Center for Systems Biology, Freiburg, Germany

Department of Systems Biology, Harvard Medical School, Boston, USA

Epilepsy Center, University Hospital Freiburg, Germany

Department of Neuroscience, Children's Hospital, Harvard Medical School, Boston, USA

Division Systems Biology of Signal Transduction, DKFZ-ZMBH Alliance, German Cancer Research Center, Heidelberg, Germany

College of Pharmacy, University of Florida, Gainesville, USA

Bioquant, Heidelberg University, Germany

Freiburg Institute for Advanced Studies, University of Freiburg, Germany

BIOSS Centre for Biological Signalling Studies, University of Freiburg, Germany

Department of Clinical and Experimental Medicine, Linköping University, Sweden

Abstract

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

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 (

Background

Motivation

An increasing number of biological phenomena are described by mathematical models, specifically on the basis of biochemical reaction networks

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

Mathematically, the _{1/2 }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 vs. label half-life

**Species vs. label half-life**. Panel A: The species half-life of a substrate

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

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

We here introduce the

Extended Reaction Network

To determine the label half-life, it is important to distinguish entities residing in the source pool at

In case of label half-life calculations, it is sufficient to create labeled reactions for all reactions in which the source species is a reactant. In fact, labeled reactions are prohibited if the source species is a product; this is to avoid double-counting the labeled species. In the case of transit-time calculations, for all original reactions in which labeled species are involved, a new labeled reaction is added. In all labeled reactions with the target species being the product, the label is removed and accumulated in an artificial pool which is used to determine when 50% of the existing label has reached the target.

The label stays virtually attached to a species throughout all modifications of the species, such as phosphorylation or relocalizations, e.g. shuttling into the nucleus. While the suggested approach can be implemented in a straightforward way for monomeric reaction networks with only up to one labeled reactant and product, for the general case where the reactions involve multiple reactants and products or where labeled species may form a polymer, a systematic book-keeping of all possible combinations of labeled and unlabeled species is required.

As motivated by laboratory tracer experiments the fluxes of the additional system are based on the corresponding fluxes in the original one, which is explained in detail in the methods section.

Profile Likelihood-based Confidence Intervals

Recently, we suggested a profile likelihood-based approach to determine the confidence intervals on calibrated parameter values in mechanistic mathematical models

Implementation

All concepts have been implemented within the PottersWheel modeling and parameter estimation framework that is available from

**Application within PottersWheel**. This additional file contains MATLAB scripts to run various tasks related to the in silico labeling approach.

Click here for file

In the next section, the proposed labeling method is illustrated for the JAK-STAT signal transduction pathway and afterwards described in detail. After proving the equality of species and label half-life for isolated or linear processes, a fitted model of the JAK-STAT pathway is used to determine the label half-life of unphosphorylated STAT and its label transit-time when cycling through the nucleus of a cell.

Methods

Illustration of the method

Figure ^{L}/S^{F}

In silico labeled JAK-STAT signaling pathway

**In silico labeled JAK-STAT signaling pathway**. STAT molecules S (blue) are phosphorylated by an active receptor-kinase complex (pR) and form dimers (pS_{-}pS). These dimers enter the nucleus, dissociate, and are subsequently dephosphorylated. Finally, the single STAT molecules re-enter the cytoplasm, where they can again be phosphorylated and thus continue the nuclear-cytoplasmic shuttling. The labeling approach is visualized by red spheres attached to the STAT molecules. At

The label half-life of STAT at time-point

The label transit-time from STAT to STAT at time-point

This procedure is repeated for a series of time-points

Terminology

In the following we assume that the biological system is mathematically described by a set of reactions _{j}
_{i}

Here, _{j }
_{ij }
_{i }
_{ij }
_{i }
_{i}
_{i}
_{k}
_{j }
_{i }
_{k}
_{i}
_{i}

Analytical and numerical half-life calculation

The half-life of a species _{i }
_{i}

with initial value _{i }
_{1/2 }for which

Note that a half-life characterizes the decay of a quantity, independent of any production rates. Therefore, all influx contributions are neglected in equation (5). In general, only linear processes possess a constant half-life. Otherwise, the half-life depends on the initial concentration _{0}. In a numerical integration, it is important to limit the maximum integrator step size for an accurate approximation of the ^{0}/2 threshold crossing.

The half-life of a species _{i }
_{i }
_{i }

In silico labeling half-life for isolated processes

For simplicity, we assume that the species of interest _{1}, . . . , _{m}

The _{0}
_{in }
_{0}
_{0 }∈ ℝ^{+}:

Proof:

Let _{out }
_{in }
_{out}

Then, the kinetics of the label species

It can be shown that the factor

Since this relation holds also true for

Both processes _{1/2}, since

This relation does not hold for processes with _{in }

In silico labeling for linear processes

In this section, we prove that the label half-life coincides with the half-life of a species _{in }

Proof:

Let _{in }

Then, the analytical half-life of

For the labeled system

Creating the Extended Reaction Network

Some entities _{i }
_{1}, . . . , _{α }
_{
α+1}, . . . , _{m}
_{1}, . . . , _{
γ ≤ α }are not complexes consisting of two or more tracked single entities, and (2) that the tracked single entities within each complex _{
γ+1}, . . . , _{α }
_{1}, . . . , _{γ}
_{1}, . . . , _{α }
_{
α+1},. . . , _{m }

Creating additional entities x^{LF}

A new set of labeled or free entities **x**
^{LF }
**x**, by applying the following rules:

• Start with an empty set, **x**
^{LF }

• Single entities: For each _{i }
_{1}, . . . , _{γ}
**x**
^{LF }

• Complex entities: Each complex _{i }
_{
γ+1}, . . . , _{α}} is decomposed into

possible combinations using labeled _{i}
**x**
^{LF}
_{_}
^{F}_pS^{F}, pS^{L}_pS^{F}, pS^{F}_pS^{L}
^{L}_pS^{L}

Creating additional reactions r^{LF}

In order to create a new set of reactions **r**
^{LF }
_{i}
_{1 ≤ i≤α
}with possible repetition, as for example the reactants of the reaction ^{p }

Without loss of generality, only the first **r**
^{LF }
**r**
^{LF }
_{i }
**{**
_{1}, . . . , _{δ}
**} **with one or more reactants of tracked entities,

1. all reactants and products not belonging to the group of tracked entities are removed,

2. the combinatorial multiplicity approach is applied to the ordered list

3. 2^{p }
**r**
^{LF }

4. the fluxes

Note that again the same symbol has been used for the entity name and its concentration. The sum over all weighting factors is 1.

Reactions _{i }
_{1}, . . . , _{δ}
_{i }
**r**
^{LF}
_{i }

When calculating the label half-life, products that coincide with the initially labeled entity are replaced by the corresponding free entity. This corresponds to removing the label and is necessary to avoid double-counting and to exclude upstream fluxes.

In order to calculate the label transit-time, entities entering the target pool must be released from their labeling, again, to avoid double-counting. Therefore, all labeled target entities are replaced in the reaction network **r**
^{LF }

Calculating the Label Half-Life and Transit-Time

Since the label half-life and transit-time characteristics are time-dependent, the label is not only _{i }

1. Set all initial values for labeled entities and RL, if available, to 0. Set the initial value of free entities to the value of their counterpart in the original network.

2. Numerically integrate the ordinary differential equations corresponding to the extended reaction network {**r**, **r**
^{LF}

3. Apply a complete labeling of the source species: Set

4. Continue the numerical integration.

Threshold crossing at

Profile Likelihood-based Confidence Intervals

We recently suggested a profile likelihood-based approach to determine simultaneous and separate confidence intervals for calibrated unknown model parameters

Analytic half-lives for simple, isolated processes

The half-life _{1/2}(_{0 }= _{0}) at the time-point of interest _{0 }and is therefore time-dependent:

The half-life calculation for a process of order

In order to calculate the half-life for Michaelis-Menten kinetics, _{0 }at _{0}:

Panel A of Figure

Results

In this section, the

A smoothing spline approximation of the phosphorylated receptor served as the input function _{-}pS_{1 }= 1.37, _{2 }= 0.22, _{3 }= 0.63, _{4 }= 0.59, and _{5 }= 0.59. The initial value of _{-}obs _{-}obs

Labeled system

In order to determine the label half-life and transit-time of STAT,

Time-courses of the original and labeled system

**Time-courses of the original and labeled system**. At ^{L}_{-}^{F }^{F}_{-}^{L}

Time-dependent label half-life and transit-time of cytoplasmic unphosphorylated STAT

**Time-dependent label half-life and transit-time of cytoplasmic unphosphorylated STAT**. A: In order to determine the label half-life and transit-time, a family of labeled STAT trajectories is calculated. For trajectory _{i}^{L }_{i}_{i }

Label half-life and transit-time

Figure ^{L}

Profile likelihood-based confidence intervals

In order to investigate how uncertainties in calibrated model parameter values propagate to the estimated time-characteristics, we applied the profile likelihood approach on an identifiable version of the model. The kinetic parameters involved in phosphorylation (_{1}), dimerization (_{2}), nuclear import (_{3}) and export (_{5}) were systematically varied consecutively within four orders of magnitude between 0.01_{fit }
_{fit}
_{fit }
^{2}-threshold of the separate 95% confidence interval are used to recalculate the label half-life and transit-time. Figure

Discussion and Conclusions

In this paper, the half-life of a species has been compared conceptually, analytically, and numerically to the half-life of a label in a hypothetical tracer experiment. Two time-characteristics, the label half-life and label transit time have been introduced, which capture the kinetics of a dynamical system on a higher level than e.g. single rate constants. Calculation of the time-characteristics and their profile likelihood-based confidence intervals for an identifiable pathway model showed that the approach is robust against parameter uncertainties. The quantities are calculated based on the novel

The proposed method provides important information for a wide spectrum of biological applications ranging from cell biology and pharmacokinetics to population dynamics. We applied it to a non-linear model of the cellular JAK-STAT signaling pathway, which allowed for calculating the time-dependent label half-life and transit-time of cytoplasmic STAT.

In summary our approach enables to calculate the amount of time a molecule spends in a certain state or compartment and therefore provides novel insights into the temporal scale of networks. This knowledge will have profound impact on drug design, as it offers the possibility to predict the life-time of a specific molecule and provides a basis to improve drug targeting.

Authors' contributions

TM and JB: Definition of the biological question, initiation, development, and implementation of the proposed method, writing of the manuscript. AR: Development of the proposed method, writing of the manuscript. SH, MS, SS, VB, UK, JT: Definition of the biological question, initiation of the method, critical discussion and contribution to manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the German Virtual Liver Network of the German Federal Ministry of Education and Research (BMBF, FKZ 0315766), the German Federal Ministry for Economy, the European Social Fund (BMWi, ESF, 03EGSBW-004), the Initiative and Networking Fund of the Helmholtz Association within the Helmholtz Alliance on Systems Biology/SBCancer Helmholtz, the NIH grant GM68762, and the German Federal Ministry for Education and Research (BMBF, FRISYS 0313921). This work was also supported by the Excellence Initiative of the German Federal and State Governments.