Computational methods in synthetic biology

We discuss how a theoretical synthetic biology research programme may liberate empiricism in biological sciences beyond the unaided human brain. Because synthetic biological systems are relatively small and largely independent of evolutionary contexts, they can be represented with mathematical models strongly founded on first principles of molecular biology and laws of statistical thermodynamics. A universal mathematical formalism for describing synthetic constructs may then be plausibly used to explain in unambiguous, quantitative terms how biological phenotypic complexity emerges as a result of well-defined biomolecular interactions. SynBioSS, a publicly available software package, is described that implements this mathematical formalism.     

Natural selection and self-organization

The Darwinian concept of natural selection was conceived within a set of Newtonian background assumptions about systems dynamics. Mendelian genetics at first did not sit well with the gradualist assumptions of the Darwinian theory. Eventually, however, Mendelism and Darwinism were fused by reformulating natural selection in statistical terms. This reflected a shift to a more probabilistic set of background assumptions based upon Boltzmannian systems dynamics. Recent developments in molecular genetics and paleontology have put pressure on Darwinism once again. Current work on self-organizing systems may provide a stimulus not only for increased problem solving within the Darwinian tradition, especially with respect to origins of life, developmental genetics, phylogenetic pattern, and energy-flow ecology, but for deeper understanding of the very phenomenon of natural selection itself. Since self-organizational phenomena depend deeply on stochastic processes, self-organizational systems dynamics advance the probability revolution. In our view, natural selection is an emergent phenomenon of physical and chemical selection. These developments suggest that natural selection may be grounded in physical law more deeply than is allowed by advocates of the autonomy of biology, while still making it possible to deny, with autonomists, that evolutionary explanations can be modeled in terms of a deductive relationship between laws and cases. We explore the relationship between, chance, self-organization, and selection as sources of order in biological systems in order to make these points.     

Challenges for the identification of biological systems from in vivo time series data

Modern methods of high-throughput molecular biology render it possible to generate time series of metabolite concentrations and the expression of genes and proteins in vivo. These time profiles contain valuable information about the structure and dynamics of the underlying biological system. This information is implicit and its extraction is a challenging but ultimately very rewarding task for the mathematical modeler. Using a well-suited modeling framework, such as Biochemical Systems Theory (BST), it is possible to formulate the extraction of information as an inverse problem that in principle may be solved with a genetic algorithm or nonlinear regression. However, two types of issues associated with this inverse problem make the extraction task difficult. One type pertains to the algorithmic difficulties encountered in nonlinear regressions with moderate and large systems. The other type is of an entirely different nature. It is a consequence of assumptions that are often taken for granted in the design and analysis of mathematical models of biological systems and that need to be revisited in the context of inverse analyses. The article describes the extraction process and some of its challenges and proposes partial solutions.     

Temperature-induced complementarity as a mechanism for biomolecular assembly

Recent advances in the measurement and theory of “hydration” interactions between biomolecules provide a basis on which to formulate mechanisms of biomolecular recognition. In this paper we have developed a mathematical formalism for analyzing specificity encoded in dynamic distributions of surface polar groups, a formalism that incorporates newly recognized properties of directly measured “hydration” forces. As expected, attraction between surfaces requires complementary patterns of surface polar groups. In contrast to usual expectations, thermal motion can create these complementary surface configurations. We have demonstrated that assembly can occur with an increase in conformational entropy of polar residues. Elevated temperature then facilitates recognition rather than hinders it. This mechanism might underlie some temperature-favored assembly reactions common in biological systems that are usually associated with the “hydrophobic effect” only. © 1994 Wiley-Liss, Inc.     

Thermodynamically feasible kinetic models of reaction networks

The dynamics of biological reaction networks are strongly constrained by thermodynamics. An holistic understanding of their behavior and regulation requires mathematical models that observe these constraints. However, kinetic models may easily violate the constraints imposed by the principle of detailed balance, if no special care is taken. Detailed balance demands that in thermodynamic equilibrium all fluxes vanish. We introduce a thermodynamic-kinetic modeling (TKM) formalism that adapts the concepts of potentials and forces from irreversible thermodynamics to kinetic modeling. In the proposed formalism, the thermokinetic potential of a compound is proportional to its concentration. The proportionality factor is a compound-specific parameter called capacity. The thermokinetic force of a reaction is a function of the potentials. Every reaction has a resistance that is the ratio of thermokinetic force and reaction rate. For mass-action type kinetics, the resistances are constant. Since it relies on the thermodynamic concept of potentials and forces, the TKM formalism structurally observes detailed balance for all values of capacities and resistances. Thus, it provides an easy way to formulate physically feasible, kinetic models of biological reaction networks. The TKM formalism is useful for modeling large biological networks that are subject to many detailed balance relations.     

Extending and expanding the Darwinian synthesis: the role of complex systems dynamics

Darwinism is defined here as an evolving research tradition based upon the concepts of natural selection acting upon heritable variation articulated via background assumptions about systems dynamics. Darwin's theory of evolution was developed within a context of the background assumptions of Newtonian systems dynamics. The Modern Evolutionary Synthesis, or neo-Darwinism, successfully joined Darwinian selection and Mendelian genetics by developing population genetics informed by background assumptions of Boltzmannian systems dynamics. Currently the Darwinian Research Tradition is changing as it incorporates new information and ideas from molecular biology, paleontology, developmental biology, and systems ecology. This putative expanded and extended synthesis is most perspicuously deployed using background assumptions from complex systems dynamics. Such attempts seek to not only broaden the range of phenomena encompassed by the Darwinian Research Tradition, such as neutral molecular evolution, punctuated equilibrium, as well as developmental biology, and systems ecology more generally, but to also address issues of the emergence of evolutionary novelties as well as of life itself.     

Metabolic Engineering with power-law and linear-logarithmic systems

Metabolic Engineering aims to improve the performance of biotechnological processes through rational manipulation rather than random mutagenesis of the organisms involved. Such a strategy can only succeed when a mathematical model of the target process is available. Simplifying assumptions are often needed to cope with the complexity of such models in an efficient way, and the choice of such assumptions often leads to models that fall within a certain structural template or formalism. The most popular formalisms can be grouped in two categories: power-law and linear-logarithmic. As optimization and analysis of a model strongly depends on its structure, most methods in Metabolic Engineering have been defined within a given formalism and never used in any other.In this work, the four most commonly used formalisms (two power-law and two linear-logarithmic) are placed in a common framework defined within Biochemical Systems Theory. This framework defines every model as matrix equations in terms of the same parameters, enabling the formulation of a common steady state analysis and providing means for translating models and methods from one formalism to another. Several Metabolic Engineering methods are analysed here and shown to be variants of a single equation. Particularly, two problem solving philosophies are compared: the application of the design equation and the solution of constrained optimization problems. Generalizing the design equation to all the formalisms shows it to be interchangeable with the direct solution of the rate law in matrix form. Furthermore, optimization approaches are concluded to be preferable since they speed the exploration of the feasible space, implement a better specification of the problem and exclude unrealistic results.Beyond consolidating existing knowledge and enabling comparison, the systematic approach adopted here can fill the gaps between the different methods and combine their strengths.     

Center Finding in E. coli and the Role of Mathematical Modeling: Past,Present and Future

We review the key role played by mathematical modeling in elucidating two center-finding patterning systems in Escherichia coli: midcell division positioning by the MinCDE system and DNA partitioning by the ParABS system. We focus particularly on how, despite much experimental effort, these systems were simply too complex to unravel by experiments alone, and instead required key injections of quantitative, mathematical thinking. We conclude the review by analyzing the frequency of modeling approaches in microbiology over time. We find that while such methods are increasing in popularity, they are still probably heavily under-utilized for optimal progress on complex biological questions.     

Linear analysis of switching nets

Summary The switching net that has been used as a model of several biological systems is studied using the mathematical concept of finite field. Since every function over a finite field can be represented as a terminating series, all switching nets can be linearized and thus represented by a matrix. In fact two matrices—the function matrix and the transition matrix—both represent the net. These matrices are related by an orthogonal transformation. The autonomous behavior of switching nets is obtained from a consideration of the characteristic equation and eigenvectors of either of the matrices. A simple criterion for the invertibility of a set of switching functions is obtained from this formalism. A test for invertibility proposed by Huffman is discussed, and it is show that the Jacobian test is invalid for finite fields.This paper is based on a thesis submitted to The University of Chicago in partial fulfillment of the requirements for the Doctor of Philosophy in Mathematical Biology. The research was supported by USPHS Training Grant 1-TO1 GM-2037.     

Transmission assumptions generate conflicting predictions in host-vector disease models: a case study in West Nile virus

This review synthesizes the conflicting outbreak predictions generated by different biological assumptions in host–vector disease models. It is motivated by the North American outbreak of West Nile virus, an emerging infectious disease that has prompted at least five dynamical modelling studies. Mathematical models have long proven successful in investigating the dynamics and control of infectious disease systems. The underlying assumptions in these epidemiological models determine their mathematical structure, and therefore influence their predictions. A crucial assumption is the host–vector interaction encapsulated in the disease-transmission term, and a key prediction is the basic reproduction number, R ₀. We connect these two model elements by demonstrating how the choice of transmission term qualitatively and quantitatively alters R ₀ and therefore alters predicted disease dynamics and control implications. Whereas some transmission terms predict that reducing the host population will reduce disease outbreaks, others predict that this will exacerbate infection risk. These conflicting predictions are reconciled by understanding that different transmission terms apply biologically only at certain population densities, outside which they can generate erroneous predictions. For West Nile virus, R ₀ estimates for six common North American bird species indicate that all would be effective outbreak hosts.     

Accuracy of alternative representations for integrated biochemical systems

The Michaelis-Menten formalism often provides appropriate representations of individual enzyme-catalyzed reactions in vitro but is not well suited for the mathematical analysis of complex biochemical networks. Mathematically tractable alternatives are the linear formalism and the power-law formalism. Within the power-law formalism there are alternative ways to represent biochemical processes, depending upon the degree to which fluxes and concentrations are aggregated. Two of the most relevant variants for dealing with biochemical pathways are treated in this paper. In one variant, aggregation leads to a rate law for each enzyme-catalyzed reaction, which is then represented by a power-law function. In the other, aggregation produces a composite rate law for either net rate of increase or net rate of decrease of each system constituent; the composite rate laws are then represented by a power-law function. The first variant is the mathematical basis for a method of biochemical analysis called metabolic control, the latter for biochemical systems theory. We compare the accuracy of the linear and of the two power-law representations for networks of biochemical reactions governed by Michaelis-Menten and Hill kinetics. Michaelis-Menten kinetics are always represented more accurately by power-law than by linear functions. Hill kinetics are in most cases best modeled by power-law functions, but in some cases linear functions are best. Aggregation into composite rate laws for net increase or net decrease of each system constituent almost always improves the accuracy of the power-law representation. The improvement in accuracy is one of several factors that contribute to the wide range of validity of this power-law representation. Other contributing factors that are discussed include the nonlinear character of the power-law formalism, homeostatic regulatory mechanisms in living systems, and simplification of rate laws by regulatory mechanisms in vivo.     

Designing lymphocyte functional structure for optimal signal detection: voilà, T cells

One basic task of immune systems is to detect signals from unknown "intruders" amidst a noisy background of harmless signals. To clarify the functional importance of many observed lymphocyte properties, I ask: What properties would a cell have if one designed it according to the theory of optimal detection, with minimal regard for biological constraints? Sparse and reasonable assumptions about the statistics of available signals prove sufficient for deriving many features of the optimal functional structure, in an incremental and modular design. The use of one common formalism guarantees that all parts of the design collaborate to solve the detection task. Detection performance is computed at several stages of the design. Comparison between design variants reveals e.g. the importance of controlling the signal integration time. This predicts that an appropriate control mechanism should exist. Comparing the design to reality, I find a striking similarity with many features of T cells. For example, the formalism dictates clonal specificity, serial receptor triggering, (grades of) anergy, negative and positive selection, co-stimulation, high-zone tolerance, and clonal production of cytokines. Serious mismatches should be found if T cells were hindered by mechanistic constraints or vestiges of their (co-)evolutionary history, but I have not found clear examples. By contrast, fundamental mismatches abound when comparing the design to immune systems of e.g. invertebrates. The wide-ranging differences seem to hinge on the (in)ability to generate a large diversity of receptors.     

Efficacy of artesunate-amodiaquine for the treatment of acute uncomplicated falciparum malaria in southern Mauritania

Mathematical analyses and modelling have an important role informing malaria eradication strategies. Simple mathematical approaches can answer many questions, but it is important to investigate their assumptions and to test whether simple assumptions affect the results. In this note, four examples demonstrate both the effects of model structures and assumptions and also the benefits of using a diversity of model approaches. These examples include the time to eradication, the impact of vaccine efficacy and coverage, drug programs and the effects of duration of infections and delays to treatment, and the influence of seasonality and migration coupling on disease fadeout. An excessively simple structure can miss key results, but simple mathematical approaches can still achieve key results for eradication strategy and define areas for investigation by more complex models.     

A note on the timing of tradeoffs in discrete life history models

We discuss the timing of tradeoffs in discrete life history models. With a simple mathematical example we show that different assumptions about the temporal order of costs and benefits resulting from a reproductive effort can lead to qualitatively different predictions. We examine two models taken from the literature, in which an implicit assumption is that benefits from reproductive efforts are received before the corresponding costs are paid. We show that the reverse assumptions would have led to very different results. Since there is no biological basis for a bias towards a particular set of assumptions, we conclude that a more flexible approach should be used when studying optimality problems that are based on discrete life histories.     

Mathematical assumptions versus biological reality: myths in affected sib pair linkage analysis

Affected sib pair (ASP) analysis has become common ever since it was shown that, under very specific assumptions, ASPs afford a powerful design for linkage analysis. In 2003, Vieland and Huang, on the basis of a "fundamental heterogeneity equation," proved that heterogeneity and epistasis are confounded in ASP linkage analysis. A much more serious limitation of ASP linkage analysis is the implicit assumption that randomly sampled sib pairs share half their alleles identical by descent at any locus, whereas a critical assumption underlying Vieland and Huang's proof is that of joint Hardy-Weinberg equilibrium proportions at two trait loci. These are considered as examples of mathematical assumptions that may not always reflect biological reality. More-robust sib-pair designs and appropriate methods for their analysis have long been available.     

Modeling and simulation of biological systems with stochasticity

Mathematical modeling is a powerful approach for understanding the complexity of biological systems. Recently, several successful attempts have been made for simulating complex biological processes like metabolic pathways, gene regulatory networks and cell signaling pathways. The pathway models have not only generated experimentally verifiable hypothesis but have also provided valuable insights into the behavior of complex biological systems. Many recent studies have confirmed the phenotypic variability of organisms to an inherent stochasticity that operates at a basal level of gene expression. Due to this reason, development of novel mathematical representations and simulations algorithms are critical for successful modeling efforts in biological systems. The key is to find a biologically relevant representation for each representation. Although mathematically rigorous and physically consistent, stochastic algorithms are computationally expensive, they have been successfully used to model probabilistic events in the cell. This paper offers an overview of various mathematical and computational approaches for modeling stochastic phenomena in cellular systems.