Thomas P. Prescott

Guaranteed error bounds for structured complexity reduction of biochemical networks

Biological systems are typically modelled by nonlinear differential equations. In an effort to produce high fidelity representations of the underlying phenomena, these models are usually of high dimension and involve multiple temporal and spatial scales. However, this complexity and associated stiffness makes numerical simulation difficult and mathematical analysis impossible. In order to understand the functionality of these systems, these models are usually approximated by lower dimensional descriptions. These can be analysed and simulated more easily, and the reduced description also simplifies the parameter space of the model. This model reduction inevitably introduces error: the accuracy of the conclusions one makes about the system, based on reduced models, depends heavily on the error introduced in the reduction process. In this paper we propose a method to calculate the error associated with a model reduction algorithm, using ideas from dynamical systems. We first define an error system, whose output is the error between observables of the original and reduced systems. We then use convex optimisation techniques in order to find approximations to the error as a function of the initial conditions. In particular, we use the Sum of Squares decomposition of polynomials in order to compute an upper bound on the worst-case error between the original and reduced systems. We give biological examples to illustrate the theory, which leads us to a discussion about how these techniques can be used to model-reduce large, structured models typical of systems biology.

Synthetic biology: A control engineering perspective

Synthetic Biology is a new, rapidly developing field at the interface of Engineering and Biology. It aims to design new, or redesign existing biological systems for a particular purpose. The early years have seen the design of simple devices and parts (such as switches and oscillators); Synthetic Biology is now entering a new phase of development as the successfully designed devices of recent years are exploited to create systems of increasing sophistication. Control theoretic techniques play an important part in the design of these networks, as well as for allowing increasing levels of complexity to be engineered into synthetic biological systems. At the same time, the implementation of feedback control in these networks will allow them to sense, process and actuate on environmental and internal cues.

Signal propagation across layered biochemical networks

Biochemical reaction networks typically consist of a complicated structure with many interacting species and components. Techniques for the analysis of such complex systems commonly use decompositions into simpler subsystems. These decompositions are often modular, representing the state vector as a concatenation of component vectors. Without transformation, modular decompositions may lead to system parameters directly influencing the dynamics of many subsystems at once. When parameters are the control inputs, this complicates analysis and design. This paper investigates an alternative decomposition, termed layering, which partitions parameters between layers. This allows for hierarchical analysis, where the steady state response of the integrated system to the perturbation of a parameter is calculated in stages. The first stage is to calculate the local response of the steady state of a layer, considered in isolation from other layers; the second is to calculate the perturbed layers effect on the others when connected back into the full system. This analysis results in a strategy for detecting the layered structure of a biochemical network based on preserving cycles of mass flow within layers. Additionally, by expressing how the local response propagates through the system we uncover the paths by which the direct control of a certain layer may indirectly control others, giving insights into how to exploit their dependencies.

Layering in networks: The case of biochemical systems

Networked systems are characterised by their scale and structure. In particular, biochemical reaction networks involve complicated interconnections of chemical reaction pathways and cycles, occurring on a number of different time and space scales even within a cell. This paper seeks to formalise a method of layering the dynamics of a biochemical network by decomposing its stoichiometric matrix into a sum of stoichiometric matrices, each of which we identify with a layer. We derive a condition to test when a given layer directly communicates with another. We also examine singular perturbation by considering decomposition into fast and slow layers, characterising the approximate dynamics through the quasi-steady state approximation in terms of a perturbation of the dynamics of the slow layer.

Quantification of Interactions between Dynamic Cellular Network Functionalities by Cascaded Layering

Large, naturally evolved biomolecular networks typically fulfil multiple functions. When modelling or redesigning such systems, functional subsystems are often analysed independently first, before subsequent integration into larger-scale computational models. In the design and analysis process, it is therefore important to quantitatively analyse and predict the dynamics of the interactions between integrated subsystems; in particular, how the incremental effect of integrating a subsystem into a network depends on the existing dynamics of that network. In this paper we present a framework for simulating the contribution of any given functional subsystem when integrated together with one or more other subsystems. This is achieved through a cascaded layering of a network into functional subsystems, where each layer is defined by an appropriate subset of the reactions. We exploit symmetries in our formulation to exhaustively quantify each subsystem’s incremental effects with minimal computational effort. When combining subsystems, their isolated behaviour may be amplified, attenuated, or be subject to more complicated effects. We propose the concept of mutual dynamics to quantify such nonlinear phenomena, thereby defining the incompatibility and cooperativity between all pairs of subsystems when integrated into any larger network. We exemplify our theoretical framework by analysing diverse behaviours in three dynamic models of signalling and metabolic pathways: the effect of crosstalk mechanisms on the dynamics of parallel signal transduction pathways; reciprocal side-effects between several integral feedback mechanisms and the subsystems they stabilise; and consequences of nonlinear interactions between elementary flux modes in glycolysis for metabolic engineering strategies. Our analysis shows that it is not sufficient to just specify subsystems and analyse their pairwise interactions; the environment in which the interaction takes place must also be explicitly defined. Our framework provides a natural representation of nonlinear interaction phenomena, and will therefore be an important tool for modelling large-scale evolved or synthetic biomolecular networks.

Designing Conservation Relations in Layered Synthetic Biomolecular Networks

In Synthetic Biology, biomolecular networks are designed and constructed to perform specified tasks. Design strategies for these networks tend to center on tuning the parameters of mathematical models to achieve a specified behavior, and implementing these parameters experimentally. This design strategy often assumes a fixed network structure that defines the possible behaviors, which may be too restrictive for our purposes. This paper investigates the extent to which the state space of a synthetic network can also be designed and shaped by parametric tuning. We exploit timescale separation to implement new, nonlinear, tunable conservation relations that hold for all times beyond a fast transient. We demonstrate an application of this design strategy by flexibly constraining the possible behaviors of a gene regulatory network through the design of fast protein interactions.

Ribo-attenuators: novel elements for reliable and modular riboswitch engineering

Riboswitches are structural genetic regulatory elements that directly couple the sensing of small molecules to gene expression. They have considerable potential for applications throughout synthetic biology and bio-manufacturing as they are able to sense a wide range of small molecules and regulate gene expression in response. Despite over a decade of research they have yet to reach this considerable potential as they cannot yet be treated as modular components. This is due to several limitations including sensitivity to changes in genetic context, low tunability, and variability in performance. To overcome the associated difficulties with riboswitches, we have designed and introduced a novel genetic element called a ribo-attenuator in Bacteria. This genetic element allows for predictable tuning, insulation from contextual changes, and a reduction in expression variation. Ribo-attenuators allow riboswitches to be treated as truly modular and tunable components, thus increasing their reliability for a wide range of applications.

Structured storage functions for cascaded systems

Dissipativity analysis is an important tool for the analysis of the dynamic response of systems of Ordinary Differential Equations to structural and parametric perturbations. In order to certify dissipativity, semi-definite programming is commonly used for the computation of storage functions of polynomial systems, but is currently not a practical solution for large-scale systems. This paper formulates the computation of a class of structured storage functions that exploit the structure of systems that can be decomposed into cascades. Structured storage functions allow the decomposition of the semi-definite programs used to prove dissipativity, thereby reducing the computational cost of SOS programming and making its application to large-scale systems more practical. Thus structured storage functions deliver additional speed and flexibility to the dissipativity approach to parametric and structural sensitivity analysis.

A Synthetic Recombinase-Based Feedback Loop Results in Robust Expression

Accurate control of a biological process is essential for many critical functions in biology, from the cell cycle to proteome regulation. To achieve this, negative feedback is frequently employed to provide a highly robust and reliable output. Feedback is found throughout biology and technology, but due to challenges posed by its implementation, it is yet to be widely adopted in synthetic biology. In this paper we design a synthetic feedback network using a class of recombinase proteins called integrases, which can be re-engineered to flip the orientation of DNA segments in a digital manner. This system is highly orthogonal, and demonstrates a strong capability for regulating and reducing the expression variability of genes being transcribed under its control. An excisionase protein provides the negative feedback signal to close the loop in this system, by flipping DNA segments in the reverse direction. Our integrase/excisionase negative feedback system thus provides a modular architecture that can be tuned to suit applications throughout synthetic biology and biomanufacturing that require a highly robust and orthogonally controlled output.

Multi-scale design in layered synthetic biological systems

Synthetic biologists rely on mathematical modelling to predict and tune the behaviour of experimentally implementable biomolecular systems. In addition to electronic engineering and control/system theoretic ideas, other characteristic properties of biological systems can be used to achieve complex behaviours. This paper considers the use of timescale separation and layered architectures in synthetic biomolecular systems. We discuss how, by constructing a biomolecular reaction network on two timescales, we are able (a) to implement nonlinear, tunable constraints on a networks state independently of its slow-scale dynamics, and (b) to control the slow dynamics of a networks state in a constrained state space.