(155d) Timescale Analysis of Cytosolic Calcium Dynamics

Biological systems are characterized by multiple-scales of temporal behavior and spatial distribution. Timescale analysis is useful for understanding the hierarchical functionality of living cells exhibiting complex interactions among myriad components. As discussed by others before (1, 2), faster timescales correspond to the chemical equilibrium among metabolites and slower ones correspond to more physiologically relevant transformations. Given an ordinary differential equations (ODE)-based model, timescale is computed through Eigen-value Eigen-vector analysis of the Jacobian matrix (J) of the ODE system at a steady-state point of the system.

Eigen-values are the inverse of the characteristic time-constants or timescales of the dynamic system. This means that for small-enough perturbations from the chosen steady-state, the dynamic response of the system can be expressed as a linear combination of the exponential functions of the product of the Eigen-values with time, with the coefficients being related to the Eigen-vectors. For stable systems, all the Eigen-values must have negative real-part. Larger Eigen-value yields shorter time-constant and hence, the Eigen-component will exhibit faster dynamics. It is noteworthy that the timescales and the Eigen-components are systemic properties. Thus, every state variable (species) has some contribution from every timescale. Hence, it is useful to consider the dominant timescale for a species. For a particular timescale, we determine what species are most dominant by selecting the largest element in the corresponding Eigen-vector. For a specific species, if there are more than one dominant timescales associated with its dynamics, then during the initial short time-horizon, the dynamics is determined by the fast timescale (small time-constant) and the long-term response is dependent on the slow timescale (large time-constant), with possible intermediate ranges. Hence, depending upon the time-frame of interest, one can use different values.

We have performed timescale analysis of a complex model of the regulation of cytosolic calcium dynamics in mouse macrophage RAW 264.7 cells (3). For the calcium model, the largest Eigen-value (fastest times-scale) corresponds to a time-constant of 0.01227 second. In the corresponding Eigen-vector, the largest element is in the direction of inositol 1,4,5- trisphosphate (IP₃). It means that IP₃ is one of the fastest metabolites in the system. However, results of this analysis have a few shortcomings. First, levels of some species such as cytosolic calcium (Ca_i) and calcium in the mitochondria (Ca_mit) do not appear as the dominant component in any Eigen-vector. Second, some species, such as IP₃, phosphatidylinositol 4,5-bisphosphate (PIP₂), receptor (R) and calcium in the endoplasmic reticulum (Ca_ER) are classified as dominant elements many times (i.e. in many Eigen-vectors). It is because their numerical values at steady state are much larger as compared to others. For instance, concentration of Ca_i is about 0.05 micro-mole/liter (uM), but, that of Ca_ER is about 225 uM, several hundred times larger than that of Ca_i.

To circumvent the problem highlighted above, the differential equations are written in terms of normalized variables and a normalized Jacobian matrix (J') is computed. We show that the ordinary Jacobian (J) and the normalized Jacobian (J') are related through a similarity transformation. Due to the similarity transformation, the Eigen-values of J and J' are exactly the same. However, their Eigen-vectors are different.

Results of normalized Jacobian show improved partition of the timescale. Still, some species do not have a large weight in any eigenvector (examples are: ligand (L), R and the internalized phosphorylated receptor-ligand complex (LR_i). A more detailed inspection of the Eigen-vectors is needed for these cases. In some Eigen-vectors, more than one component is large. For example, in an eigenvector corresponding to a time-constant of 200s, there are two dominant components: internalized phosphorylated receptor (R_p,i) and LR_i. Thus, LR_i also can be considered as a medium timescale metabolite (the slowest time-scale being of the order of several hours). Similarly, the receptor is a slow metabolite.

Our study has demonstrated the challenges in species-specific interpretation of the timescales. The need for a broad and systems level interpretation of timescale analysis has been highlighted. The utility of normalization is also clearly demonstrated through its application to the calcium model.

References:

1.         Stephanopoulos, G., A. Aristidou, and J. Nielsen. 1998. Review of cellular metabolism. In Metabolic engineering: Principles and methodologies. Academic Press, San Diego, USA. 21-79.

2.         Jamshidi, N., and B. O. Palsson. 2008. Top-down analysis of temporal hierarchy in biochemical reaction networks. PLoS Comput Biol. 4:e1000177.

3.         Maurya, M. R., and S. Subramaniam. 2007. A kinetic model for calcium dynamics in RAW 264.7 cells: 1. Mechanisms, parameters, and subpopulational variability. Biophys J. 93:709-28.