Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations

I enjoy taking a scientific paper and reducing it to a homework problem. For example, one of the new homework problems in the 4th edition of Intermediate Physics for Medicine and Biology is Problem 23 of Chapter 4 (Transport in an Infinite Medium), based on a paper by John Wagner and Joel Keizer.

Problem 23 Calcium ions diffuse inside cells. Their concentration is also controlled by a buffer:

Ca + B ⇐⇒ CaB.

The concentrations of free calcium, unbound buffer, and bound buffer ([Ca], [B], and [CaB]) are governed, assuming the buffer is immobile, by the differential equations

∂[Ca]/∂t= D∇²[Ca] − k⁺[Ca][B] + k⁻[CaB],
∂[B]/∂t= −k⁺[Ca][B] + k⁻[CaB],
∂[CaB]/∂t= k⁺[Ca][B] − k⁻[CaB].

(a) What are the dimensions (units) of k⁺ and k⁻ if the concentrations are measured in mole l⁻¹ and time in s?

(b) Derive differential equations governing the total calcium and buffer concentrations, [Ca]_T = [Ca]+[CaB] and [B]_T= [B] + [CaB] . Show that [B]_T is independent of time.

(c) Assume the calcium and buffer interact so rapidly that they are always in equilibrium:

[Ca][B]/[CaB]= K,

where K = k⁻/k⁺.Write [Ca]_T in terms of [Ca] , [B]_T , and K (eliminate [B] and [CaB]).

(d) Differentiate your expression in (c) with respect to time and use it in the differential equation for [Ca]_T found in (b). Show that [Ca] obeys a diffusion equation with an “effective” diffusion constant that depends on [Ca]:

D_eff = D/(1 + K [B]_T/(K+[Ca])²) .

(e) If [Ca] < < K and [B]_T = 100K (typical for the endoplasmic reticulum), determine D_eff/D.

For more about diffusion with buffers, see Wagner and Keizer (1994).

The reference and abstract of the paper is given below.

John Wagner and Joel Keizer (1994) “Effects of Rapid Buffers on Ca2+ Diffusion and Ca2+ Oscillations,” Biophysical Journal, Volume 67, Pages 447–456.

Based on realistic mechanisms of Ca²⁺ buffering that include both stationary and mobile buffers, we derive and investigate models of Ca²⁺ diffusion in the presence of rapid buffers. We obtain a single transport equation for Ca²⁺ that contains the effects caused by both stationary and mobile buffers. For stationary buffers alone, we obtain an expression for the effective diffusion constant of Ca²⁺ that depends on local Ca²⁺ concentrations. Mobile buffers, such as fura^-2, BAPTA, or small endogenous proteins, give rise to a transport equation that is no longer strictly diffusive. Calculations are presented to show that these effects can modify greatly the manner and rate at which Ca²⁺ diffuses in cells, and we compare these results with recent measurements by Allbritton et al. (1992). As a prelude to work on Ca²⁺ waves, we use a simplified version of our model of the activation and inhibition of the IP3 receptor Ca²⁺ channel in the ER membrane to illustrate the way in which Ca²⁺ buffering can affect both the amplitude and existence of Ca²⁺ oscillations.

John Wagner is currently with the Functional Genomics and Systems Biology Group of the IBM T. J. Watson Research Center. In the mid 1990s he was a research assistant with Joel Keizer.

Joel Keizer was a long-time member of the University of California at Davis. A UC Davis website states

Joel’s scientific legacy encompassed several fields. Joel originally trained as a chemist at the University of Oregon under Terrell Hill, where he received his doctorate in theoretical physical chemistry, and did postdoctoral work in chemical physics at the Battelle Institute in Columbus, Ohio. He began his career in 1971 at the University of California, Davis, as an assistant professor of chemistry. He pioneered an approach to the thermodynamics of non-equilibrium steady states, which culminated in the monograph, Statistical Thermodynamics of Nonequilibrium Processes in 1987. By this time, he had over 60 journal publications to his credit.

In the 1980s, Joel gradually shifted his research program and focused his powerful intellect on problems within the biological sciences, first on mathematical models of insulin secretion, and later on intracellular calcium oscillations and diffusion. He subsequently transferred his appointment to the Division of Biological Sciences, where both theoreticians and empiricists respected and admired Joel for his strong modeling work and his insightful collaborations with experimental biologists.

I never met Joel Keizer, but I did know a couple of his collaborators, John Rinzel and Arthur Sherman, both at NIH when I was there in the early 1990s. They worked on bursting in pancreatic beta-cells, and published some influential papers with Keizer (for example, see: Sherman, Rinzel, and Keizer (1988) “Emergence of Organized Bursting in Clusters of Pencreatic Beta-Cells by Channel Sharing,” Biophysical Journal, Volume 54, Pages 411–425).

Finally, the paper by Allbritton et al. cited in the Wagner and Keizer paper is:

Allbritton, Meyer, Stryer (1992) “Range of Messenger Action of Calcium-Ion and Inositol 1,4,5-Trisphosphate,” Science, Volume 258, Pages 1812–1815.