In silico labeling reveals the time-dependent label half-life and transit-time in dynamical systems

Thomas Maiwald Julie Blumberg Andreas Raue Stefan Hengl Marcel Schilling Sherwin KB Sy Verena Becker Ursula Klingmller Jens Timmer - Methods, software and technology Open Access In silico labeling reveals the time-dependent label half-life and transit-time in dynamical systems Background: Mathematical models of dynamical systems facilitate the computation of characteristic properties that are not accessible experimentally. In cell biology, two main properties of interest are (1) the time-period a protein is accessible to other molecules in a certain state - its half-life - and (2) the time it spends when passing through a subsystem - its transit-time. We discuss two approaches to quantify the half-life, present the novel method of in silico labeling, and introduce the label half-life and label transit-time. The developed method has been motivated by laboratory tracer experiments. To investigate the kinetic properties and behavior of a substance of interest, we computationally label this species in order to track it throughout its life cycle. The corresponding mathematical model is extended by an additional set of reactions for the labeled species, avoiding any doublecounting within closed circuits, correcting for the influences of upstream fluxes, and taking into account combinatorial multiplicity for complexes or reactions with several reactants or products. A profile likelihood approach is used to estimate confidence intervals on the label half-life and transit-time. Results: Application to the JAK-STAT signaling pathway in Epo-stimulated BaF3-EpoR cells enabled the calculation of the time-dependent label half-life and transit-time of STAT species. The results were robust against parameter uncertainties. Conclusions: Our approach renders possible the estimation of species and label half-lives and transit-times. It is applicable to large non-linear systems and an implementation is provided within the PottersWheel modeling framework (http://www.potterswheel.de). Background Motivation An increasing number of biological phenomena are described by mathematical models, specifically on the basis of biochemical reaction networks [1,2]. The dynamic properties of these networks are given by their model structure, kinetic parameters, initial values of the involved species, and externally specified input functions. The interpretation of an isolated element of the network, e.g. a certain rate constant, has only a limited meaning, because its effect can only be understood when taking the whole network context into account. We therefore seek to introduce two dynamical characteristics which have a physiological meaning, are * Correspondence: Contributed equally 1Center for Systems Biology, Freiburg, Germany Full list of author information is available at the end of the article intuitive to understand, and capture the system kinetics on a higher level of abstraction. The first characteristic, the label half-life, applies the half-life concept not to a species, but to a virtual label attached to the species. The second one, the label transit-time, is the time-period it takes for a fraction of labeled entities to pass through a subsystem of the network. Both quantities are calculated using a novel approach called in silico labeling, which is also introduced in the present work. In Silico Labeling and Species vs. Label Half-Life In a laboratory tracer experiment, a substance is marked to better understand the kinetic properties of the dynamical system [3]. Different tracer substances have been used, e.g. radioactive iodine-125 [4,5] or green fluorescent protein-tagged proteins in combination with fluorescence recovery after photobleaching (FRAP) [6]. A good tracer does not hamper the flux of the substance, therefore one can assume that the flux of the tracer within a certain reaction is proportional to the flux of the original species. This is the key property of the in silico labeling approach, where an additional set of reactions is added to an existing mathematical model describing the kinetic behavior of a tracer, called the label. In contrast to real tracer experiments, the in silico method offers the opportunity to define dead-ends, avoid double-counting of cycling label, and to restrict the label to a sub-network of reactions. This allows asking specific questions about the original system, like how long it takes for 50% of the molecules of a substance to travel along a certain path, while in reality an alternative path may exist. In addition, predominant paths can be identified in deterministic models as has been done previously for stochastic systems [7]. Mathematically, the half-life T1/2 of a species is defined as the time-period until it reaches half of its initial amount assuming no influx. For clarity, we denote this time-period as the species half-life (SHL). In nonisolated and non-linear processes, this time-period differs from the amount of time required for 50% of initially existing molecules to be processed. For this, we introduce the label half-life (LHL), defined as the halflife of the label of a species. Equalities and differences between the species and label half-life are displayed in Figure 1 and proven in the methods section. While for simple systems the species half-life can be determined analytically, the symbolic integration of a Michaelis-Menten kinetics leads to advanced mathematical calculations including the Lambert W function [8]. We therefore also provide an automatic and generally applicable numerical method to determine the species half-life. Label Transit-Time Transit-times are discussed in a variety of fields and they are, for example, used to quantify how quickly food moves through the gastrointestinal tract [9]. When describing the dynamics of Markovian particles, the mean transit-time denotes the time spent on average in a subsystem [10], while the mean sojourn-time also takes into account the probability that the subsystem is entered at all [11]. In pharmacokinetics, the so-called mean residence time values [12] are estimated based on empirical data assuming linear kinetics [13]. Apart from linearity, no influx for the species of interest is permitted. Eventually, the estimation is only applicable to observable species. The computation of the mean residence time is accomplished by the ratio of the area under the first moment curve (AUMC) to the area under the curve (AUC) of the concentration-time profile of a drug [14]. We here introduce the label transit-time (LTT) from a source to a target pool in a chemical reaction network as the time-period after which 50% of all entities residing in the source pool at t = 0 have reached the target pool at least once. The exact path from source to target pool is not important in the unconditioned case. The Figure 1 Species vs. label half-life. Panel A: The species half-life of a substrate S in the reaction S P is plotted for different reaction types (solid lines). Except for processes of (...truncated)