A simplified model for calcium signaling including calcium influx, ER, and mitochondrial exchange and storage [diagram in panel B taken from Maurya and Subramaniam (Ref. 3) with permission from Biophysical Journal]. (a) Ligand Complement 5a (C5a) binds to its receptor on plasma membrane (PM) and activates G protein . The free binds to and increases its activity which accelerates the phosphorylation of into and DAG. binds to its receptor on the ER membrane. Thus, calcium from the ER is released into the cytosol. Other fluxes between cytosol and mitochondria or ECM are also shown. (b) Receptor module (box 1), GTPase cycle module (box 2), generation module (box 3), and feedback module (box 4); ECM, extracellular matrix; , phosphatidylinositol 4,5-bisphosphate; , inositol 1,4,5-trisphosphate; , a lumped product of phosphorylation; , cytosolic ; Pr, proteins; ER, endoplasmic reticulum; ATP, adenosine triphosphate; ADP, adenosine diphosphate; SERCA, sarco(endo) plasmic reticulum calcium ATPase; PMCA, plasma membrane calcium ATPase; NCX, exchanger; L, ligand C5a; R, receptor C5aR; GRK, G-protein-coupled receptor kinase; CaM, calmodulin; , phospholipase C-; GAP, GTPase activating protein; RGS, regulator of G-protein signaling; DAG, diacylglycerol; PKC, protein kinase C; Pi, phosphate.
Temporal evolution of the concentrations of substrate and product computed using the Gillespie, tau-leap, and CLE approaches. (a) shows time-course of one realization from each method. (b) and (c) show the time-course of mean and standard deviation from 1024 realizations, which show excellent agreement among the three different methods. (d)–(f) show histograms and probability distribution of the number of molecules of S sampled at . The shapes of the three histograms are very similar.
Dose response. (a) Comparison between ensemble average of 16 realizations and deterministic simulation. For better contrast, the time-course from deterministic simulation is shifted by 100 s. (b) Comparison between ensemble average and individual realizations in stochastic simulation for 0.1% (of 30 nM) strength of the ligand C5a. (c) Comparison of the dose response (peak-heights): The difference is quite small as compared to the scale of peak-height. (d) At lower doses, the NRD is larger indicating the stochastic effects. The NRD decreases with increasing dose as the number of the molecules of C5a becomes several hundreds or more.
Revelation of stochastic effects at low doses. (a)–(d) Distributions of peak-height for the 0.1% dose of C5a computed from 16, 64, 256, and 512 realizations, respectively. The dotted vertical line represents the mean value and the solid curves denote theoretical Gaussian distributions. As the number of realizations increases, the shape of the histogram approaches a Gaussian distribution. (e)–(h) The mean of 4, 8, 16, or 32 realizations was computed and 1024 such mean values were generated. All the four histograms are similar to a Gaussian distribution and the standard deviation from these distributions indeed decreased proportional to , being the number of realizations used to compute the mean. (i) The standard deviation computed from 16 realizations for several doses. Contrary to the expectation, higher doses result in larger absolute standard deviations. (j) The normalized standard deviation decreases as the dose is increased, signifying the effect of randomness at lower doses.
Sensitivity analysis. (a)–(c) Response of to changes in . The decrease in the peak-height due to decrease in is much more pronounced than that caused by the same decrease of IC:[R]. (c) NRD is extremely high at very low , suggesting significant stochastic effects at low numbers of molecules of . (b) and (c) also show the effect of using different numbers of realizations for computing the mean. Such differences are small (see text) indicating that 16 realizations are sufficient for computing the mean.
Knockdown response of . [(a)–(b)] The response to the 50%, 80%, 90%, and 99% knockdown of for 0.1% and 10% levels of IC:[R], respectively. As the knockdown rate of increases, both the basal level and peak-height of decrease because the production decreases due to decrease in . (c) Peak-height of response corresponding to different combinations of the and IC:[R] levels. Peak-height increases with high amount of IC:[R] and . (d) NRD is insignificant and decreases as doses of R and increase.
Knockdown response of GRK. [(a) and (b)] The response to the 50%, 80%, 90%, and 99% knockdown of GRK for 0.1% and 10% levels of IC:[R], respectively. (c) Peak-height of response corresponding to different combinations of [GRK] and IC:[R] levels. (d) NRD is insignificant, reaching its maximum of about 1.5% at low IC:[R].
The response to the simultaneous knockdown of GRK and gene/protein related to . Knockdown of GRK and reduction of have opposite effects on the response. The response is much more sensitive to knockdown of GRK than to decrease in .
The run-time scalability of the Gillespie, tau-leap, and chemical Langevin equation algorithms as a function of the number of molecules.
Criteria used to identify slow and fast reactions and corresponding numerical method. Columns 2 and 3 list the scale and simulation method in the scale (method) format.
Summary of results of KD response. The change in the features of calcium response listed is for increase in KD-level (decrease in IC:[.] of the protein). Qualitative nature of the features is mostly independent of the level of [R].
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