Plasmodium knowlesi: a superb in vivo nonhuman primate model of antigenic variation in malaria.

Antigenic variation in malaria was discovered in Plasmodium knowlesi studies involving longitudinal infections of rhesus macaques (M. mulatta). The variant proteins, known as the P. knowlesi Schizont Infected Cell Agglutination (SICA) antigens and the P. falciparum Erythrocyte Membrane Protein 1 (PfEMP1) antigens, expressed by the SICAvar and var multigene families, respectively, have been studied for over 30 years. Expression of the SICA antigens in P. knowlesi requires a splenic component, and specific antibodies are necessary for variant antigen switch events in vivo. Outstanding questions revolve around the role of the spleen and the mechanisms by which the expression of these variant antigen families are regulated. Importantly, the longitudinal dynamics and molecular mechanisms that govern variant antigen expression can be studied with P. knowlesi infection of its mammalian and vector hosts. Synchronous infections can be initiated with established clones and studied at multi-omic levels, with the benefit of computational tools from systems biology that permit the integration of datasets and the design of explanatory, predictive mathematical models. Here we provide an historical account of this topic, while highlighting the potential for maximizing the use of P. knowlesi - macaque model systems and summarizing exciting new progress in this area of research.

The neurochemical mobile with non-linear interaction matrix: an exploratory computational model.

Several years ago, the "neurochemical mobile" was introduced as a visual tool for explaining the different balances between neurotransmitters in the brain and their role in mental disorders. Here we complement this concept with a non-linear computational systems model representing the direct and indirect interactions between neurotransmitters, as they have been described in the "neurochemical interaction matrix." The model is constructed within the framework of biochemical systems theory, which facilitates the mapping of numerically ill-characterized systems into a mathematical and computational construct that permits a variety of analyses. Simulations show how short- and long-term perturbations in any of the neurotransmitters migrate through the entire system, thereby affecting the balances within the mobile. In cases of short-term alterations, transients are of particular interest, whereas long-term changes shed light on persistently altered, allostatic states, which in mental diseases and sleep disorders could be due to a combination of unfavorable factors, resulting from a specific genetic predisposition, epigenetic effects, disease, or the repeated use of drugs, such as opioids and amphetamines.

Mesoscopic models of neurotransmission as intermediates between disease simulators and tools for discovering design principles.

Two grand challenges have been declared as premier goals of computational systems biology. The first is the discovery of network motifs and design principles that help us understand and rationalize why biological systems are organized in the manner we encounter them rather than in a different fashion. The second goal is the development of computational models supporting the investigation of complex systems, in particular, as simulation platforms in personalized medicine and predictive health. Interestingly, most published systems models in biology contain between a handful and a few dozen variables. They are usually too complicated for systemic analyses of organizing principles, but they are at the same time too coarse to allow reliable simulations of diseases. While it may thus appear that the modeling efforts of the past have missed the declared targets of systems biology, we argue in this article that midsized mesoscopic models are excellent starting points for pursuing both goals in computational systems biology.

Effects of dopamine and glutamate on synaptic plasticity: a computational modeling approach for drug abuse as comorbidity in mood disorders.

Major depressive disorder (MDD) affects about 16% of the general population and is a leading cause of death in the United States and around the world. Aggravating the situation is the fact that "drug use disorders" are highly comorbid in MDD patients, and VICE VERSA. Drug use and MDD share a common component, the dopamine system, which is critical in many motivation and reward processes, as well as in the regulation of stress responses in MDD. A potentiating mechanism in drug use disorders appears to be synaptic plasticity, which is regulated by dopamine transmission. In this article, we describe a computational model of the synaptic plasticity of GABAergic medium spiny neurons in the nucleus accumbens, which is critical in the reward system. The model accounts for effects of both dopamine and glutamate transmission. Model simulations show that GABAergic medium spiny neurons tend to respond to dopamine stimuli with synaptic potentiation and to glutamate signals with synaptic depression. Concurrent dopamine and glutamate signals cause various types of synaptic plasticity, depending on input scenarios. Interestingly, the model shows that a single 0.5 mg/kg dose of amphetamine can cause synaptic potentiation for over 2 h, a phenomenon that makes synaptic plasticity of medium spiny neurons behave quasi as a bistable system. The model also identifies mechanisms that could potentially be critical to correcting modifications of synaptic plasticity caused by drugs in MDD patients. An example is the feedback loop between protein kinase A, phosphodiesterase, and the second messenger cAMP in the postsynapse. Since reward mechanisms activated by psychostimulants could be crucial in establishing addiction comorbidity in patients with MDD, this model might become an aid for identifying and targeting specific modules within the reward system and lead to a better understanding and potential treatment of comorbid drug use disorders in MDD.

Computational modeling of synaptic neurotransmission as a tool for assessing dopamine hypotheses of schizophrenia.

Schizophrenia is a severe and complex mental disorder that causes an enormous societal and financial burden. Following the identification of dopamine as a neurotransmitter and the invention of antipsychotic drugs, the dopamine hypothesis was formulated to suggest hyperdopaminergia as the cause of schizophrenia. Over time there have been modifications and improvements to the dopamine-based model of schizophrenia, as well as models that do not implicate dopamine dysregulation as a primary cause of the disease. It seems clear by now that disruption of dopamine homeostasis occurs in schizophrenia and likely plays a major contributory role to its symptoms. Three primary versions of the dopamine hypothesis of schizophrenia have been proposed. In this article, we review these hypotheses and subject their assumptions to a computational model of dopamine signaling. Based on this review and analysis, we propose slight revisions to the existing hypotheses. Although we are still at the beginning of a comprehensive modeling effort to capture relevant phenomena associated with schizophrenia, our preliminary models have already yielded intriguing results and identified the systems biological approach as a beneficial complement to clinical and experimental research and a powerful method for exploring human diseases like schizophrenia. It is hoped that the past, present and future models will support and guide refined experimentation and lead to a deeper understanding of schizophrenia.

Approximation of delays in biochemical systems.

In the past metabolic pathway analyses have mostly ignored the effects of time delays that may be due to processes that are slower than biochemical reactions, such as transcription, translation, translocation, and transport. We show within the framework of biochemical systems theory (BST) that delay processes can be approximated accurately by augmenting the original variables and non-linear differential equations with auxiliary variables that are defined through a system of linear ordinary differential equations. These equations are naturally embedded in the structure of S-systems and generalized mass action systems within BST and can be interpreted as linear signaling pathways or cascades. We demonstrate the approximation method with the simplest generic modules, namely single delayed steps with and without feedback inhibition. These steps are representative though, because they are easily incorporated into larger systems. We show that the dynamics of the approximated systems reflects that of the original delay systems well, as long as the systems do not operate in very close vicinity of threshold values where the systems lose stability. The accuracy of approximation furthermore depends on the selected number of auxiliary variables. In the most relevant situations where the systems operate at states away from their critical thresholds, even a few auxiliary variables lead to satisfactory approximations.

Canonical modeling: review of concepts with emphasis on environmental health.

The article reviews concepts of canonical modeling in the context of environmental health. Based on biochemical systems theory, the canonical approach was developed over the past thirty years and applied to complex systems primarily in biochemistry and the regulation of gene expression. Canonical modeling is based on nonlinear ordinary differential equations whose right-hand sides consist of products of power-law functions. This structure results from the linearization of complex processes in logarithmic space. The canonical structure has many intriguing features. First, almost any system of smooth functions or ordinary differential equations can be recast equivalently in a canonical model, which demonstrates that the model structure is rich enough to deal with all relevant nonlinearities. Second, a large body of successful applications suggests that canonical models are often valid and accurate representations of quite complex, real-world systems. Third, a set of guidelines supports the modeler in all phases of analysis. These guidelines address model design, algebraic and numerical analysis, and the interpretation of results. Fourth, the structure of canonical models, especially those in S-system form, facilitates algebraic and numerical analyses. Of particular importance is the derivation of steady-state solutions in an explicit symbolic or numerical form, which allows further assessments of stability and robustness. The homogeneous structure of canonical models has also led to the development of very efficient, customized computer algorithms for all steps of a typical analysis. Fifth, a surprising number of models currently used in environmental health research are special cases of canonical models. The traditional models are thus subsumed in one modeling framework, which offers new avenues of analysis and interpretation.

Encounters in predator-prey systems: a simple discrete model.

Predator-prey systems are often described by exploitation models. These models can mimic experimental data very accurately, but it is sometimes difficult to realize the relationships between the models and the behavior of individual predator and prey animals. A simple discrete model is proposed here that tries to elucidate the connections between: the animals' movements, the predator/prey encounters; and the dynamics in the system as globally represented by the exploitation models. In these models, the term "area of discovery" plays an essential role. This term is shown to be a predictable coefficient that is composed of measurable physical properties of the analyzed predator-prey system. The model takes into account that predators and prey in experimental systems often do not search randomly but prefer some parts of the test area. The model is applied to the mite system Phytoseiulus persimilis/Tetranychus urticae under simple artificial conditions.

Symmetries of S-systems.

An S-system is a set of first-order nonlinear differential equations that all have the same structure: The derivative of a variable is equal to the difference of two products of power-law functions. S-systems have been used as models for a variety of problems, primarily in biology. In addition, S-systems possess the interesting property that large classes of differential equations can be recast exactly as S-systems, a feature that has been proven useful in statistics and numerical analysis. Here, simple criteria are introduced that determine whether an S-system possesses certain types of symmetries and how the underlying transformation groups can be constructed. If a transformation group exists, families of solutions can be characterized, the number of S-system equations necessary for solution can be reduced, and some boundary value problems can be reduced to initial value problems.

Mathematical models of purine metabolism in man.

Experimental and clinical data on purine metabolism are collated and analyzed with three mathematical models. The first model is the result of an attempt to construct a traditional kinetic model based on Michaelis-Menten rate laws. This attempt is only partially successful, since kinetic information, while extensive, is not complete, and since qualitative information is difficult to incorporate into this type of model. The data gaps necessitate the complementation of the Michaelis-Menten model with other functional forms that can incorporate different types of data. The most convenient and established representations for this purpose are rate laws formulated as power-law functions, and these are used to construct a Complemented Michaelis-Menten (CMM) model. The other two models are pure power-law-representations, one in the form of a Generalized Mass Action (GMA) system, and the other one in the form of an S-system. The first part of the paper contains a compendium of experimental data necessary for any model of purine metabolism. This is followed by the formulation of the three models and a comparative analysis. For physiological and moderately pathological perturbations in metabolites or enzymes, the results of the three models are very similar and consistent with clinical findings. This is an encouraging result since the three models have different structures and data requirements and are based on different mathematical assumptions. Significant enzyme deficiencies are not so well modeled by the S-system model. The CMM model captures the dynamics better, but judging by comparisons with clinical observations, the best model in this case is the GMA model. The model results are discussed in some detail, along with advantages and disadvantages of each modeling strategy.

So, you want to be a systems biologist? Determinants for creating graduate curricula in systems biology.

Systems biology is uniquely situated at the interface of computing, mathematics, engineering and the biological sciences. This positioning creates unique challenges and opportunities over other interdisciplinary studies when developing academic curricula. Integrative systems biology attempts to span the field from observation to innovation, and thus requires successful students to gain skills from mining to manipulation. The authors outline examples of graduate program structures, as well as curricular aspects and assessment metrics that can be customised around the environmental niche of the academic institution towards the formalisation of effective educational opportunities in systems biology. Some of this material was presented at the 2009 Foundations of Systems Biology in Engineering (FOSBE 2009) Conference in Denver, August 2009.

Estimation of metabolic pathway systems from different data sources.

Parameter estimation is the main bottleneck of metabolic pathway modelling. It may be addressed from the bottom up, using information on metabolites, enzymes and modulators, or from the top down, using metabolic time series data, which have become more prevalent in recent years. The authors propose here that it is useful to combine the two strategies and to complement time-series analysis with kinetic information. In particular, the authors investigate how the recent method of dynamic flux estimation (DFE) may be supplemented with other types of estimation. Using the glycolytic pathway in Lactococcus lactis as an illustration example, the authors demonstrate some strategies of such supplementation.

Steps of modeling complex biological systems.

A disease like schizophrenia results from the malfunctioning of a complex, multi-faceted biological system. As a consequence, the root causes of such a disease and the trajectories from health toward the disease are very difficult to comprehend with simple cause-and-effect reasoning. Similarly, reductionistic investigations are crucial for the discovery of specific disease mechanisms, but they are not sufficient for comprehensive assessments and explanations. A promising option for advancing the field is the utilization of mathematical models that can quantitatively account for hundreds of components and their interactions and thus have the potential of truly explaining complex diseases. While the potential of mathematical models is quite evident in principle, their practical implementation is a daunting task. On the one hand, many distinctly different approaches are possible. For instance, in the case of schizophrenia, models could focus on neurological aspects, physiological features, or the biochemical malfunctioning within some cell complexes in the brain, and each model would ultimately be very different. On the other hand, it seems that there are no rules or recommendations that guide the development of a new mathematical model from scratch. We discuss here that, even though mathematical models in biology and medicine may ultimately have a very different appearance, their development can be structured as a sequence of generic steps. Major drivers for many of the details of model development are the goals and objectives of the modeling task and the availability and quality of data that can be used for model design and validation.

A mathematical model of presynaptic dopamine homeostasis: implications for schizophrenia.

Several lines of evidence implicate altered dopamine neurotransmission in schizophrenia. Current drugs for schizophrenia focus on postsynaptic sites of the dopamine signaling pathways, but do not target presynaptic dopamine metabolism. We have begun to develop a mathematical model of dopamine homeostasis, which will aid our understanding of how genetic, environmental, and pharmacological factors alter the functioning of the presynaptic dopamine neuron. Formulated within the modeling framework of BIOCHEMICAL SYSTEMS THEORY, the mathematical model integrates relevant metabolites, enzymes, transporters, and regulators involved in the control of the biochemical environment within the dopamine neuron. In this report we use the model to assess several components and factors that affect the dopamine neuron and have been implicated in schizophrenia. These include the enzymes COMT, MAO, and TH, different dopamine transporters, as well as administration of amphetamine or cocaine. We also investigate scenarios that could increase (or decrease) dopamine neurotransmission and thus exacerbate (or alleviate) symptoms of schizophrenia. Our results indicate that the model predicts the effects of various factors related to schizophrenia on the homeostasis of the presynaptic dopamine neuron rather well. Upon further refinements and testing, the model has the potential of serving as a tool for screening novel therapeutics aimed at altering presynaptic dopamine function and thereby potentially ameliorating some of the symptomology of schizophrenia.

Challenges in lin-log modelling of glycolysis in Lactococcus lactis.

The performance of the lin-log method for modelling the glycolytic pathway in Lactococcus lactis using in vivo time-series data is investigated. The network structure of this pathway has been studied in previous reports and the authors concentrate here on the challenge of fitting the lin-log model parameters to experimental data. To calibrate the estimation methods, the performance of the lin-log method on a simpler model of a small gene regulatory system was first investigated, which has become a benchmark in the field. Two families of optimisation algorithms were employed. One computes the objective function by solving a system of ordinary differential equations (ODEs), whereas the other discretises the ODEs and incorporates them as nonlinear equality constraints in the optimisation problem. Gradient-based, simplex-based and stochastic search algorithms were used to solve the former, whereas only a gradient-based algorithm was used to solve the latter. Although the estimation methods succeeded in determining the parameter values for the small gene network model, they did not yield a satisfactory lin-log model for the glycolytic pathway. The main reasons are apparently that several system variables approach low, and ultimately zero concentrations, which are intrinsically problematic for lin-log models, and that this pathway does not offer a natural non-zero reference state. [Includes supplementary material.].

Regulation of glycolysis in Lactococcus lactis: an unfinished systems biological case study.

The unexpectedly long, and still unfinished, path towards a reliable mathematical model of glycolysis and its regulation in Lactococcus lactis is described. The model of this comparatively simple pathway was to be deduced from in vivo nuclear magnetic resonance time-series measurements of the key glycolytic metabolites. As to be expected from any nonlinear inverse problem, computational challenges were encountered in the numerical determination of parameter values of the model. Some of these were successfully solved, whereas others are still awaiting improved techniques of analysis. In addition, rethinking of the model formulation became necessary, because some generally accepted assumptions during model design are not necessarily valid for in vivo models. Examples include precursor-product relationships and the homogeneity of cells and their responses. Finally, it turned out to be useful to model only some of the metabolites, while using time courses of ubiquitous compounds such as adenosine triphosphate, inorganic phosphate, nicotinamide adenine dinucleotide (oxidised) and nicotinamide adenine dinucleotide (reduced) as unmodelled input functions. With respect to our specific application, the modelling process has come a long way, but it is not yet completed. Nonetheless, the model analysis has led to interesting insights into the design of the pathway and into the principles that govern its operation. Specifically, the widely observed feedforward activation of pyruvate kinase by fructose 1,6-bisphosphate is shown to provide a crucial mechanism for positioning the starving organism in a holding pattern that allows immediate uptake of glucose, as soon as it becomes available.

Smooth bistable S-systems.

S-systems have been used as models of biochemical systems for over 30 years. One of their hallmarks is that, although they are highly non-linear, their steady states are characterised by linear equations. This allows streamlined analyses of stability, sensitivities and gains as well as objective, mathematically controlled comparisons of similar model designs. Regular S-systems have a unique steady state at which none of the system variables is zero. This makes it difficult to represent switching phenomena, as they occur, for instance, in the expression of genes, cell cycle phenomena and signal transduction. Previously, two strategies were proposed to account for switches. One was based on a technique called recasting, which permits the modelling of any differentiable non-linearities, including bistability, but typically does not allow steady-state analyses based on linear equations. The second strategy formulated the switching system in a piece-wise fashion, where each piece consisted of a regular S-system. A representation gleaned from a simplified form of recasting is proposed and it is possible to divide the characterisation of the steady states into two phases, the first of which is linear, whereas the other is non-linear, but easy to execute. The article discusses a representative pathway with two stable states and one unstable state. The pathway model exhibits strong separation between the stable states as well as hysteresis.

Cell cycles and growth laws: the CCC model.

In the cell-cycle-with-control model (CCC model), cells have to satisfy a condition before they are allowed to pass a control point during G1. Different cycle durations within a cell population are explained by individual time spans needed to satisfy the passing condition. If the distribution of cycle durations is time invariant, the population will grow exponentially. However, if the average cycle duration becomes longer, while the population grows, non-exponential population growth results. Simple functions for the lengthening of the average cycle duration, like linear or exponential ones, yield the well-known growth laws found in the biological literature. The same functions can be represented by an "S-system" differential equation that was derived earlier as an approximation for biochemical systems with many fast reactions (metabolism) and one slow process (e.g. ageing).

Optimization in integrated biochemical systems.

As of yet, steady-state optimization in biochemical systems has been limited to a few studies in which networks of fluxes were optimized. These networks of fluxes are represented by linear (stoichiometric) equations that are used as constraints in a linear program, and a flux or a sum of weighted fluxes is used as the objective function. In contrast to networks of fluxes, systems of metabolic processes have not been optimized in a comparable manner. The primary reason is that these types of integrated biochemical systems are full of synergisms, antagonisms, and regulatory mechanisms that can only be captured appropriately with nonlinear models. These models are mathematically complex and difficult to analyze. In most cases it is not even possible to compute, let alone optimize, steady-state solutions analytically. Rare exceptions are S-system representations. These are nonlinear and able to represent virtually all types of dynamic behaviors, but their steady states are characterized by linear equations that can be evaluated both analytically and numerically. The steady-state equations are expressed in terms of the logarithms of the original variables, and because a function and its logarithms of the original variables, and because a function and its logarithm assume their maxima for the same argument, yields or fluxes can be optimized with linear programs expressed in terms of the logarithms of the original variables.

Utility of Biochemical Systems Theory for the analysis of metabolic effects from low-dose chemical exposure.

Adverse health outcomes from exposure to chemical agents are of increasing interest in human and ecological risk assessment and require the development of new analytical methods. Such methods must be able to capture the essence of integrated networks of biochemical pathways in a mathematically feasible fashion. Over the past three decades, Biochemical Systems Theory has been successfully applied to numerous biological systems. It is suggested here that S-system models derived from BST can provide the means for assessing chemical exposures and their effects at the metabolic level. This article briefly reviews essential concepts of S-systems and provides generic examples of chemical exposure scenarios. S-system models can be considered mechanistic, since their components are measurable quantities (e.g., concentrations, fluxes, enzyme activities, and rates). As dynamic models, they can be used to assess immediate and long-term metabolic responses to environmental stimuli. Direct mathematical analysis for low exposures leads to simple dose-response relationships, which have the form of power-law functions. Thus, if the S-system model yields an appropriate description of chemical exposure and its metabolic effects, the dose-response relationship for low exposures is linear in logarithmic coordinates. This result includes as a special case the standard linear relationship in Cartesian coordinates with zero intercept.