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Background.
The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level.
Results.
We introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space.
Discussion.
In order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.
Conclusions.
The proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability.
Electronic version: http://www.biomedcentral.com/1752-0509/9/67

High performing dairy cows require a particular composition of nutritional ingredients, adapted to their individual requirements and depending on their production status. The optimal dimensioning of minerals in the diet, one of them being potassium, is indispensable for the prevention of imbalances. The potassium balance in cows is the result of potassium intake, distribution in the organism, and excretion, it is closely related with the glucose and electrolyte metabolism. In this paper, we present a dynamical model for the potassium balance in lactating and non-lactating dairy cows based on ordinary differential equations. Parameter values are obtained from clinical trial data and from the literature. To verify the consistency of the model, we present simulation outcomes for three different scenarios: potassium balance in (i) non-lactating cows with varying feed intake, (ii) non-lactating cows with varying potassium fraction in the diet, and (iii) lactating cows with varying milk production levels. The results give insights into the short and long term potassium metabolism, providing an important step towards the understanding of the potassium network, the design of prophylactic feed additives, and possible treatment strategies.

One of the main goals of mathematical modelling in systems biology related to medical applications is to obtain patient-specific parameterisations and model predictions.
In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Using these population data, we propose an iterative algorithm for contructing an informative prior distribution, which then serves as the basis for computing patient-specific posteriors and obtaining individual predictions. We demonsrate the performance of our method by applying it to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modelling.

When estimating a probability density within the empirical Bayes framework, the non-parametric maximum likelihood estimate (NPMLE) usually tends to overfit the data. This issue is usually taken care of by regularization - a penalization term is subtracted from the marginal log-likelihood before the maximization step, so that the estimate favors smooth solutions, resulting in the so-called maximum penalized likelihood estimation (MPLE).
The majority of penalizations currently in use are rather arbitrary brute-force solutions, which lack invariance under transformation of the parameters(reparametrization) and measurements.
This contradicts the principle that, if the underlying model
has several equivalent formulations, the methods of inductive inference should lead to consistent results. Motivated by this principle and using an information-theoretic point of view, we suggest an entropy-based penalization term that guarantees this kind of invariance. The resulting density estimate can be seen as a generalization of reference priors. Using the reference prior as a hyperprior, on the other hand, is argued to be a poor choice for regularization. We also present an insightful connection between the NPMLE, the cross entropy
and the principle of minimum discrimination information suggesting another method of inference that contains the doubly-smoothed maximum likelihood estimation as a special case.

One of the main goals of mathematical modelling in systems medicine related to medical applications is to obtain patient-specific parameterizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Therefore, before applying Bayes’ rule separately to the data of each patient (which is typically performed using a non-informative prior), it is meaningful to use empirical Bayes methods in order to construct an informative prior from all available data. We compare the performance of four priors - a non-informative prior and priors chosen by nonparametric maximum likelihood estimation (NPMLE), by maximum penalized lilelihood estimation (MPLE) and by doubly-smoothed maximum likelihood estimation (DS-MLE) - by applying them to a low-dimensional parameter estimation problem in a toy model as well as to a high-dimensional ODE model of the human menstrual cycle, which represents a typical example from systems biology modelling.

Particle methods have become indispensible in conformation dynamics to
compute transition rates in protein folding, binding processes and
molecular design, to mention a few.
Conformation dynamics requires at a decomposition of a molecule's position
space into metastable conformations.
In this paper, we show how this decomposition
can be obtained via the design of either ``soft'' or ``hard''
molecular conformations.
We show, that the soft approach results in a larger metastabilitiy of
the decomposition and is thus more advantegous. This is illustrated
by a simulation of Alanine Dipeptide.

Mathematical modeling of the Smad and Non-Smad BMP signaling pathways in context of cell density
(2013)

Nutrition plays a crucial role in regulating reproductive hormones and follicular
development in cattle. This is visible particularly during the time of negative
energy balance at the onset of milk production after calving. Here, elongated
periods of anovulation have been observed, resulting from alterations in luteiniz-
ing hormone concentrations, likely caused by lower glucose and insulin concen-
trations in the blood. The mechanisms that result in a reduced fertility are
not completely understood, although a close relationship to the glucose-insulin
metabolism is widely supported. Following this idea, a mathematical model of
the hormonal network combining reproductive hormones and hormones that are
coupled to the glucose compartments within the body of the cow was developed.
The model is built on ordinary differential equations and relies on previously
introduced models on the bovine estrous cycle and the glucose-insulin dynam-
ics. Necessary modifications and coupling mechanisms are thoroughly discussed.
Depending on the composition and the amount of food, in particular the glu-
cose content in the dry matter, the model quantifies reproductive hormones and
follicular development over time. Simulation results for different nutritional
regimes in lactating and non-lactating dairy cows are examined and compared
with experimental studies. Regarding its applicability, this work is an early
attempt towards developing in silico feeding strategies and may eventually help
refining and reducing animal experiments.

We present a mechanistic pharmacokinetic-pharmacodynamic model to simulate the effect of dexamethasone on the glucose metabolism in dairy cows.
The coupling of the pharmacokinetic model to the pharmacodynamic model
is based on mechanisms underlying homeostasis regulation by dexamethasone.
In particular, the coupling takes into account the predominant role of dexamethasone in stimulating glucagon secretion, glycogenolysis and lipolysis and in
impairing the sensitivity of cells to insulin. Simulating the effect of a single
dose of dexamethasone on the physiological behaviour of the system shows that
the adopted mechanisms are able to induce a temporary hyperglycemia and
hyperinsulinemia, which captures the observed data in non-lactating cows. In
lactating cows, the model simulations show that a single dose of dexamethasone
reduces the lipolytic effect, owing to the reduction of glucose uptake by the
mammary gland.