The fundamental equations of change in statistical ensembles and biological populations

A recent article in Nature Physics unified key results from thermodynamics, statistics, and information theory. The unification arose from a general equation for the rate of change in the information content of a system. The general equation describes the change in the moments of an observable quantity over a probability distribution. One term in the equation describes the change in the probability distribution. The other term describes the change in the observable values for a given state. We show the equivalence of this general equation for moment dynamics with the widely known Price equation from evolutionary theory, named after George Price. We introduce the Price equation from its biological roots, review a mathematically abstract form of the equation, and discuss the potential for this equation to unify diverse mathematical theories from different disciplines. The new work in Nature Physics and many applications in biology show that this equation also provides the basis for deriving many novel theoretical results within each discipline

Exploration of the spontaneous fluctuating activity of single enzyme molecules

Single enzyme molecules display inevitable, stochastic fluctuations in their catalytic activity. In metabolism, for instance, the stochastic activity of individual enzymes is averaged out due to their high copy numbers per single cell. However, many processes inside cells rely on single enzyme activity, such as transcription, replication, translation, and histone modifications. Here we introduce the main theoretical concepts of stochastic single-enzyme activity starting from the Michaelis–Menten enzyme mechanism. Next, we discuss stochasticity of multi-substrate enzymes, of enzymes and receptors with multiple conformational states and finally, how fluctuations in receptor activity arise from fluctuations in signal concentration. This paper aims to introduce the exciting field of single-molecule enzyme kinetics and stochasticity to a wider audience of biochemists and systems biologists

Zin en onzin van het onderzoek van fecesstalen van honden met diarree op de aanwezigheid van Escherichia coli

Bacteriologisch onderzoek van fecesstalen van honden met diarree resulteert steevast in de isolatie van Escherichia coli. Het verband met de waargenomen klinische symptomen is tot nu toe niet duidelijk. Daarom werd in deze studie nagegaan of de aanwezigheid van bepaalde E. coli virulentiegenen geassocieerd is met de aanwezigheid van diarree bij honden. Hiertoe werden E. coli-isolaten uit fecesstalen van 34 gezonde honden en 25 honden met diarree (leeftijd: 7,5 maanden tot 10 jaar) onderzocht op de aanwezigheid van 17 virulentiegenen. Er konden geen virulentiegenen aangetoond worden in 15 van de 34 E. coli-isolaten van gezonde dieren en in 15 van de 25 E. coli-isolaten van dieren met diarree. In de overige isolaten werden genen teruggevonden die coderen voor een of meerdere van de toxinen cytotoxisch necrotiserende factor (CNF)l, CNF2, verotoxine (VT)1 en VT2 en/of voor een of meerdere van de adhesinen intimine, F5 fimbriae en F41 fimbriae. Er werden evenwel geen significante verschillen aangetoond in het voorkomen van deze virulentiegenen tussen de isolaten afkomstig van de onderzochte volwassen honden met of zonder diarree. Zo lang geen diagnostische merkers worden gevonden voor E. coli. starnmen die geassocieerd zijn met diarree bij honden is het dus weinig zinvol om de aanwezigheid van deze bacterie in de mest te laten bepalen

A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems.

Biochemical systems are bound by two mathematically-relevant restrictions. First, state variables in such systems represent non-negative quantities, such as concentrations of chemical compounds. Second, biochemical systems conserve mass and energy. Both properties must be reflected in results of an integration scheme applied to biochemical models. This paper first presents a mathematical framework for biochemical problems, which includes an exact definition of biochemical conservation: elements and energy, rather than state variable units, are conserved. We then analyze various fixed-step integration schemes, including traditional Euler-based schemes and the recently published modified Patankar schemes, and conclude that none of these deliver unconditional positivity and biochemical conservation in combination with higher-order accuracy. Finally, we present two new fixed-step integration schemes, one first-order and one second-order accurate, which do guarantee positivity and (biochemical) conservatio

A Data Integration and Visualization Resource for the Metabolic Network of Synechocystis sp. PCC 6803

Data integration is a central activity in systems biology. The integration of genomic, transcript, protein, metabolite, flux, and computational data yields unprecedented information about the system level functioning of organisms. Often, data integration is done purely computationally, leaving the user with little insight besides statistical information. In this article, we present a visualization tool for the metabolic network of Synechocystis PCC6803, an important model cyanobacterium for sustainable biofuel production. We illustrate how this metabolic map can be used to integrate experimental and computational data for Synechocystis systems biology and metabolic engineering studies. Additionally, we discuss how this map, and the software infrastructure that we supply with it, can be used in the development of other organism-specific metabolic network visualizations. Besides a Python console package VoNDA (http://vonda.sf.net), we provide a working demonstration of the interactive metabolic map and the associated Synechocystis genome-scale stoichiometric model, as well as various ready-to-visualize microarray data sets, at http://f-a-m-e.org/synechocystis/

Understanding start-up problems in yeast glycolysis

Yeast glycolysis has been the focus of research for decades, yet a number of dynamical aspects of yeast glycolysis remain poorly understood at present. If nutrients are scarce, yeast will provide its catabolic and energetic needs with other pathways, but the enzymes catalysing upper glycolytic fluxes are still expressed. We conjecture that this overexpression facilitates the rapid transition to glycolysis in case of a sudden increase in nutrient concentration. However, if starved yeast is presented with abundant glucose, it can enter into an imbalanced state where glycolytic intermediates keep accumulating, leading to arrested growth and cell death. The bistability between regularly functioning and imbalanced phenotypes has been shown to depend on redox balance. We shed new light on these phenomena with a mathematical analysis of an ordinary differential equation model, including NADH to account for the redox balance. In order to gain qualitative insight, most of the analysis is parameter-free, i.e., without assigning a numerical value to any of the parameters. The model has a subtle bifurcation at the switch between an inviable equilibrium state and stable flux through glycolysis. This switch occurs if the ratio between the flux through upper glycolysis and ATP consumption rate of the cell exceeds a fixed threshold. If the enzymes of upper glycolysis would be barely expressed, our model predicts that there will be no glycolytic flux, even if external glucose would be at growth-permissable levels. The existence of the imbalanced state can be found for certain parameter conditions independent of the mentioned bifurcation. The parameter-free analysis proved too complex to directly gain insight into the imbalanced states, but the starting point of a branch of imbalanced states can be shown to exist in detail. Moreover, the analysis offers the key ingredients necessary for successful numerical continuation, which highlight the existence of this bistability and the influence of the redox balance

Effective bet-hedging through growth rate dependent stability

Microbes in the wild face highly variable and unpredictable environments and are naturally selected for their average growth rate across environments. Apart from using sensory regulatory systems to adapt in a targeted manner to changing environments, microbes employ bet-hedging strategies where cells in an isogenic population switch stochastically between alternative phenotypes. Yet, bet-hedging suffers from a fundamental trade-off: Increasing the phenotype-switching rate increases the rate at which maladapted cells explore alternative phenotypes but also increases the rate at which cells switch out of a well-adapted state. Consequently, it is currently believed that bet-hedging strategies are effective only when the number of possible phenotypes is limited and when environments last for sufficiently many generations. However, recent experimental results show that gene expression noise generally decreases with growth rate, suggesting that phenotype-switching rates may systematically decrease with growth rate. Such growth rate dependent stability (GRDS) causes cells to be more explorative when maladapted and more phenotypically stable when well-adapted, and we show that GRDS can almost completely overcome the trade-off that limits bet-hedging, allowing for effective adaptation even when environments are diverse and change rapidly. We further show that even a small decrease in switching rates of faster-growing phenotypes can substantially increase long-term fitness of bet-hedging strategies. Together, our results suggest that stochastic strategies may play an even bigger role for microbial adaptation than hitherto appreciated

The omnistat:A flexible continuous-culture system for prolonged experimental evolution

1. Microbial evolution experiments provide a powerful tool to unravel the molecular basis of adaptive evolution but their outcomes can be difficult to interpret, unless the selective forces that are applied during the experiment are carefully controlled. In this respect, experimental evolution in continuous cultures provides advantages over commonly used sequential batch-culture protocols because continuous cultures allow for more accurate control over the induced selective environment. However, commercial continuous-culture systems are large and expensive, while available DIY continuous-culture systems are not versatile enough to allow for multiple sensors and rigorous stirring. 2. We present a modular continuous-culture system that adopts the commonly used GL45 glass laboratory bottle as a bioreactor vessel. Our design offers three advantages: first, it is equipped with a large head plate, fitting two sensors and seven input/output ports, enabling the customization of the system for many running modes (chemostat, auxostat, etc.). Second, the bioreactor is small (25-250 ml), which makes it feasible to run many replicates in parallel. Third, bioreactor modules can be coupled by uni- or bi-directional flows to induce spatiotemporal variation in selection. These features result in a particularly flexible culturing platform that facilitates the investigation of a broad range of evolutionary and ecological questions. 3. To illustrate the versatility of our culturing system, we outline two evolution experiments that impose a temporally or spatially variable regime of selection. The first experiment illustrates how controlled temporal variation in resource availability can be utilized to select for anticipatory switching. The second experiment illustrates a spatially structured morbidostat setup that is designed to probe epistatic interactions between adaptive mutations. Furthermore, we demonstrate how sensor data can be used to stabilize selection pressures or track evolutionary adaptation. 4. Evolution experiments in which populations are exposed to controlled spatiotemporal variation, are essential to gain insight into the process of adaptation and the mechanisms that constrain evolution. Continuous-culture systems, like the one presented here, offer control over key environmental parameters and establish a well-defined regime of selection. As such, they create the opportunity to expose evolutionary constraints in the form of phenotypic trade-offs, contributing to a mechanistic understanding of adaptive evolution