Verena Wolf

In Vivo Control of CpG and Non-CpG DNA Methylation by DNA Methyltransferases

The enzymatic control of the setting and maintenance of symmetric and non-symmetric DNA methylation patterns in a particular genome context is not well understood. Here, we describe a comprehensive analysis of DNA methylation patterns generated by high resolution sequencing of hairpin-bisulfite amplicons of selected single copy genes and repetitive elements (LINE1, B1, IAP-LTR-retrotransposons, and major satellites). The analysis unambiguously identifies a substantial amount of regional incomplete methylation maintenance, i.e. hemimethylated CpG positions, with variant degrees among cell types. Moreover, non-CpG cytosine methylation is confined to ESCs and exclusively catalysed by Dnmt3a and Dnmt3b. This sequence position–, cell type–, and region-dependent non-CpG methylation is strongly linked to neighboring CpG methylation and requires the presence of Dnmt3L. The generation of a comprehensive data set of 146,000 CpG dyads was used to apply and develop parameter estimated hidden Markov models (HMM) to calculate the relative contribution of DNA methyltransferases (Dnmts) for de novo and maintenance DNA methylation. The comparative modelling included wild-type ESCs and mutant ESCs deficient for Dnmt1, Dnmt3a, Dnmt3b, or Dnmt3a/3b, respectively. The HMM analysis identifies a considerable de novo methylation activity for Dnmt1 at certain repetitive elements and single copy sequences. Dnmt3a and Dnmt3b contribute de novo function. However, both enzymes are also essential to maintain symmetrical CpG methylation at distinct repetitive and single copy sequences in ESCs.

Comparative branching-time semantics for Markov chains

This paper presents various semantics in the branching-time spectrum of discrete-time and continuous-time Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation preorders are covered and are logically characterized in terms of the temporal logics Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic (CSL). Apart from presenting various existing branching-time relations in a uniform manner, this paper presents the following new results: (i) strong simulation for CTMCs, (ii) weak simulation for CTMCs and DTMCs, (iii) logical characterizations thereof (including weak bisimulation for DTMCs), (iv) a relation between weak bisimulation and weak simulation equivalence, and (v) various connections between equivalences and pre-orders in the continuous-and discrete-time setting. The results are summarized in a branching-time spectrum for DTMCs and CTMCs elucidating their semantics as well as their relationship.

Three-valued abstraction for continuous-time Markov chains

This paper proposes a novel abstraction technique for continuous-time Markov chains (CTMCs). Our technique fits within the realm of three-valued abstraction methods that have been used successfully for traditional model checking. The key idea is to apply abstraction on uniform CTMCs that are readily obtained from general CTMCs, and to abstract transition probabilities by intervals. It is shown that this provides a conservative abstraction for both true and false for a three-valued semantics of the branching-time logic CSL (Continuous Stochastic Logic). Experiments on an infinite-state CTMC indicate the feasibility of our abstraction technique.

Don’t know in probabilistic systems

In this paper the abstraction-refinement paradigm based on 3-valued logics is extended to the setting of probabilistic systems. We define a notion of abstraction for Markov chains. To be able to relate the behavior of abstract and concrete systems, we equip the notion of abstraction with the concept of simulation. Furthermore, we present model checking for abstract probabilistic systems (abstract Markov chains) with respect to specifications in probabilistic temporal logics, interpreted over a 3-valued domain. More specifically, we introduce a 3-valued version of probabilistic computation-tree logic (PCTL) and give a model checking algorithm w.r.t. abstract Markov chains.

Sliding Window Abstraction for Infinite Markov Chains

We present an on-the-fly abstraction technique for infinite-state continuous -time Markov chains. We consider Markov chains that are specified by a finite set of transition classes. Such models naturally represent biochemical reactions and therefore play an important role in the stochastic modeling of biological systems. We approximate the transient probability distributions at various time instances by solving a sequence of dynamically constructed abstract models, each depending on the previous one. Each abstract model is a finite Markov chain that represents the behavior of the original, infinite chain during a specific time interval. Our approach provides complete information about probability distributions, not just about individual parameters like the mean. The error of each abstraction can be computed, and the precision of the abstraction refined when desired. We implemented the algorithm and demonstrate its usefulness and efficiency on several case studies from systems biology.

Hybrid numerical solution of the chemical master equation

We present a numerical approximation technique for the analysis of continuous-time Markov chains that describe networks of biochemical reactions and play an important role in the stochastic modeling of biological systems. Our approach is based on the construction of a stochastic hybrid model in which certain discrete random variables of the original Markov chain are approximated by continuous deterministic variables. We compute the solution of the stochastic hybrid model using a numerical algorithm that discretizes time and in each step performs a mutual update of the transient probability distribution of the discrete stochastic variables and the values of the continuous deterministic variables. We implemented the algorithm and we demonstrate its usefulness and efficiency on several case studies from systems biology.

Fast adaptive uniformisation of the chemical master equation

Within systems biology there is an increasing interest in the stochastic behavior of biochemical reaction networks. An appropriate stochastic description is provided by the chemical master equation, which represents a continuous-time Markov chain (CTMC). Standard Uniformization (SU) is an efficient method for the transient analysis of CTMCs. For systems with very different time scales, such as biochemical reaction networks, SU is computationally expensive. In these cases, a variant of SU, called adaptive uniformization (AU), is known to reduce the large number of iterations needed by SU. The additional difficulty of AU is that it requires the solution of a birth process. In this paper we present an on-the-fly variant of AU, where we improve the original algorithm for AU at the cost of a small approximation error. By means of several examples, we show that our approach is particularly well-suited for biochemical reaction networks.

Bounding the equilibrium distribution of Markov population models

SUMMARY We propose a bounding technique for the equilibrium probability distribution of continuous-time Markov chains with population structure and infinite state space. We use Lyapunov functions to determine a finite set of states that contains most of the equilibrium probability mass. Then we apply a refinement scheme based on stochastic complementation to derive lower and upper bounds on the equilibrium probability for each state within that set. To show the usefulness of our approach, we present experimental results for several examples from biology. Copyright

A numerical aggregation algorithm for the enzyme-catalyzed substrate conversion

Computational models of biochemical systems are usually very large, and moreover, if reaction frequencies of different reaction types differ in orders of magnitude, models possess the mathematical property of stiffness, which renders system analysis difficult and often even impossible with traditional methods. Recently, an accelerated stochastic simulation technique based on a system partitioning, the slow-scale stochastic simulation algorithm, has been applied to the enzyme-catalyzed substrate conversion to circumvent the inefficiency of standard stochastic simulation in the presence of stiffness. We propose a numerical algorithm based on a similar partitioning but without resorting to simulation. The algorithm exploits the connection to continuous-time Markov chains and decomposes the overall problem to significantly smaller subproblems that become tractable. Numerical results show enormous efficiency improvements relative to accelerated stochastic simulation.

Trace Machines for Observing Continuous-Time Markov Chains

In this paper, we study several linear-time equivalences (Markovian trace equivalence, failure and ready trace equivalence) for continuous-time Markov chains that refer to the probabilities for timed execution paths. Our focus is on testing scenarios by means of push-button experiments with appropriate trace machines and a discussion of the connections between the equivalences. For Markovian trace equivalence, we provide alternative characterizations, including one that abstracts away from the time instances where actions are observed, but just reports on the average sojourn times in the states. This result is used for a reduction of the question whether two finite-state continuous-time Markov chains are Markovian trace equivalent to the probabilistic trace equivalence problem for discrete-time Markov chains (and the latter is known to be solvable in polynomial time).