A Statistical View on Calcium Oscillations

Abstract

Transient rises and falls of the intracellular calcium concentration have been observed in numerous cell types and under a plethora of conditions. There is now a growing body of evidence that these whole-cell calcium oscillations are stochastic, which poses a significant challenge for modelling. In this review, we take a closer look at recently developed statistical approaches to calcium oscillations. These models describe the timing of whole-cell calcium spikes, yet their parametrisations reflect subcellular processes. We show how non-stationary calcium spike sequences, which e.g. occur during slow depletion of intracellular calcium stores or in the presence of time-dependent stimulation, can be analysed with the help of so-called intensity functions. By utilising Bayesian concepts, we demonstrate how values of key parameters of the statistical model can be inferred from single cell calcium spike sequences and illustrate what information whole-cell statistical models can provide about the subcellular mechanistic processes that drive calcium oscillations. In particular, we find that the interspike interval distribution of HEK293 cells under constant stimulation is captured by a Gamma distribution.

Appendices

Appendix 1

We here show the equivalence of Eqs. (32.3) and (32.6). For this, it is convenient to introduce

$$\displaystyle \begin{aligned} F(t,s)=1-\int_s^t f(u,s) {\mathrm d} u\,. \end{aligned} $$

(32.18)

The right hand side of Eq. (32.6) can be written as a full derivative in the form

$$\displaystyle \begin{aligned} q(t|s)=-\frac{{\mathrm d} }{{\mathrm d} t} \ln F(t,s) \end{aligned} $$

(32.19)

Multiplying through by (−1) and integrating both sides with respect to t yields

$$\displaystyle \begin{aligned} -\int_s^t q(u|s) {\mathrm d} u= \ln F(t,s)\,, \end{aligned} $$

(32.20)

noting that $\ln F(s,s)=0$. When we exponentiate both sides and use the fact that F(t, s) = f(t, s)∕q(t|s) as per Eq. (32.6) we arrive at Eq. (32.3).

Appendix 2

Here, we demonstrate how to practically apply Eq. (32.10) when f is given by the density for the Gamma distribution as in Eq. (32.5). We obtain

$$\displaystyle \begin{aligned} g_{\mathrm{G}}(u_i,u_{i-1}|x)=x(y_i)\frac{\beta^\alpha}{\varGamma(\alpha)}X_{i,i-1}^{\alpha-1}{\mathrm{e}}^{-\beta X_{i,i-1}}\,, {} \end{aligned} $$

(32.21)

where

$$\displaystyle \begin{aligned} X_{i,i-1}=\int_{y_{i-1}}^{y_i} x(v) {\mathrm d} v\,, {} \end{aligned} $$

(32.22)

since the difference (t − s) in the transformed time u is given by

$$\displaystyle \begin{aligned} u(t)-u(s)=u(y_i)-u(y_{i-1})=\int_0^{y_i} x(v) {\mathrm d} v- \int_0^{y_{i-1}} x(v) {\mathrm d} v=\int_{y_{i-1}}^{y_i} x(v) {\mathrm d} v\,. \end{aligned} $$

(32.23)

A common choice for f ₁ and f _n is a Poisson distribution, which leads to

$$\displaystyle \begin{aligned} g_1(u_1,0|x)=x(y_1) {\mathrm{e}}^{-X_{1,0}}\,,\qquad g_n(U,u_n|x)=e^{-X_{n+1,n}}\,, \end{aligned} $$

(32.24)

in the transformed time, where we have set y ₀ = 0 and y _n+1 = T.

Appendix 3

To generate the Ca²⁺ spikes that underlie the histogram in Fig. 32.3 we use inverse sampling [84]. Since x(t) is non-constant, we need to use the time-dependent ISI density g(u _i, u _i−1|x) from Eq. (32.10). For ease of presentation, we rewrite Eq. (32.10) in terms of the non-transformed Ca²⁺ spike times y _i using the results from Appendix 2 as

$$\displaystyle \begin{aligned} f_{\mathrm{G}}(y_i,y_{i-1} |x)=x(y_i)\frac{\beta^\alpha}{\varGamma(\alpha)}\left[\int_{y_{i-1}}^{y_i} x(v) {\mathrm d} v \right]^{\alpha-1}\exp \left\{-\beta \int_{y_{i-1}}^{y_i} x(v) {\mathrm d} v \right\}\,, {}\end{aligned} $$

(32.25)

which we use in the definition of the cumulative probability function

$$\displaystyle \begin{aligned} F_{\mathrm G}(t,y_{i-1}|x)=\int_{y_{i-1}}^t f_{\mathrm G}(s,y_{i-1}|x) {\mathrm d} s\,. \end{aligned} $$

(32.26)

Suppose now that the last Ca²⁺ spike occurred at time y _i−1. We find the next Ca²⁺ spike time as y _i = y _i−1 + Δ, where

$$\displaystyle \begin{aligned} \varDelta=\inf \left\{t | F_{\mathrm G}(t,y_{i-1}|x)>\omega \right\}\,, \end{aligned} $$

(32.27)

and ω is a random number that is uniformly distributed between 0 and 1. To put it another way, we need to integrate the ISI probability density f _G from y _i−1 until we obtain a value of ω for the integral and then add the corresponding upper bound of the integral to the previous Ca²⁺ spike time y _i−1.