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Conceived and designed the experiments: JEP RR SLD. Performed the experiments: JEP. Analyzed the data: JEP RR SLD. Wrote the paper: JEP RR SLD.

The authors have declared that no competing interests exist.

A defining characteristic of living cells is the ability to respond dynamically to external stimuli while maintaining homeostasis under resting conditions. Capturing both of these features in a single kinetic model is difficult because the model must be able to reproduce both behaviors using the same set of molecular components. Here, we show how combining small, well-defined steady-state networks provides an efficient means of constructing large-scale kinetic models that exhibit realistic resting and dynamic behaviors. By requiring each kinetic module to be homeostatic (at steady state under resting conditions), the method proceeds by (_{1} signaling can cause widespread compensatory effects on cellular resting states.

Cells respond to extracellular signals through a complex coordination of interacting molecular components. Computational models can serve as powerful tools for prediction and analysis of signaling systems, but constructing large models typically requires extensive experimental datasets and computation. To facilitate the construction of complex signaling models, we present a strategy in which the models are built in a stepwise fashion, beginning with small “resting” networks that are combined to form larger models with complex time-dependent behaviors. Interestingly, we found that only a minor fraction of potential model configurations were compatible with resting behavior in an example signaling system. These reduced sets of configurations were used to limit the search for more complicated solutions that also captured the dynamic behavior of the system. Using an example model constructed by this approach, we show how a cell's resting behavior adjusts to changes in the kinetic rate processes of the system. This strategy offers a general and biologically intuitive framework for building large-scale kinetic models of steady-state cellular systems and their dynamics.

Computational models help quantify the reaction dynamics and regulatory modes in complex biochemical systems

To address these challenges, we propose a strategy for assembling large kinetic networks that retain the nonlinear dynamics governing individual reactions in the system. The key features of the method are: (

The method exploits three properties common to many biological systems: modularity, homeostasis, and known quantitative kinetic relationships among interacting molecular components. Interestingly, this physiology-inspired approach enforces natural constraints on the range of allowable system states and allows one to monitor shifts in steady states due to kinetic perturbations. To illustrate the method with an example, we show how 77 reactions from 17 primary data sources were integrated to construct an accurate model of intracellular calcium and phosphoinositide metabolism in the resting and activated human platelet. Finally, we extend our analysis of this modeling approach by examining the steady-state characteristics of a system that is affected by changes in kinetic rate constants.

Our method builds upon a common representation of biochemical reaction networks

(A) The model topology defines the state transitions (arrows) and rate equations (

Often, the topology of a biological system is better characterized than its CV

A special situation arises when

The first phase of the method involves generating a compact representation of the steady-state solutions for each module. The steps for module reduction are outlined in

(A) Steps in dimensionality reduction of kinetic modules: (1) Restrict value ranges for each ^{9} initial guesses for

Once the sampling distribution for

In the third step, a large collection of steady-state solutions for each module is subjected to principal component analysis (PCA). A sample size of 1000 points per unknown concentration is generally sufficient to minimize error due to over-fitting

The reduction procedure is illustrated with an example of a human platelet model comprising 4 interlinked signaling modules (^{9} sets of initial guesses (^{2+} ([Ca^{2+}]_{i}^{2+}]_{i}^{2+} balance module. This procedure was used to generate 10,000 steady state solutions for each module for subsequent reduction by PCA. A minimal set of principal component (PC) vectors (those capturing 90% or more of the variance in the solution set) were used as search directions in the final estimation step, in which the transient behavior of the perturbed steady-state was compared to experimental time-series data.

Interestingly, only a small fraction of initial guesses produce steady-state solutions that are also consistent with known concentration values. For example, it was previously shown that only 50,000 of 10^{9} initial guesses (0.005%) in the Ca^{2+} balance module (_{3} molecules/cell, although initial guesses were sampled uniformly between 1 and 10^{6} molecules/cell. This observation shows that the kinetic topology of these molecular networks places very strong constraints on the range of concentrations that can exist at steady state. In biological terms, this suggests that fixed kinetic properties at the molecular level (e.g., IP_{3}R and SERCA kinetics) can affect not only the dynamical features of a biochemical system but can also determine the abundance of chemical species and the compartmental structures that contain them.

In the final step of the method, the full model is assembled by combining PCA-reduced, steady-state solution spaces from each module into a combined steady-state solution space for the entire system (^{2+} release (^{2+} and IP_{3} response of platelets exposed to ADP (0–100 µM). Additionally, rigid and flexible nodes (steady-state concentrations) in this 72-dimensional space were readily identified when a set of allowable steady-state solution vectors are compared

(A) The full model is assembled by combining PCA-reduced, steady-state solution spaces from each module into a combined steady-state solution space. This global space is searched for full-length, steady-state solution vectors that satisfy both the steady-state requirements of each module and the desired time-dependent properties when the steady-state is perturbed (in this example, by increasing the concentration of the signaling molecule ADP and measuring the change in intracellular Ca^{2+} concentration). A simple linear constraint is imposed for every pair of modules that share a common molecule ^{2+} release. Species are grouped according to compartment. Color values correspond to molar concentrations (mol/L or mol/m^{2}) or as indicated: *DTS species (mol L^{−1}). †Extracellular species (mol L^{−1}). ‡DTS volume (L). §PM leak conductance/area (S m^{−2}).

Resting systems remain in a steady state by the coordinated action of opposing but balanced kinetic processes. Thus, in general, altering one ore more of these rate processes (e.g., increasing the catalytic rate of a reaction) should upset the balance of the system and cause it to adopt a new steady state. Various cell types have been shown to have altered steady-state properties because of mutations that affect the constitutive rates of reactions. For example, patients with type 1 diabetes harbor more Ca^{2+} ATPase activity in their platelets than healthy volunteers and experience high resting levels of intracellular Ca^{2+}

The steady-state platelet model was perturbed by changing selected kinetic parameters (±10-fold) and simulating for 1 h (^{2+} release through open IP_{3}R channels (_{2} hydrolysis (10-fold increase in _{cat} of hydrolysis). Reactions with perturbed rate constants are circled and correspond to reaction mechanisms from ^{2+}_{dts}^{2+}_{i}_{2} → PLC-β*+IP_{3}+DAG.

As expected, increasing the rate of Ca^{2+} release from intracellular stores resulted in higher cytosolic Ca^{2+} levels (7-fold increase) and 10-fold greater pumping activity by plasma membrane Ca^{2+} pumps (PMCA), although the new steady-state Ca^{2+} release flux remained relatively unchanged (_{2}) hydrolysis, elevated (inositol 1,4,5-trisphosphate) IP_{3} concentration, and accelerated Ca^{2+} release. Interestingly, the same reaction that was initially perturbed with a 10-fold decrease experienced a 10-fold increase in steady-state flux. This was a compensatory effect caused by the negative feedback loop involving Ca^{2+}-regulated activity of PKC, a resulting new hypothesis that can be probed experimentally. In a third example, increasing the hydrolytic activity of PLC-β for the substrate PIP_{2} by 10-fold caused an expected stimulatory effect, raising intracellular calcium and steady-state levels of cytosolic inositol phosphates (IP_{3}, IP_{2}, and IP) between 2- and 3-fold. Interestingly, reaction fluxes for phosphoinositide hydrolysis were diminished, possibly due to substrate depletion. Taken together, these examples illustrate the system-wide effects of perturbations in the kinetic rate processes. The procedure could easily be extended to examine multiple simultaneous perturbations in both reaction rates and steady-state concentrations.

We have presented a novel strategy for enumerating permissible steady-state solutions to fixed kinetic topologies and combining these solutions spaces to form large kinetic models. This is a practical strategy because kinetic parameters are commonly reported whereas absolute concentrations are not (see, for example,

The proposed method requires the model to fulfill a steady-state assumption (i.e., the model must contain nontrivial steady states) even if the system is typically characterized by transient behavior. It is precisely this requirement that allows the model to have the dual functional behavior observed in many biological contexts, such as in cellular signaling responses. At very low levels of activating signal, the model remains at rest by quenching the low level of activating signal through feedback mechanisms or futile cycling. When activating signals are increased, the system responds with the appropriate transient signaling behavior. As an example, a human platelet must remain quiescent under normal circulating conditions, tolerating a number of fluctuations in its surrounding chemical and physical environment. In the presence of the appropriate stimulus, however, it must be able to respond rapidly to bleeding conditions and trigger a precise program of molecular signaling events. Developing a mathematical model that is consistent with two or more biological behaviors is analogous to writing a set of equations that has multiple solutions, each dependent on a given set of initial conditions and parameter values.

Our approach differs critically from metabolic flux analysis and previous genome-scale metabolic network reconstructions

Additional computational savings are provided through modularization. When estimating modules of modest size (5 or less unknown concentrations), we use a brute-force Monte Carlo approach to densely sample the feasible space of initial conditions. Larger networks (20 or more unknowns) cannot be efficiently searched in this brute-force manner, but can be built piecewise by combining subspaces of smaller size that have been densely sampled. Using the naïve Monte Carlo approach, estimating

As presented, the method exploits known kinetic parameters to restrict unknown concentrations due to kinetic interactions. However, the method is equally valid for estimating unknown kinetic parameters and/or utilizing known concentrations. Both concentrations and kinetic parameters appear indistinguishably as nonlinear terms in the ordinary differential equations that describe the system (^{9} steady-state solutions representing calcium balance in a resting platelet were divided into 3 groups, according to their qualitative response to increased IP_{3} concentration (low, mild, and high response). Using this technique, the functional testing of steady-state modules may be used to eliminate a large subset of the original steady-state solution set. As another example, one may use data from a Western blot to establish the relative abundance between two proteins in the model. This qualitative information may be used to filter the steady-state solutions to a reduced set that is consistent with experimental results. This kinetically-driven, constraint-based approach, which combines a homeostasis requirement with known kinetic parameters and cellular concentrations, naturally enforces numerical limits on unknown system quantities.

_{1}activation.

^{2+}-ATPase expression in diabetes: a novel signature of abnormal megakaryocytopoiesis.