Mathematical modeling plays an important and often indispensable role in synthetic biology because it serves as a crucial link between the concept and realization of a biological circuit. We review mathematical modeling concepts and methodologies as relevant to synthetic biology, including assumptions that underlie a model, types of modeling frameworks (deterministic and stochastic), and the importance of parameter estimation and optimization in modeling. Additionally we expound mathematical techniques used to analyze a model such as sensitivity analysis and bifurcation analysis, which enable the identification of the conditions that cause a synthetic circuit to behave in a desired manner. We also discuss the role of modeling in phenotype analysis such as metabolic and transcription network analysis and point out some available modeling standards and software. Following this, we present three case studies—a metabolic oscillator, a synthetic counter, and a bottom-up gene regulatory network—which have incorporated mathematical modeling as a central component of synthetic circuit design.

Synthetic biology aims to design novel biological circuits for desired applications, implemented through the assembly of biological parts including natural components of cells and artificial molecules that emulate biological behavior [

This stumbling block in synthetic biology can be alleviated by the use of computer-aided mathematical modeling. Modeling is a powerful and often indispensable link between design and realization in engineering. It can predict the dynamics of a network under several different conditions and combinations thereof. Due to this, a user can search large parameter spaces

The role of mathematical modeling in synthetic biology. Computer-aided mathematical modeling bridges a design concept to realization in synthetic biology. Solid lines depict typical steps that have to be performed while developing a model; dashed lines depict unusual scenarios or conditions under which the steps shown by the corresponding solid lines are trivial or can be bypassed. A concept or ideas for designing a circuit for a particular function may be inspired by data from experiments or the literature. A mathematical model is then formulated on the basis of certain assumptions. The framework of a model could be deterministic or stochastic. The development of a model generally begins with the estimation of parameters that govern the model; this is a process that involves sensitivity analysis, bifurcation analysis, and, under certain circumstances, metabolic and transcription (regulatory) network analysis. The dashed line from design concepts to deterministic model indicates that, in some cases, parameter estimation is trivial or can be bypassed for this type of model. A stochastic model is developed by employing statistical functions to mimic system dynamics and considering fluctuations in the data. The dashed line from parameter estimation to stochastic model indicates that in some cases, parameter estimation may offer information in choosing statistical functions when constructing a stochastic model. Optimization is required for both models and is complete when the model exhibits an agreement (goodness of fit) with experimental data. A good agreement enables reliable prediction of system behavior and further biological realization, whereas unsatisfactory agreement requires the revision of the initial assumptions and the beginning of the next modeling cycle. See text and Figure

In this review we present mathematical modeling concepts as relevant to synthetic biology and illustrate their application through the discussion of three case studies [

Figure

Biological systems are difficult to model and simulate despite a wealth of data on the structure and function of biomolecules and on cellular mechanisms. This is because biological systems exhibit complexity on several scales. Firstly metabolites, metabolic fluxes, proteins, RNA, and genes network in a highly complex manner; furthermore, their interconnections could constitute feedback or feedforward loops that respond at various time scales [

However, a biological system can often be simplified to a level that permits a user to obtain insights toward synthetic circuit construction [

Terms used in mathematical modeling and an example of a simple (deterministic) mathematical model.

Mathematical models of biological systems can be categorized into two major types: deterministic and stochastic [

Figure

To analyze steady states of a time-dependent biological system, the time derivatives in (

Numerous deterministic models have been developed for biological systems, including several for synthetic circuits. Case studies I and II discussed in this article [

In deterministic models, every interaction and every parameter value is certain. Therefore, such models predict identical system dynamics for the same set of parameter values and initial conditions. However, real systems are characterized by unexpected and irreproducible fluctuations. To capture these fluctuations and their consequences on the behavior of the system, an alternative type of model, the stochastic model, is used. Such models mimic a system as a collection of interacting particles, with the reaction rates being governed by probabilistic rate laws [

One approach in stochastic modeling is to assume that a system is comprised of randomly interacting biomolecules, wherein the reactions between the molecules are modeled as Poisson processes with a probabilistically determined rate parameter [_{n}

Several stochastic models have been developed for synthetic biological circuits and related simple biological systems [

Any model contains several variables that do not represent the system state, but whose values govern the dynamics of the equations in the model. Such variables include reaction rate constants, equilibrium constants, diffusivities, and other physical properties. These are termed “parameters” of the model, as opposed to “state variables” such as species concentrations that represent the state of the system. To make useful predictions from a model, the parameters in the model have to be accurately estimated.

Mechanistic models, which are based on physical and chemical laws, include parameters that carry physical, chemical, or biological meaning. However, there could be many instances where not much information is available about a system, and constructing a “black box” model is the only option available. The parameters of such a model do not carry physical or biological meaning, but their estimation is nevertheless indispensable to the success of the model. Occasionally, information about a system could be so meager that even a black box model cannot be constructed. In such cases, a reverse engineering approach is employed to translate observable information to not only parameters but also model equations. This approach involves searching through (discrete) topological space instead of (continuous) numerical parameter space. Sometimes, combining the topological and numerical parameters for a system and simultaneously searching for both types of parameters has many advantages in understanding systems that are sparsely known [

Parameter estimation is known as the “inverse problem” or “model calibration” and is both a key step and a limiting step of model construction [

Parameter estimation is generally an optimization problem that involves locating the optimum (minimum or maximum) of an objective function that represents how well the model simulations agree with experimental data. This can be expressed as

The calculation of the first and second partial derivatives of the objective function is sometimes useful in optimization. Gradient search optimization algorithms depend on the partial derivatives of the objective function (or the partial derivative matrix, the Jacobian) for their success [

In gradient search methods it is not always possible to reach the global optimum of the objective function, especially for nonlinear objective functions that may have several local optima far away from the global optimum. Therefore, it may become necessary to sacrifice the speed of the gradient search methods for the exhaustive searching abilities of probabilistic methods [

After a satisfactory model is constructed, the analysis of the model and its predictions provides crucial input toward designing synthetic circuits that exhibit a given behavior. Here, we discuss two analysis techniques: sensitivity analysis and bifurcation analysis.

Sensitivity analysis, which analyzes how sensitive a system is with respect to changes in parameter values [

For local sensitivity analysis a sensitivity

Bifurcation analysis is crucial to understanding and analyzing steady states, oscillations, and other dynamic features of a system and has found use in numerous modeling studies [

Synthetic biology can also benefit from metabolic flux and transcription network analyses, which combine high-throughput experimental observations (such as metabolome, isotopomer, and gene expression profiles) with mathematical modeling to quantitatively describe the phenotype of a biological system. This type of analysis could be particularly useful for highly complex systems. For example, Noirel et al. [

Isotope-assisted [

Another set of powerful techniques for modeling metabolic networks includes flux balance analysis (FBA) [

Another valuable technique, metabolic control analysis [

Determining how genes are controlled by regulatory motifs is an important problem in biology. Because synthetic circuits are composed of well-characterized components, they can be used to investigate and quantify transcription networks. Such an investigation would employ a combinatorial technique to construct a circuit comprised of numerous genes and a smaller number of regulatory motifs [

Several standards and software are now available to simplify the process of building mathematical models and thereby bridge the gap between model description and prediction of the system’s behavior. System biology markup language (SBML) [

Below we present three case studies that illustrate several of the previously discussed modeling methodologies. The case studies feature two deterministic models [

Modeling techniques used in the case studies.

Case study | Model types | Modeling techniques | |||||

Deterministic | Stochastic | Parameter estimation | Optimization | Sensitivity analysis | Bifurcation analysis | ||

Case study I | Fung et al. [ | ||||||

Case study II | Friedland et al. [ | ||||||

Case study III | Guido et al. [ |

Fung et al. [_{1} (Acetyl-CoA) and M_{2} (lumped pool of Ack, AcP, _{1} and M_{2} while equation (10) and Michaelis-Menten rate laws were used to describe the kinetics of the enzymes driving the M_{1}-M_{2} interconversions. Using parameter values or ranges typical for this system, the authors implemented the model with a fourth-order Runge-Kutta algorithm. Parameter sensitivity analysis showed that increasing glycolytic flux increases the oscillatory capability of the system (Figure

Conceptual design and biological implementation of the oscillatory circuit metabolator in Fung et al. [_{1} and M_{2}; their interconversions are catalyzed by the enzymes _{2} at the transcriptional or translational level; the accumulation of M_{2} represses _{1}, which is converted to M_{2} by _{1} is high and M_{2} is low. However, M_{2} gradually accumulates causing _{2} to M_{1}, which then starts a new cycle. (b) Biological implementation. The design of the metabolator was implemented using the acetate pathway in _{1} pool is acetyl-CoA; the M_{2} pool consists of AcP, OAc^{-}, and HOAc. Pta and Acs correspond to enzymes ^{-} by Ack. The protonated form of OAc^{-} (HOAc) is permeable across the cell membrane. AcP is used as a signaling molecule and functions as follows. When AcP builds up, it will activate promoter ^{-}: acetate; Pta: phosphate acetyltransferase (adapted from Fung et al. [

Sensitivity and bifurcation analyses of the metabolator model in Fung et al. [

Hopf bifurcation was then used to characterize the dynamics of the model and determine the transition point at which the steady state would turn to a periodic state. Figure

This work represents a universal approach to construct a synthetic biological circuit with interesting dynamics and beautifully demonstrates the key role played by mathematical modeling in realizing a design concept. Modeling offered valuable insights on the analysis of experiment data and made nontrivial predictions of the system dynamics. The use of bifurcation analysis was particularly useful as it facilitated the determination of the points at which the system transitions between stable and periodic states. We expect that as oscillator design develops [

Another example of deterministic modeling is that of Friedland et al. [

Design concepts for the RTC three-counter in Friedland et al. [

The authors constructed and analyzed a mathematical model for the two- and three-counters. The model used equations of the form of (10) to describe the dynamics of the species in the circuit. The degradation terms in these equations were assumed to be simple exponential decays with a different rate constant for each species whereas the synthesis rates were rate laws that reflected how each species was synthesized. The Hill function was used to describe arabinose induction and the dynamics of GFP. Arabinose pulse dynamics were modeled with two differential equations as follows. Arabinose consumption from the medium was modeled as a zero order rate law:

Modeling predictions and analysis of the RTC three-counter in Friedland et al. [

The RTC two- and three-counters constitute an elegant example showing how synthetic circuit elements can be combined to recognize sequential events. Here too, the mathematical model was important in investigating the system dynamics and identifying the pulse length and interval that yielded the most effective response. Parameter estimation (and thereby optimization) provided crucial insights toward improving counter performance. As these counters are expanded to become capable of counting larger numbers of events (and thereby increasing in complexity), the role played by mathematical modeling in design will become increasingly important.

Guido et al. [

Design concept for the repressor-activator system in Guido et al. [

The authors first developed a baseline deterministic model for the repressor-activator system and then extended it to include stochastic effects. A quasi-equilibrium state was assumed for the promoter

The six possible binding states in the repressor-activator model in Guido et al. [

Steps in the repressor-activator mathematical model in Guido et al. [

Model predictions and analysis of the repressor-activator system in Guido et al. [

To further verify the predictive power of the model and test its stochastic aspects, the authors expanded the circuit to include a positive feedback. This was accomplished adding the

Although other researchers similarly built larger regulatory systems from simple ones (e.g., [

Mathematical modeling is an often indispensable tool in synthetic biology. The mathematical techniques of parameter estimation as well as sensitivity and bifurcation analyses can be crucial to the development of a model intended to mimic a complex system. Modeling also plays an important role in phenotypic analyses such as metabolic flux analysis or transcription network analysis.

A mathematical model is akin to a road map that provides a visualization of a geographical area. Although the map may not describe every detail of the landscape, it contains adequate information to enable users to plan a journey; a mathematical model is similar in scope [

This work was funded by the Department of Chemical and Biomolecular Engineering, University of Maryland (faculty startup grant to GS), the A. James Clark School of Engineering, University of Maryland (Minta Martin award to GS), and Maryland Industrial Partnerships (MIPS) (award no. 4426). Y. Zheng was cofunded by a Jan and Anneke Sengers fellowship.