robust framework for designing various kinds of biological networks.

In complex dynamical systems, precise and accurate estimation of parameters and quality of predictions depends on the information contained in the experimental data. Choosing experimental schemes that maximize information contained in the data is known as Optimal Experimental Design (OED). Fisher Information Matrix and variance-covariance matrix are the central ideas of OED. However, using OED in a class of models known as sloppy models renders the model less predictive, even though the parameters are estimated with substantial precision. This work introduces a new information gain index as an experiment design criterion in the Bayesian framework. The proposed design criterion is based on what is known as the Bhattacharyya coefficient. Our previous studies show that the information gain index indicates a loss of practical identifiability. Further, it is also an indication of sloppy and stiff parameters. Hence, we extend the information index and its interpretation to joint Gaussian distributions; then, using simulations, we demonstrate that the new experiment design criterion selects experiments that minimize prediction and parameter uncertainty in sloppy models.

The deduction of design principles for complex biological functionalities has been a source of constant interest in the fields of systems and synthetic biology. A number of approaches have been adopted, to identify the space of network structures or topologies that can demonstrate a specific desired functionality, ranging from brute force to systems theory-based methodologies. The former approach involves performing a search among all possible combinations of network structures, as well as the parameters underlying the rate kinetics for a given form of network. In contrast to the search-oriented approach in brute force studies, the present chapter introduces a generic approach inspired by systems theory to deduce the network structures for a particular biological functionality. As a first step, depending on the functionality and the type of network in consideration, a measure of goodness of attainment is deduced by defining performance parameters. These parameters are computed for the most ideal case to obtain the necessary condition for the given functionality. The necessary conditions are then mapped as specific requirements on the parameters of the dynamical system underlying the network. Following this, admissible minimal structures are deduced. The proposed methodology does not assume any particular rate kinetics in this case for deducing the admissible network structures notwithstanding a minimum set of assumptions on the rate kinetics. The problem of computing the ideal set of parameter/s or rate constants, unlike the problem of topology identification, depends on the particular rate kinetics assumed for the given network. In this case, instead of a computationally exhaustive brute force search of the parameter space, a topology–functionality specific optimization problem can be solved. The objective function along with the feasible region bounded by the motif specific constraints amounts to solving a non-convex optimization program leading to non-unique parameter sets. To exemplify our approach, we adopt the functionality of adaptation, and demonstrate how network topologies that can achieve adaptation can be identified using such a systems-theoretic approach. The outcomes, in this case, i.e., minimum network structures for adaptation, are in agreement with the brute force results and other studies in literature.

Biological adaptation, the tendency of every living organism to regulate its essential activities in environmental fluctuations, is a well-studied functionality in systems and synthetic biology. In this work, we present a generic methodology inspired by systems theory to discover the design principles for robust adaptation, perfect and imperfect, in two different contexts: (1) in the presence of deterministic external and parametric disturbances and (2) in a stochastic setting. In all the cases, firstly, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as design requirements for the underlying networks. Thus, contrary to the existing approaches, the proposed methodologies provide an exhaustive set of admissible network structures without resorting to computationally burdensome brute-force techniques. Further, the proposed frameworks do not assume prior knowledge about the particular rate kinetics, thereby validating the conclusions for a large class of biological networks. In the deterministic setting, we show that unlike the incoherent feed-forward network structures (IFFLP or opposer modules), the modules containing negative feedback with buffer action (NFBLB or balancer modules) are robust to parametric fluctuations when a specific part of the network is assumed to remain unaffected. To this end, we propose a sufficient condition for imperfect adaptation and show that adding negative feedback in an IFFLP topology improves the robustness concerning parametric fluctuations. Further, we propose a stricter set of necessary conditions for imperfect adaptation. Turning to the stochastic scenario, we adopt a Wiener–Kolmogorov filter strategy to tune the parameters of a given network structure towards minimum output variance. We show that both NFBLB and IFFLP can be used as a reduced-order W–K filter. Further, we define the notion of nearest neighboring motifs to compare the output variances across different network structures. We argue that the NFBLB achieves adaptation at the cost of a variance higher than its nearest neighboring motifs whereas the IFFLP topology produces locally minimum variance while compared with its nearest neighboring motifs. We present numerical simulations to support the theoretical results. Overall, our results present a generic, systematic, and robust framework for advancing the understanding of complex biological networks.

Computational modelling of biological processes poses multiple challenges in each stage of the modelling exercise. Some significant challenges include identifiability, precisely estimating parameters from limited data, informative experiments and anisotropic sensitivity in the parameter space. One of these challenges’ crucial but inconspicuous sources is the possible presence of large regions in the parameter space over which model predictions are nearly identical. This property, known as sloppiness, has been reasonably well-addressed in the past decade, studying its possible impacts and remedies. However, certain critical unanswered questions concerning sloppiness, particularly related to its quantification and practical implications in various stages of system identification, still prevail. In this work, we systematically examine sloppiness at a fundamental level and formalise two new theoretical definitions of sloppiness. Using the proposed definitions, we establish a mathematical relationship between the parameter estimates’ precision and sloppiness in linear predictors. Further, we develop a novel computational method and a visual tool to assess the goodness of a model around a point in parameter space by identifying local structural identifiability and sloppiness and finding the most sensitive and least sensitive parameters for non-infinitesimal perturbations. We demonstrate the working of our method in benchmark systems biology models of various complexities. The pharmacokinetic HIV infection model analysis identified a new set of biologically relevant parameters that can be used to control the free virus in an active HIV infection.