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Inferring Proteolytic Processes from Mass Spectrometry Time Series Data Using Degradation Graphs
(2012)
Mathematical modeling of the Smad and Non-Smad BMP signaling pathways in context of cell density
(2013)
Markov Decision Processes (MDP) or Partially Observable MDPs (POMDP) are
used for modelling situations in which the evolution of a process is partly random and
partly controllable. These MDP theories allow for computing the optimal control policy
for processes that can continuously or frequently be observed, even if only partially.
However, they cannot be applied if state observation is very costly and therefore rare
(in time). We present a novel MDP theory for rare, costly observations and derive the
corresponding Bellman equation. In the new theory, state information can be derived
for a particular cost after certain, rather long time intervals. The resulting information
costs enter into the total cost and thus into the optimization criterion. This approach
applies to many real world problems, particularly in the medical context, where the
medical condition is examined rather rarely because examination costs are high. At the
same time, the approach allows for efficient numerical realization. We demonstrate the
usefulness of the novel theory by determining, from the national economic perspective,
optimal therapeutic policies for the treatment of the human immunodefficiency virus
(HIV) in resource-rich and resource-poor settings. Based on the developed theory and
models, we discover that available drugs may not be utilized efficiently in resource-poor
settings due to exorbitant diagnostic costs.
We propose the first, to our knowledge, coarse-grained modeling strategy for peptides where the effect of changes of the pH can be efficiently described. The idea is based on modeling the effects of the pH value on the main driving interactions. We use reference data from atomistic simulations and experimental databases and transfer their main physical features to the coarse-grained resolution according to the principle of “consistency across the scales”. The coarse-grained model is refined by finding a set of parameters that, when applied to peptides with different sequences and experimental properties, reproduces the experimental and atomistic data of reference. We use such a parameterized model for performing several numerical tests to check its transferability to other systems and to prove the universality of the related modeling strategy. We have tried systems with rather different responses to pH variations, showing a highly satisfactory performance of the model.
We present the theory of “Markov decision processes (MDP) with rare state observation” and apply it to optimal treatment scheduling and diagnostic testing to mitigate HIV-1 drug resistance development in resource-poor countries. The developed theory assumes that the state of the process is hidden and can only be determined by making an examination. Each examination produces costs which enter into the considered cost functional so that the resulting optimization problem includes finding optimal examination times. This is a realistic ansatz: In many real world applications, like HIV-1 treatment scheduling, the information about the disease evolution involves substantial costs, such that examination and control are intimately connected. However, a perfect compliance with the optimal strategy can rarely be achieved. This may be particularly true for HIV-1 resistance testing in resource-constrained countries. In the present work, we therefore analyze the sensitivity of the costs with respect to deviations from the optimal examination times both analytically and for the considered application. We
discover continuity in the cost-functional with respect to the examination times. For the HIV-application, moreover, sensitivity towards small deviations from the optimal examination rule depends on the disease state. Furthermore, we compare the optimal rare-control strategy to (i) constant control strategies (one action for the remaining time) and to (ii) the permanent control of the original, fully observed MDP. This comparison is
done in terms of expected costs and in terms of life-prolongation. The proposed rare-control strategy offers a clear benefit over a constant control, stressing the usefulness of medical testing and informed decision making. This indicates that lower-priced medical tests could improve HIV treatment in resource-constrained settings and warrants further investigation.
A good deal of molecular dynamics simulations aims at predicting and quantifying rare events, such as the folding of a protein or a phase transition. Simulating rare events is often prohibitive, especially if the equations of motion are high-dimensional, as is the case in molecular dynamics. Various algorithms have been proposed for efficiently computing mean first passage times, transition rates or reaction pathways. This article surveys and discusses recent developments in the field of rare event simulation and outlines a new approach that combines ideas from optimal control and statistical mechanics. The optimal control approach described in detail resembles the use of Jarzynski's equality for free energy calculations, but with an optimized protocol that speeds up the sampling, while (theoretically) giving variance-free estimators of the rare events statistics. We illustrate the new approach with two numerical examples and discuss its relation to existing methods.
Rare but important transition events between long lived states are a key feature of many molecular systems. In many cases the computation of rare event statistics by direct molecular dynamics (MD) simulations is infeasible even on the most powerful computers because of the immensely long simulation timescales needed. Recently a technique for spatial discretization of the molecular state space designed to help overcome such problems, so-called Markov State Models (MSMs), has attracted a lot of attention. We review the theoretical background and algorithmic realization of MSMs and illustrate their use by some numerical examples. Furthermore we introduce a novel approach to using MSMs for the efficient solution of optimal control problems that appear in applications where one desires to optimize molecular properties by means of external controls.
Linear response theory and optimal control for a molecular system under nonequilibrium conditions
(2013)
In this paper, we propose a straightforward generalization of linear
response theory to systems in nonequilibrium that are subject to
nonequilibrium driving. We briefly revisit the standard linear response
result for equilibrium systems, where we consider Langevin dynamics
as a special case, and then give an alternative derivation using a
change-of-measure argument that does not rely on any stationarity or
reversibility assumption. This procedure moreover easily enables us
to calculate the second order correction to the linear response formula
(which may or may not be useful in practice). Furthermore, we outline
how the novel nonequilibirum linear response formula can be used to
compute optimal controls of molecular systems for cases in which one
wants to steer the system to maximize a certain target expectation
value. We illustrate our approach with simple numerical examples.
Grand-canonical-like molecular-dynamics simulations by using an adaptive-resolution technique
(2013)
Linear response theory and optimal control for a molecular system under nonequilibrium conditions
(2013)
We employ the adaptive resolution approach AdResS, in its recently developed Grand Canonicallike version (GC-AdResS) [Wang et al. Phys.Rev.X 3, 011018 (2013)], to calculate the excess chemical potential, $μ^{ex}$, of various liquids and mixtures. We compare our results with those obtained from full atomistic simulations using the technique of thermodynamic integration and show a satisfactory agreement. In GC-AdResS the procedure to calculate $μ^{ex}$ corresponds to the process of standard initial equilibration of the system; this implies that, independently of the specific aim of the study, $μ^{ex}$, for each molecular species, is automatically calculated every time a GC-AdResS simulation is performed.
The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarsegrained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits all necessary information about modular structures from the original network.
Techniques for finding metastable or almost invariant sets have been investigated, e.g., for deterministic dynamical systems in set-oriented numerics, for stochastic processes in molecular dynamics, and for random walks on complex networks. Most prominent algorithms are based on spectral apporaches and identify metastable sets via the doimant eigenvalues of the transfer operator associated with the dynamical system under consideration. These algorithms require the dominant eigenvalues to be real-valued. However, for many types of dynamics, e.g. for non-reversible Markov chains, this condition is not met. In this paper we utilize the hitting time apporach to metastable sets and demonstrate how the wellknown statements about optimal metastable decompositions of reversible chains can be reformulated for non-reversible chains if one switches from a spectral approach to an exit time approach. The performance of the resulting algorithm is illustrated by numerical experiments on random walks on complex networks.
In recent years Markov State Models (MSMs) have attracted a consid-
erable amount of attention with regard to modelling conformation changes
and associated function of biomolecular systems. They have been used
successfully, e.g., for peptides including time-resolved spectroscopic ex-
periments, protein function and protein folding , DNA and RNA, and
ligand-receptor interaction in drug design and more complicated multi-
valent scenarios. In this article a novel reweighting scheme is introduced
that allows to construct an MSM for certain molecular system out of an
MSM for a similar system. This permits studying how molecular proper-
ties on long timescales differ between similar molecular systems without
performing full molecular dynamics simulations for each system under con-
sideration. The performance of the reweighting scheme is illustrated for
simple test cases including one where the main wells of the respective en-
ergy landscapes are located differently and an alchemical transformation
of butane to pentane where the dimension of the state space is changed.
We investigate the problem of finding modules (or clusters, communities) in directed networks. Until now, most articles on this topic have been oriented towards finding complete network partitions despite the fact that this often is unwanted. We present a novel random walk based approach for non-complete partitions of the directed network into modules in which some nodes do not belong to only one of the modules but to several or to none at all. The new random walk process is reversible even for directed networks but inherits all necessary information about directions and structure of the original network. We demonstrate the performance of the new method in application to a real-world earthquake network.
Life Sciences
(2014)
Markov Model Theory
(2014)
This section reviews the relation between the continuous dynamics of a molecular system in thermal equilibrium and the kinetics given by a Markov State Model (MSM). We will introduce the dynamical propagator, an error-less, alternative description of the continuous dynamics, and show how MSMs result from its discretization. This allows for an precise understanding of the approximation quality of MSMs in comparison to the continuous dynamics. The results on the approximation quality are key for the design of good MSMs. While this section is important for understanding the theory of discretization and related systematic errors, practitioners wishing only to learn how to construct MSMs may skip directly to the discussion of Markov model estimation.
In this paper, we investigate the conformational dynamics of alanine dipeptide under an external electric field by nonequilibrium molecular dynamics simulation. We consider the case of a constant and of an oscillatory field. In this context, we propose a procedure to implement the temperature control, which removes the irrelevant thermal effects of the field. For the constant field different time-scales are identified in the conformational, dipole moment, and orientational dynamics. Moreover, we prove that the solvent structure only marginally changes when the external field is switched on. In the case of oscillatory field, the conformational changes are shown to be as strong as in the previous case, and nontrivial nonequilibrium circular paths in the conformation space are revealed by calculating the integrated net probability fluxes.
Markov Decision Processes (MDP) or Partially Observable MDPs (POMDP) are used for modelling situations in which the evolution of a process is partly random and partly controllable. These MDP theories allow for computing the optimal control policy for processes that can continuously or frequently be observed, even if only partially. However, they cannot be applied if state observation is very costly and therefore rare (in time). We present a novel MDP theory for rare, costly observations and derive the corresponding Bellman equation. In the new theory, state information can be derived for a particular cost after certain, rather long time intervals. The resulting information costs enter into the total cost and thus into the optimization criterion. This approach applies to many real world problems, particularly in the medical context, where the medical condition is examined rather rarely because examination costs are high. At the same time, the approach allows for efficient numerical realization. We demonstrate the usefulness of the novel theory by determining, from the national economic perspective, optimal therapeutic policies for the treatment of the human immunodeficiency virus (HIV) in resource-rich and resource-poor settings. Based on the developed theory and models, we discover that available drugs may not be utilized efficiently in resource-poor settings due to exorbitant diagnostic costs.
SAIMeR: Self-adapted method for the identification of metastable states in real-world time series
(2014)
In the framework of time series analysis with recurrence networks, we introduce SAIMeR, a heuristic self-adapted method that determines the elusive recurrence threshold and identifies metastable states in complex time series. To identify metastable states as well as the transitions between them, we use graph theory concepts and a fuzzy partitioning clustering algorithm. We illustrate SAIMeR by applying it to three real-world time series and show that it is able to identify metastable states in real-world data with noise and missing data points. Finally, we suggest a way to choose the embedding parameters used to construct the state space in which this method is performed, based on the analysis of how the values of these parameters affect two recurrence quantitative measurements: recurrence rate and entropy.
Applications of the cross-entropy method to importance sampling and optimal control of diffusions
(2014)
We study the cross-entropy method for diffusions. One of the results is a versatile cross-entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems. The approach is based on the minimization of a suitable cross-entropy functional, with a parametric family of exponentially tilted probability distributions. We illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method.
Applications of the cross-entropy method to importance sampling and optimal control of diffusions
(2014)
Recent years have seen an increased interest in non-equilibrium molecular dynamics (NEMD) simulations, especially for molecular systems with periodic forcing by external fields, e.g., in the context of studying effects of electromagnetic radiation on the human body tissue. Lately, an NEMD methods with local thermostating has been proposed that allows for studying non-equilibrium processes in a statistically reliable and thermodynamically consistent way. In this article, we demonstrate how to construct Markov State Models (MSMs) for such NEMD simulations. MSM building has been well-established for systems in equilibrium where MSMs with just a few (macro-)states allow for accurate reproduction of the essential kinetics of the molecular system under consideration. Non-equilibrium MSMs have been lacking so far. The article presents how to construct such MSMs and illustrates their validity and usefulness for the case of conformation dynamics of alanine dipeptide in an external electric field.
In recent years Markov State Models (MSMs) have attracted a consid-
erable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g., for peptides including time-resolved spectroscopic experiments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multivalent scenarios. In this article a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular properties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under con-
sideration. The performance of the reweighting scheme is illustrated for simple test cases including one where the main wells of the respective energy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed.
Markov State Modelling as a concept for a coarse grained description of the essential kinetics of a molecular system in equilibrium has gained a lot of atten- tion recently. The last 10 years have seen an ever increasing publication activity on how to construct Markov State Models (MSMs) for very different molecular systems ranging from peptides to proteins, from RNA to DNA, and via molecu- lar sensors to molecular aggregation. Simultaneously the accompanying theory behind MSM building and approximation quality has been developed well be- yond the concepts and ideas used in practical applications. This article reviews the main theoretical results, provides links to crucial new developments, outlines the full power of MSM building today, and discusses the essential limitations still to overcome.
An estimated 2.7 million new HIV-1 infections occurred in 2010. `Treatment-for-prevention’ may strongly prevent HIV-1 transmission. The basic idea is that immediate treatment initiation rapidly decreases virus burden, which reduces the number of transmittable viruses and thereby the probability of infection. However, HIV inevitably develops drug resistance, which leads to virus rebound and nullifies the effect of `treatment-for-prevention’ for the time it remains unrecognized. While timely conducted treatment changes may avert periods of viral rebound, necessary treatment options and diagnostics may be lacking in resource-constrained settings. Within this work, we provide a mathematical platform for comparing different treatment paradigms that can be applied to many medical phenomena. We use this platform to optimize two distinct approaches for the treatment of HIV-1: (i) a diagnostic-guided treatment strategy, based on infrequent and patient-specific diagnostic schedules and (ii) a pro-active strategy that allows treatment adaptation prior to diagnostic ascertainment. Both strategies are compared to current clinical protocols (standard of care and the HPTN052 protocol) in terms of patient health, economic means and reduction in HIV-1 onward transmission exemplarily for South Africa. All therapeutic strategies are assessed using a coarse-grained stochastic model of within-host HIV dynamics and pseudo-codes for solving the respective optimal control problems are provided. Our mathematical model suggests that both optimal strategies (i)-(ii) perform better than the current clinical protocols and no treatment in terms of economic means, life prolongation and reduction of HIV-transmission. The optimal diagnostic-guided strategy suggests rare diagnostics and performs similar to the optimal pro-active strategy. Our results suggest that ‘treatment-for-prevention’ may be further improved using either of the two analyzed treatment paradigms.