The Influence of Ca2+ Buffers on Free [Ca2+] Fluctuations and the ...

Biophysical Journal Volume 106 June 2014 2693–2709

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The Influence of Ca2D Buffers on Free [Ca2D] Fluctuations and the Effective Volume of Ca2D Microdomains Seth H. Weinberg and Gregory D. Smith* Department of Applied Science, The College of William & Mary, Williamsburg, Virginia

ABSTRACT Intracellular calcium (Ca2þ) plays a significant role in many cell signaling pathways, some of which are localized to spatially restricted microdomains. Ca2þ binding proteins (Ca2þ buffers) play an important role in regulating Ca2þ concentration ([Ca2þ]). Buffers typically slow [Ca2þ] temporal dynamics and increase the effective volume of Ca2þ domains. Because fluctuations in [Ca2þ] decrease in proportion to the square-root of a domain’s physical volume, one might conjecture that buffers decrease [Ca2þ] fluctuations and, consequently, mitigate the significance of small domain volume concerning Ca2þ signaling. We test this hypothesis through mathematical and computational analysis of idealized buffer-containing domains and their stochastic dynamics during free Ca2þ influx with passive exchange of both Ca2þ and buffer with bulk concentrations. We derive Langevin equations for the fluctuating dynamics of Ca2þ and buffer and use these stochastic differential equations to determine the magnitude of [Ca2þ] fluctuations for different buffer parameters (e.g., dissociation constant and concentration). In marked contrast to expectations based on a naive application of the principle of effective volume as employed in deterministic models of Ca2þ signaling, we find that mobile and rapid buffers typically increase the magnitude of domain [Ca2þ] fluctuations during periods of Ca2þ influx, whereas stationary (immobile) Ca2þ buffers do not. Also contrary to expectations, we find that in the absence of Ca2þ influx, buffers influence the temporal characteristics, but not the magnitude, of [Ca2þ] fluctuations. We derive an analytical formula describing the influence of rapid Ca2þ buffers on [Ca2þ] fluctuations and, importantly, identify the stochastic analog of (deterministic) effective domain volume. Our results demonstrate that Ca2þ buffers alter the dynamics of [Ca2þ] fluctuations in a nonintuitive manner. The finding that Ca2þ buffers do not suppress intrinsic domain [Ca2þ] fluctuations raises the intriguing question of whether or not [Ca2þ] fluctuations are a physiologically significant aspect of local Ca2þ signaling.

INTRODUCTION The regulation of intracellular calcium (Ca2þ) concentration ([Ca2þ]) is a fundamental aspect of many cell signaling pathways (1). In many types of cells, spatially localized Ca2þ signals known as Ca2þ micro- or nanodomains regulate specific cellular processes in different subcellular regions. Significant examples include pre- and postsynaptic signaling in neuronal dendrites (2–4), contraction regulation in cardiac dyadic subspaces (3,5–7), localized control of mitochondria (8–10), and nuclear gene transcription (4,8), and localized mechanical and olfactory sensing in primary cilia (11–13). Ca2þ buffers play an important role in modulating spatially localizing Ca2þ signals. The term ‘‘Ca2þ buffers’’ is generic and includes endogenous binding proteins, exogenous Ca2þ chelators (e.g., EGTA and BAPTA), indicator dyes (e.g., Fluo-4 and Rhod-2), and other nonspecific Ca2þ binding molecules (e.g., membrane phospholipids). In addition to regulating the levels of free Ca2þ in the cytoplasm, Ca2þ buffers influence the spatiotemporal dynamics of Ca2þ signaling. Because Ca2þ buffer binding rates, affinities, and diffusivities can range over several orders of magnitude (14), it is important to understand the influence of these parameters on Ca2þ-dependent signaling. Using deterministic formulations for Ca2þ and buffer dynamics,

investigators have derived analytical results pertaining to the influence of rapid mobile and immobile buffers on Ca2þ signaling, demonstrating that buffers can greatly alter the [Ca2þ] profile in the proximity of Ca2þ channels (15–19). Immobile buffers have been shown to reduce the effective diffusivity of Ca2þ, whereas mobile buffers can facilitate diffusion near an open Ca2þ channel and produce steep [Ca2þ] gradients (20). Experimental and simulation studies have shown that the decay of residual Ca2þ after Ca2þ channel closure can be longer for rapid buffers than slow buffers (21,22). The influence of buffers on Ca2þ signaling can be counterintuitive, because it depends subtly on binding rate kinetics and competition between Ca2þbinding sites. For example, analytical results have shown that increased buffer diffusivity may increase or decrease the speed of a propagating Ca2þ wave, depending on the excitability of Ca2þ dynamics and buffer properties (23). Assuming a rapidly equilibrating bimolecular association reaction between Ca2þ and buffer (B), kþ

Ca2þ þ B # CaB; k

the time evolution of [Ca2þ], denoted by c below, can be described by the ordinary differential equation (ODE) (24) U_c ¼ bJ;

Submitted December 12, 2013, and accepted for publication April 30, 2014. *Correspondence: [email protected] Editor: James Sneyd. 2014 by the Biophysical Society 0006-3495/14/06/2693/17 $2.00

(1)

(2)

where the dot indicates a time derivative, U is the compartmental volume, J ¼ Jin – Jout is the net flux of http://dx.doi.org/10.1016/j.bpj.2014.04.045

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Ca2þ molecules into the compartment, and the buffering factor b is the differential fraction of free-to-total Ca2þ that takes values between 0 and 1, 1 ; and 1þw bT k w ¼ 2: ðc þ kÞ

b ¼

(3)

In this expression, total buffer concentration bT ¼ b þ cb is the sum of free and Ca2þ-bound buffer concentration, b and cb, respectively; k ¼ k /kþ is the dissociation constant for Ca2þ; and the ratio w ¼ bTk/(c þ k)2 is the buffering capacity, i.e., the differential fraction of bound-to-free Ca2þ. Concentration balance equations such as Eq. 2 are often written in the form Ueff c_ ¼ J;

(4)

where the effective volume Ueff is defined as Ueff ¼ U/b R U. It is well known that the number of molecules in biochemical reaction networks randomly fluctuate, and the resulting concentration fluctuations for each chemical species are larger amplitude when the system size is small (25,26). For a sufficiently large system, i.e., large volume and number of molecules, concentration fluctuations become negligible and deterministic modeling is appropriate (26). However, in physiological settings of local Ca2þ signaling, i.e., Ca2þ microdomains, the system size is often small and concentration fluctuations may be significant. Intracellular [Ca2þ] is very low in resting cells, typically 100 nM, and the volume of Ca2þ microdomains or subspaces are ~1017–1015 liters, values that correspond to 0.6–60 Ca2þ ions at rest. A large amount of prior work has focused on the influence of stochastic gating of Ca2þ-regulated Ca2þ channels on [Ca2þ] (19,27–32), but only a few have considered the role of concentration fluctuations associated with the small number of Ca2þ ions typically present in domains (30,33,34). The relative size of [Ca2þ] fluctuations can be characterized by their coefficient of variation (cv), defined as the standard deviation of [Ca2þ] divided by the mean [Ca2þ]. In a spatially restricted domain and in the absence of Ca2þ buffers, the relative size of [Ca2þ] fluctuations is given by (35) 1 c0v ¼ pffiffiffiffiffiffiffiffiffi; css U

(5)

where css is the steady-state [Ca2þ] and U is the physical volume of the domain (the superscripted 0 indicates no buffers). Because cssU is the number of Ca2þ ions in the domain, this expression is consistent with the influence of system size on molecular fluctuations well known in statisBiophysical Journal 106(12) 2693–2709

tical physics (26). For the small volume domains mentioned above, this c0v is in the range of 1.3–0.13. Our prior work suggests that fluctuations in [Ca2þ] of this magnitude could significantly influence the gating of Ca2þ-regulated Ca2þ channels in Ca2þ microdomains, such as dyadic subspaces or dendritic spines (35). Although prior work makes it clear that [Ca2þ] fluctuations are a nonnegligible aspect of Ca2þ signaling, the influence of Ca2þ buffers on the dynamics of [Ca2þ] fluctuations has not previously been studied. Based on the notion of effective volume, one might conjecture that buffers decrease [Ca2þ] fluctuations and, consequently, mitigate the significance of small domain volume vis-a-vis downstream Ca2þ signaling (e.g., Ca2þ-triggered events). Do Ca2þ buffers influence [Ca2þ] fluctuations? If so, how do these fluctuations depend on Ca2þ buffer properties such as dissociation constant and total concentration (Ca2þ free plus bound)? To answer these questions, we develop and analyze a minimal stochastic model that includes Ca2þ and buffer in a domain of small volume. Our analysis focuses on Ca2þ microdomains that arise from Ca2þ influx into a spatially restricted subcellular compartment that may contain Ca2þ buffers, e.g., a dendritic spine or cardiac dyadic subspace (Fig. 1, A and B). The theory is most directly applicable to Ca2þ signaling complexes with membrane configurations or other obstructions to diffusion (e.g., molecular crowding) resulting in spatially confined Ca2þ signaling, for example, the primary cilium (Fig. 1 C), mitochondriaendoplasmic reticulum (ER) junctions, and presynaptic boutons (2–13). Free Ca2þ and buffer (Ca2þ-bound and -free) may exchange between domain and bulk via diffusion represented as a first-order reaction (Fig. 1 D). We specify the volume but not the detailed geometry of the spatially confined region that delimits the domain. Spatial gradients between domain and bulk are minimally represented in the model, but the microdomain itself is treated as a well-stirred compartment. The compartmental model formulation is most applicable and relevant to microdomains for which the time constant for domain escape is longer than the characteristic time for diffusion across the domain (36). For clarity and readability, we first present the stochastic model representing [Ca2þ] fluctuations in a microdomain with no Ca2þ buffer. This preliminary calculation illustrates our approach and provides a reference point in our interpretation of results that include influx and efflux of both Ca2þ and buffer. We perform parameter studies to determine how Ca2þ buffers influence the size of [Ca2þ] fluctuations, derive the fluctuating rapid buffer approximation to the dynamics of domain [Ca2þ] fluctuations, and illustrate how the influence of buffers on the effective system size and the time course of [Ca2þ] fluctuations depend on buffer parameters. We conclude with a discussion of our findings.

Influence of Ca2þ Buffers on [Ca2þ] Fluctuations

2695 FIGURE 1 The analysis of the effect of Ca2þ buffers on domain [Ca2þ] fluctuations is applicable to microdomains that arise from Ca2þ influx into a spatially restricted subcellular compartment. (A) In a dendritic spine (U denotes volume of spine head), Ca2þ influx occurs via ionotropic receptors (e.g., NMDA) or plasma membrane (PM) Ca2þ channels downstream of metabotropic receptors (e.g., mGluRs) (2). (B) In the cardiac dyad, Ca2þ influx via PM L-type Ca2þ channels trigger Ca2þ influx from Ca2þ-activated sarcoplasmic reticulum (SR) Ca2þ channels (ryanodine receptors) (7). Close associations of endoplasmic reticulum (ER) and PM are observed in many cell types. During ER depletion in Jurkat T cells, luminal Ca2þ sensor STIM1 aggregates on the ER membrane and binds the Ca2þ channel Orai1 providing store-operated Ca2þ entry. (C) In olfactory primary cilium, odorant detection induces a rise in second messenger cAMP, activating cyclic nucleotidegated channels allowing entry of Ca2þ and Naþ ions (11). (D) Model components and fluxes employed in the microdomain model. The stochastic ODE model includes Ca2þ influx, association of Ca2þ and buffer, and passive exchange of Ca2þ and buffer between the domain of volume U and the bulk cytosol.

MODEL FORMULATION Domain [Ca of buffer

2D

elementary molecular events that drive concentration fluctuations (26,37,38),

] fluctuations in the absence

c_ ¼ f ðcÞ þ xc ðtÞ:

(8)

Deterministic ODE formulation

The deterministic ODE that minimally describes the time evolution of [Ca2þ] in the domain depicted in Fig. 1 without buffer, constant influx jin, and passive exchange with the bulk is c_ ¼ jin kc ðc cN Þhf ðcÞ;

(6)

where kc is the exchange rate (due to diffusion) that has physical dimensions of inverse time. Setting the left-hand side of Eq. 6 to zero, steady-state [Ca2þ] is found to be css ¼ jin =kc þ cN :

(7)

This deterministic formulation assumes that concentration fluctuations are negligible (i.e., large system size, U / N). The following section presents the corresponding stochastic ODEs that are valid for small domain volume. Stochastic ODE (Langevin) formulation

In a domain with physical volume small enough that fluctuations are not negligible, but large enough that molecular concentrations can be modeled continuously, [Ca2þ] dynamics may be described by a Langevin equation. This would be a stochastic differential equation (SDE) similar to the deterministic ODE (Eq. 6), but augmented by a fluctuating term, xc(t), with properties determined by the

In this stochastic ODE, xc(t) is Gaussian noise (specifically, a random function of time with zero mean), hxc ðtÞi ¼ 0

(9)

and two-time covariance hxc ðtÞxc ðt 0 Þi ¼ gc dðt t 0 Þ;

(10)

where d is the Dirac delta function and gc is given by gc ¼

jin þ kc ðc þ cN Þ : U

(11)

Note that the signs in the numerator are correct; gc is proportional to the sum of rates of the elementary processes leading to changes in [Ca2þ], which includes increases through influx (jin) as well as decreases and increases due to exchange with the bulk (kcc and kccN). The reader who is unfamiliar with chemical Langevin equations may review the general form of the two-time covariance (see Appendix A in the Supporting Material). Employing a linear noise approximation (26) for fluctuations around the stable steady-state css transforms Eq. 8 into _ ¼ kc dc þ xss ðtÞ; dc c

(12)

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where dc is the deviation of the fluctuating [Ca2þ] from the steady-state value, css, that is, dcðtÞ ¼ cðtÞ css ;

(13)

and we have used f 0 (css) ¼ kc and hxcss(t)i ¼ 0. The twotime covariance of the random term takes the form ss 0 ss 0 xc ðtÞxss c ðt Þ ¼ gc dðt t Þ; where the constant gss c is found by evaluating Eq. 11 at steady state, that is, gss c ¼

2kc css ; U

(14)

where we have used the balance of Ca2þ influx and efflux at steady state, jin ¼ kc(css – cN), to express the numerator as 2kccss rather than jin þ kc(css þ cN) (compare to Eq. 14).

at steady-state (cssU). This result is a reference point for our analysis of domain [Ca2þ] fluctuations in the presence of buffer. Domain [Ca2D] fluctuations in the presence of buffer Deterministic ODE formulation

In this section we characterize [Ca2þ] fluctuations in a domain model that includes Ca2þ buffers, Ca2þ influx, and exchange of both Ca2þ and buffer with the bulk (see Fig. 1). Assuming mass-action kinetics and bimolecular association of Ca2þ and buffer (Eq. 1), we write the following deterministic system of ODEs, c_ ¼ R þ jin kc ðc cN Þ;

(19a)

b_ ¼ R kb ðb bN Þ;

(19b)

c_ b ¼ R kb ðcb cbN Þ;

(19c)

Analysis of concentration fluctuations

To understand the dynamics of [Ca2þ] fluctuations implied by Eq. 12, consider an ensemble of domains with initial state c(0) ¼ css, that is, dc(0) ¼ 0. The time evolution of the ensemble variance of the deviations dc, defined by sc(t) ¼ hd2c (t)i, follows from Eq. 12, s_ c ¼ 2kc sc þ gss c ;

(15)

where kc ¼ jf 0 (css)j is the relaxation rate. By assumption, the ensemble variance is initially zero, that is, dc(0) ¼ 0 implies sc(0) ¼ 0, but as time proceeds, sc(t) is given by the solution of Eq. 15, sc ðtÞ ¼ s0c 1 e2kc t ; (16) where steady-state ensemble variance s0c is found by setting s_ c ¼ 0 in Eq. 15, s0c ¼

gss css c ¼ : 2kc U

(17)

where kc and kb are the Ca2þ and buffer exchange rates, and the reaction terms are given by R ¼ k þ c b þ k cb, with c, b, and cb denoting free Ca2þ, free buffer, and bound buffer, respectively. We assume that the free and Ca2þbound buffer have the same exchange rate kb and, furthermore, that the bulk concentrations bN and cbN are in equilibrium with the bulk free [Ca2þ], bN ¼

kbN T ; and cN þ k

cbN ¼

c N bN T : cN þ k

Setting the time derivatives of Eq. 19 to 0, we find that the steady-state domain [Ca2þ], denoted by css, solves the implicit expression jin kc ðcss cN Þ kb ðcbss cbN Þ ¼ 0;

(20a)

The first equality in Eq. 17 relating s0c and gss c is called the ‘‘fluctuation-dissipation theorem’’ (25,26,39). Thus, the relative size of the steady-state [Ca2þ] fluctuations in the Langevin formulation, given by the coefficient of variation of the fluctuating [Ca2þ], is pffiffiffiffiffi s0c 1 0 ¼ pffiffiffiffiffiffiffiffiffi: cv ¼ (18) css css U

where the steady-state Ca2þ-free and -bound buffer concentrations are given by kbN T þ k b bN k þ bss ¼ ; and css þ k þ kb =kþ (20b) css bN T þ kb cbN kþ : cbss ¼ css þ k þ kb =kþ

Using the Langevin equation for a fluctuating Ca2þ microdomain, we have derived the well-known result (35) that the relative magnitude of concentration fluctuations is inversely proportional to the square-root of the system size, i.e., the expected number of Ca2þ ions in the domain

In the presence of Ca2þ influx, jin > 0, and because vcbss/vcss > 0, css > cN, cbss > cN, and bss > bN. Also note that cbss þ bss ¼ bN T at this nonequilibrium steady state, because buffer in the bulk is at equilibrium, 2þ cbN þ bN ¼ bN T , and the exchange of Ca -bound buffer



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between domain and bulk, balances the exchange of Ca2þfree buffer, kb(cbss cbN) ¼ kb(bN bss) > 0. Stochastic ODE (Langevin) formulation

As in the previous section, the Langevin-type stochastic Ca2þ domain model is found by adding the appropriate random terms to the deterministic ODEs (Eq. 19), c_ ¼ R þ jin kc ðc cN Þ þ xc ðtÞ;

(21a)

b_ ¼ R kb ðb bN Þ þ xb ðtÞ;

(21b)

_ ¼ R kb ðcb cbN Þ þ xcb ðtÞ; cb

(21c)

where hxi(t)i ¼ 0 for i ˛ {c,b,cb}. Because the xi(t) are correlated, the two-time covariances are most easily expressed in matrix form, 0 0 xðtÞxT t ¼ Gðc; b; cbÞd t t ; (22) where x is the column vector (xc, xb, xcb)T and G is the statedependent covariance matrix (see Appendix A in the Supporting Material), 0 1 gr þ gc gr gr Gðc; b; cbÞ ¼ @ gr gr þ gb gr A; (23a) gr gr gr þ gcb where kþ c b þ k cb ; U jin þ kc ðc þ cN Þ gc ðcÞ ¼ ; U kb ðb þ bN Þ gb ðbÞ ¼ ; U kb ðcb þ cbN Þ gcb ðcbÞ ¼ : U

gr ðc; b; cbÞ ¼

(23b)

ss gss r þ gc ss Gss ¼ @ gr gss r

gss r ss gr þ gss b gss r

1 gss r A; gss r ss gr þ gss cb

(25a)

where kþ css bss þ k cbss ; U jin þ kc ðcss þ cN Þ gss ; c ¼ U kb ðbss þ bN Þ gss ; b ¼ U kb ðcbss þ cbN Þ gss : cb ¼ U

gss r ¼

(25b)

If one prefers, the Langevin system for the fluctuations (Eq. 24) can be written as _ ¼ ðkþ bss þ kc Þ dc kþ css db þ k dcb þ xss ðtÞ; (26a) dc c _ ¼ kþ bss dc ðkþ css þ kb Þ db þ k dcb þ xss ðtÞ; (26b) db b _ ¼ kþ bss dc þ kþ css db ðk þ kb Þ dcb þ xss ðtÞ: (26c) dcb cb Analysis of concentration fluctuations

Our analysis of concentration fluctuations in the presence of buffer begins by defining a symmetric 33 covariance matrix for the fluctuating concentrations, S(t) ¼ (sij) for i, j ˛{c,b,cb}, that is, 0 2 1 hdc i hdc dbi hdc dcbi T SðtÞ ¼ dðtÞd ðtÞ ¼ @ + hdb2 i hdb dcbi A; + + hdcb2 i where each star indicates a redundant entry. The timedependent dynamics of S(t) follows from Eq. 24 (26),

Denoting the fluctuations around steady state by dc ¼ c – css, db ¼ b – bss, and dcb ¼ cb – cbss, the linear SDE system for the concentration fluctuations in the presence of buffer is d_ ¼ Hss d þ xss ðtÞ;

0

(24)

where d ¼ (dc, dc, dcb)T is a column vector, Hss is the Jacobian of the full system of SDEs evaluated at steady state, 1 0 kþ css k kþ bss kc A; Hss ¼ @ kþ bss kþ css kb k kþ bss kþ css k kb hxss(t)i ¼ 0, and hxss(t) xTss(t 0 ) i ¼ Gssd(t – t 0 ) with steadystate covariance matrix (compare to Eq. 23),

S_ ¼ Hss S þ SHssT þ Gss ;

(27)

which is the matrix version of Eq. 15. Solving Eq. 27 with the initial condition Sð0Þ ¼ 0 corresponds to an ensemble of domains with initial state c(0) ¼ css, b(0) ¼ bss, and cb(0) ¼ cbss, that is, dc ¼ db ¼ dcb ¼ 0. This equation represents a linear system of 6 ODEs simultaneously solved by the six covariances hdc2i, hdcdbi,., hdcb2i. The covariances S(t) that solve Eq. 27 are the focus of this article’s mathematical and computational analysis of domain [Ca2þ] fluctuations in the presence of buffer. The solution of Eq. 27 with S(0) ¼ 0 is (26), Zt SðtÞ ¼

expðHss^tÞGss exp HssT ^t d^t:

(28)

0


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The steady-state ensemble variance is the limiting value of this expression as t / N, ZN Sss ¼

expðHss tÞGss exp HssT t dt;

(29)

8 of vec(Sss) and vec(Gss) and the corresponding rows and columns of the Hss 4 Hss. The remainder of this article communicates numerical and analytical results obtained using this Langevin formulation for domain [Ca2þ] fluctuations in the presence of buffer.

0

but the steady-state ensemble variance Sss is often more conveniently found by solving the linear algebraic system for the steady state of Eq. 27, the continuous Lyapunov equation, and the more general form of the fluctuation-dissipation theorem, Hss Sss þ Sss HssT ¼ Gss :

(30)

Equation 30 can be solved numerically using the command LYAP available in the software MATLAB (The MathWorks, Natick, MA). For symbolic and analytical calculations, we rewrite Eq. 30 as ðHss 4Hss ÞvecðSss Þ ¼ vecðGss Þ

(31)

and then expand in terms of the parameters of the problem and the unknown entries of Sss. In Eq. 31, I is the 33 identity matrix, Hss 4 Hss is a Kronecker sum given by Hss 4Hss ¼ I 5 Hss þ Hss 5 I; where 5 is the Kronecker product, the vec operation creates a column vector from a matrix by stacking its column vectors, and we have used vecðABÞ ¼ ðI 5 AÞvecðBÞ ¼ BT 5 I vecðAÞ: Although Hss 5 Hss is 99 and vec(Sss) and vec(Gss) are 91, Sss and Gss are symmetric and thus Eq. 31 can be contracted to a 66 linear system by eliminating rows 4, 7, and

RESULTS The influence of Ca2D buffers on domain [Ca2D] fluctuations Fig. 2 shows parameter studies that characterize the dependence of the relative magnitude of domain [Ca2þ] fluctuations on Ca2þ buffer parameters. Results were obtained by numerically solving the continuous Lyapunov equation (Eq. 30) for a range of total buffer concentrations (bN T ) and rate constants (kþ and k), fast and slow exchange rates for Ca2þ-free and -bound buffer with the bulk (kb), and Ca2þ influx rates leading to steady-state free [Ca2þ] of css ¼ 1, 10, and 100 mM. The relative magnitude of domain [Ca2þ] fluctuations are characterized by the coefficient of variation pffiffiffiffiffissffi sc cv ¼ ; (32) css where the steady-state domain concentration css is deter2 mined by the Ca2þ influx rate via Eq. 20 and sss c ¼ hdc iss is the germane element of Sss found by numerical solution of Eq. 30. Numerically calculated cv values that were generated from Monte Carlo simulations of an ensemble of domains using Gillespie’s stochastic simulation algorithm (40) agree with the linear noise approximation, validating our approach (see Fig. S1 in the Supporting Material). Fig. 2 reveals that the relative magnitude of domain [Ca2þ] fluctuations (cv) is a biphasic (bell-shaped) function of the total buffer concentration bN T (gray lines). For small

A

B

C


FIGURE 2 Dependence of the relative magnitude of domain [Ca2þ] fluctuations probed by plotting the coefficient of variation (cv) of the deviation of the fluctuating [Ca2þ] on buffer parameters. (Shaded lines) cv for different values of buffer binding reaction rate kþ. (Solid lines) c0v in the absence of buffer (Eq. 5). (Dotted line) cv predicted from a naive application of the concept of effective volume employed in deterministic models of Ca2þ signaling (Eq. 33). (Dashed lines) c0v , the (correctly derived) influence of buffer on cv in the rapid buffer limit (Eq. 39). Parameters: css ¼ 1 (A), 10 (B), and 100 (C) mM, cN ¼ 0.1 mM, k ¼ 0.2 mM, kc ¼ 0.2 ms1, kþ ¼ 103, and 1 mM1 ms1. The domain volume U ¼ 1017 L corresponds to 0.01 mm3 z (0.22 mm)3.


or large total buffer concentrations (bN T ), the relative fluctuation magnitude (cv) asymptotically approaches pffiffiffiffiffiffiffiffiffi the value obtained in the absence of buffer (c0v ¼ 1= css U, solid black line). This asymptotic value represents the minimum fluctuation size; cv is enhanced for intermediate total buffer concentrations. Rapid Ca2þ buffers increase the relative fluctuation magnitude, that is, for fixed k ¼ k/kþ, the cv increases with increasing kþ. High mobility Ca2þ buffer (represented in this compartmental formulation by increasing the exchange rate kb) also enhances domain [Ca2þ] fluctuations (compare left and right columns). High Ca2þ influx rates (jin) amplify domain [Ca2þ] fluctuations for intermediate, but not extreme, concentrations of buffer. Taken together, these results indicate that domain [Ca2þ] fluctuations are significantly enhanced by Ca2þ buffer and nontrivially dependent on Ca2þ buffer properties. The response of domain [Ca2þ] to buffer parameters documented in Fig. 2 are counterintuitive in the sense that our numerical calculations are precisely the opposite of what one would predict using a naive application of the concept of effective volume. As mentioned in the Introduction, the effective volume derived from the rapid buffer limit of the deterministic equations for buffered Ca2þ dynamics (Eq. 19) is Ueff ¼ U/b, where the buffering factor b is given by Eq. 3. If one replaces the physical volume U in Eq. 18 with this effective volume, one might conjecture that sffiffiffiffiffiffiffiffiffi 1 bss ? cv ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; (33) css U css Ueff where bss ¼ b(css). However, the numerical calculations presented in Fig. 2 show that this is incorrect (dotted line). In fact, this naive application of the concept of buffer-mediated increase in effective volume is quite misleading. Comparison of solid and dotted lines in Fig. 2 show that Eq. 33 incorrectly suggests that the size of [Ca2þ] fluctuations is suppressed by buffer (by increasing the system size, cssUeff > cssU). The conjecture of Eq. 33 also incorrectly suggests that domain [Ca2þ] fluctuations can be eliminated (cv / 0) as total buffer concentration bN T increases and drives the differential fraction of free to total calcium to zero (bss / 0). On the contrary, the numerical results of this section (and analytical results presented below) are definitive: during periods of Ca2þ influx, Ca2þ buffers enhance domain [Ca2þ] fluctuations. The size of [Ca2D] fluctuations in the rapid buffer limit Although closed-form analytical solution for the entries of the steady-state covariances Sss (and in particular sss c ) can be obtained beginning with Eq. 31, the algebra is tedious, the result is unwieldy, and little insight is gained from these expressions (not shown). On the other hand, numerical evaluation of symbolic solutions of Eq. 31 confirm the results of

2699

the previous section obtained by numerically solving the Lyapunov equation (Eq. 30). To obtain more insight into the enhancement of domain [Ca2þ] fluctuations mediated by Ca2þ buffers, we sought to derive a stochastic version of the rapid buffer approximation (RBA) that correctly accounts for the influence of fast Ca2þ buffering on domain [Ca2þ] fluctuations. Two distinct approaches were identified and successfully employed, both yielding the same analytical result. In the first approach, the above-mentioned cumbersome analytical expression for sss c was symbolically evaluated in the limit that kþ, k / N for fixed k ¼ k /kþ. The second approach involves no symbolic computations, but rather implements a general method for deriving a 0th-order approximation to the size of fluctuations in chemical reaction networks with widely separated timescales (26). Briefly, beginning with the stochastic ODEs for buffered Ca2þ dynamics (Eq. 26), we identify the fluctuations in total calcium (dcT ¼ dc þ dcb) and total buffer (dbT ¼ db þ dcb) as the slow variables. Differentiating these expressions and adding equations, we obtain the fast/slow system _ ¼½kþ ðcss þ bss Þ þ k þ kc dcþðkþ css þ k ÞdcT dc kþ css dbT þxss c ðtÞ; d_cT ¼ ðkc kb Þdc kb dcT þ xss cT ðtÞ; db_ T ¼ kb dbT þ xss bT ðtÞ;

(34a)

(34b) (34c)

where dc is the fast variable, dcT and dbT are slow variables, and we define ss ss ss ss ss xss cT ðtÞ ¼ xc ðtÞ þ xcb ðtÞ and xbT ðtÞ ¼ xb ðtÞ þ xcb ðtÞ:

In the rapid buffer limit (kþ, k / N with k ¼ k /kþ fixed), a quasistatic approximation for the average value of the [Ca2þ] fluctuation that is valid on the slow (outer) timescale is hdci zbss ½dcT nss dbT ;

(35)

where we have written nss ¼ css/(css þ k), and h,i* indicates a time average (as opposed to an ensemble average) and we have used the fact that bss / bTk/(css þ k) in the RBA limit (compare to Eq. 20b). Replacing dc in the slow equations by this average value gives d_cT ¼ bss ðkc þ wss kb ÞdcT þ bss ðkc kb Þnss dbT þ xss cT ðtÞ; (36a) db_ T ¼ kb dbT þ xss bT ðtÞ;

(36b)

thereby expressing the slow SDEs in terms of dcT and dbT. ss Next, the 22 covariance matrix Gslow for xss cT and xbT is ss Biophysical Journal 106(12) 2693–2709

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calculated, and the 22 matrix Sslow ss , whose entries correspond to hdc2T i, hdcTdbT i, and hdb2T i, is found by solving a 22 Lyapunov equation (Eq. 30). Finally, the values of hdc2i, hdcdcT i, and hdcdbTi consistent with Eq. 34 are found by solving a 33 Lyapunov equation for hdc2i, hdcdcTi, and hdcdbTi. From the perspective of analytical work, this two-step process is far easier than solving the Lyapunov equation for the covariance matrix Sss for dc, db, and dcb, because the fast and slow versions of the Gss and Hss matrices are simpler than those that appear in the full calculation (see Appendix B in the Supporting Material for details). Using both of the above-mentioned approaches (symbolic and analytical), we find that the variance of the free [Ca2þ] fluctuations in the presence of rapid Ca2þ buffer (denoted by s0c ) is given by css 0 ð1 þ cÞ; sc ¼ U (37) wss kb css cN c ¼ bss ; kc þ wss kb k þ cN where bss and wss are evaluated at the steady-state domain [Ca2þ], bss ¼ wss ¼

1 ; 1 þ wss bN T k

(38)

2; ðcss þ kÞ

and we note that css R cN and thus c R 0. The variance s0c implies that in the rapid buffer limit the coefficient of variation for domain [Ca2þ] fluctuations is given by pffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi s0c 1þc ¼ c0v 1 þ c; (39) ¼ c0v ¼ css U css pffiffiffiffiffiffiffiffiffi where c0v ¼ 1= css U is the result in absence of buffer. Noting that c R 0, we find pffiffiffiffiffiffiffiffiffiffiffiffi c0v ¼ 1 þ cR1 0 cv

(40)

and conclude that Ca2þ buffers may enhance but cannot suppress domain [Ca2þ] fluctuations. One way to represent the effect of rapid buffers on domain [Ca2þ] fluctuations involves rewriting Eq. 37 as 0

sc ¼ 0

Ueff

css 0 ; Ueff

U ; ¼ 1þc

(41)

where U0 eff denotes an effective volume that appropriately accounts for the influence of rapid Ca2þ buffers on [Ca2þ] Biophysical Journal 106(12) 2693–2709

fluctuations. The fluctuating RBA is readily generalizable to multiple buffers (see Eq. S1 in the Supporting Material). We show that domain [Ca2þ] fluctuations in the presence of two rapid buffers with different dissociation constants and exchange rates is given by the multiple buffer fluctuating RBA and not a naive weighted average utilizing total buffer concentration (see Fig. S2 in the Supporting Material).

Interpretation of the fluctuating rapid buffer limit Some salient observations can be made regarding domain [Ca2þ] fluctuations in the rapid buffer limit (Eqs. 37 and 41): 1. The effective volume in the fluctuating RBA is smaller 0 than the physical volume (c R 0 and thus U eff % U). This is in marked contrast to the effective volume for the deterministic RBA that is greater than the physical 0 volume (U eff ¼ U/bss > U). This is to say, we have analytically confirmed our previous numerical result demonstrating that rapid buffers typically increase the magnitude of domain [Ca2þ] fluctuations (Fig. 2). 2. The c in s0c is proportional to the concentration difference css cN (Eq. 37). That is, the buffer-mediated enhancement of domain [Ca2þ] fluctuations is an increasing function of the disequilibrium between the steady-state domain [Ca2þ] (css) and the bulk (cN). 3. The numerically observed biphasic dependence of the relative magnitude of domain [Ca2þ] fluctuations (cv in Fig. 2) corresponds to the factor bsswsskb/(kc þ wsskb) in c that is a biphasic function of bN T through bss and wss. 4. In several limiting parameter regimes, the steady-state variance of domain [Ca2þ] fluctuations is not affected by Ca2þ buffer (c ¼ 0 and thus s0c ¼ s0c ¼ css/U). These include the absence of Ca2þ influx (jin ¼ 0, css ¼ cN), immobile buffer that is unable to exchange with the bulk (kb ¼ 0), and extreme values for the total buffer conN centration (both bN T / 0 and bT / N). However, even in these limiting parameter regimes, buffers do not decrease [Ca2þ] fluctuations to a size less than that observed in the absence of buffer. 5. The total buffer concentration that maximizes domain [Ca2þ] fluctuations for a given fixed css, rffiffiffiffi 2 kc ðcss þ kÞ ; (42) ¼ bN T kb k N N is found as the bN T that zeros v(1 þ c)/vbT ¼ vc/vbT . This corresponds to pffiffiffiffiffiffiffiffiffiffiffi wss ¼ kc =kb ;

bss ¼ and

pffiffiffiffiffiffiffiffiffiffiffi. pffiffiffiffiffiffiffiffiffiffiffi kb =kc 1 þ kb =kc ;

Influence of Ca2þ Buffers on [Ca2þ] Fluctuations 0

U ; 1 þ c sffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c ¼ ; css U

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Ueff ¼

0

cv

(43)

kb =kc css cN c ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 k þ c : N 1 þ kb =kc 6. Equation 43 indicates that the maximum fluctuation magnitude is determined in part by the relative exchange rate kb/kc. Because c* is a monotone increasing function of kb/kc, domain [Ca2þ] fluctuations can, formally, be made as large as possible by choosing kb [ kc and optimal bN T (Eq. 42). However, the molecular weight of Ca2þ-bound buffer is greater than Ca2þ and, consequently, one expects Ca2þ-bound buffer to diffuse and exchange with the bulk more slowly than free Ca2þ. This physical consideration implies that kb % kc and, after imposing this constraint, the physiologically realistic parameter choice that maximizes domain [Ca2þ] fluctuations (kb z kc) leads to c* z 1/4(css –cN)/ (cN þ k) and thus rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0

cv

css cN ¼ 1þ :

0 cv 4ðcN þ kÞ 7. The buffer dissociation constant k influences buffermediated increase in [Ca2þ] fluctuations in a complex manner. For example, the variance of free [Ca2þ] fluctuations (s0c , Eqs. 37 and 38) depends on the relative (as opposed to absolute) concentrations of domain (css/k) and bulk (cN/k) Ca2þ and total buffer (bN T /k) via the dimensionless quantities bss and wss. This is because the buffering capacity can expressed as 2 wss ¼ (bN T /k)/(1 þ css/k) and, similarly, the second factor of c can be written as (css/k cN/k)/(1 þ cN/ k). For a given steady-state [Ca2þ], css, the total buffer concentration with maximal effect (bN T , Eq. 42) is a ushaped function of the buffer dissociation constant k (see Supporting Results and Fig. S3 in the Supporting Material).

Applicability of the fluctuating rapid buffer approximation In the spatial and deterministic formulation of the buffered diffusion of intracellular Ca2þ, the rapid buffer approximation is valid when the equilibration time for buffers is much smaller than the time constant for Ca2þ diffusion. In this minimal model of domain [Ca2þ] fluctuations, Ca2þ and buffer diffusion are represented as first-order reactions (rate constants kc and kb) that couple the domain with the bulk. The validity of the fluctuating RBA depends on buffer

kinetics being faster than exchange. Consequently, the physical domain volume influences the validity of this approximate result. The time constant for diffusion-mediated passive exchange between compartments can be characterized by the mean escape time for a Brownian particle with diffusion constant D leaving a domain of volume U via a small opening of radius a (36), t ¼

U : 4aD

(44)

As illustrated in Fig. 3 A, a cubical domain (U ¼ L3) with an opening whose radius scales with linear dimension (r ¼ a/L < 1) would yield rate constant k ¼ 1/t and thus ki ¼ 4rDi/L2, where i ˛{c, b} and Dc and Db are the diffusion coefficients for Ca2þ and buffer, respectively. Using parameters for the fast, mobile buffer BAPTA, Fig. 3 B plots the coefficient of variation of domain [Ca2þ] fluctuations (cv) and the buffer-mediated relative decrease in system size 0 (s0c /sss c , where sc ¼ css/U) as functions of physical domain volume U for different values of the dimensionless escape radius r (shaded lines). As expected, both cv and s0c /sss c decrease with system size U. Narrowing the dimensionless escape radius (decreasing r) leads to an increase in cv and a decrease in s0c /sss c for any physical domain volume U. Comparison of the solid and dotted lines in Fig. 3 B shows that the fluctuating RBA agrees with the full calculation when the physical domain volume U is sufficiently large and/or the escape opening is sufficiently narrow (small r). Both situations lead to exchange rates (kc and kb) that are small compared to rate of buffer equilibration (an increasing function of both kþ and k). [Ca2D] fluctuations in a physiological setting Neuronal dendritic spines are a good setting to illustrate the influence of Ca2þ buffer on domain [Ca2þ] fluctuations and explore the validity of the fluctuating RBA, in part because such spines are highly variable in shape and size. Although it is difficult to define an average spine geometry, most spines have a clearly defined neck (corresponding to the escape opening of radius a in the previous paragraph) and head (corresponding to the microdomain of volume U). Fig. 3 C illustrates three frequently observed spine shapes—thin, mushroom, and stubby—whose geometric parameters (a and U) have been experimentally quantified (41). Fig. 3 D shows that over a wide range of concentrations, the fast buffer BAPTA increases the coefficient of variation, compared with the absence of buffer (cv/c0v ), by as much as a factor of 2 (top). The largest increase occurs in the case of the mushroom-shaped dendritic spine, for which the ratio of the spine-neck opening cross-sectional area, and the spine-head volume, is smallest (a/U). Biophysical Journal 106(12) 2693–2709

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A

B

C

D

For small and large BAPTA concentrations, there is a negligible difference between the fluctuating RBA and the corresponding full calculation, even measured in relative terms (Fig. 3 D, bottom). When the BAPTA concentration induces comparatively large [Ca2þ] fluctuations, the fluctuating RBA overestimates the cv (depending on spine shape by as much as 50–150% in the worst case). When the steady-state domain [Ca2þ] (css) is decreased, the disequilibrium between domain Ca2þ and buffer is attenuated, and the relative deviation between the RBA and full calculation is smaller (not shown). The fluctuating RBA works best for the mushroomshaped dendritic spine, because a/U is smallest. Influence of buffers on the relaxation time for domain [Ca2D] fluctuations We have demonstrated that in the presence of Ca2þ influx (jin > 0), Ca2þ buffer may significantly increase the steady-state covariance of [Ca2þ] fluctuations, especially when buffers are mobile and rapid (Eq. 37). On the other hand, the fluctuating RBA shows that in the absence of Ca2þ influx (jin ¼ 0, css ¼ cN), fluctuation amplitude is not influenced by total buffer concentration pffiffiffiffiffiffiffiffiffi (cv ¼ c0v ¼ 1= css U as when bN T ¼ 0), a result that holds generally (e.g., when buffers are slow) in numerical simulations (not shown). This occurs because Gss and Hss scale in such a way that the steady-state covariance matrix Sss is not a function of bN T when jin ¼ 0 (Eq. 30). On the other hand, the time-dependent stochastic dynamics of the fluctuating concentrations (dc, db, dcb) and the relaxation of the covariance matrix S to steady state are a function of buffer propBiophysical Journal 106(12) 2693–2709

FIGURE 3 The geometric parameters of a Ca2þ domain determine the time constant for escape of Ca2þ and buffer and the range of applicability of the fluctuating rapid buffer approximation (RBA). (A) Diagram of restricted subspace with volume U, linear dimension L, and radius of domain escape a. (B) (Shaded lines) Coefficient of variation of domain [Ca2þ] fluctuations (cv, top) and buffermediated relative decrease in system size (s0c /sss c , bottom) as functions of physical domain volume U using four decades of dimensionless escape radii r ¼ a/L ¼ 104,.,101. (Dashed black lines) Fluctuating RBA (Eq. 39) well approximates the full calculation when r is small. css ¼ 1 mM. (C) Illustration of three frequent spine shapes adapted from Harris et al. (41). (D, top) Relative increase in cv in the presence of BAPTA, compared with the absence of buffer c0v , for different spine shapes. (D, bottom) Percent difference between cv as calculated using the fluctuating RBA, c0v and the full calculation, cv. css ¼ 10 mM. Parameters: cN ¼ 0.1 mM, Dc ¼ 0.2 mm2/ms. BAPTA (72): Db ¼ 0.1 mm2/ms, kþ ¼ 0.6 mM1 ms1 , k ¼ 0.2 mM, and bN T ¼ 100 mM. Spine geometries (41): thin (a ¼ 0.05 mm, U ¼ 4 1017 L), mushroom (a ¼ 0.1 mm, U ¼ 2.9 1016 L), and stubby (a ¼ 0.16 mm, U ¼ 3 1017 L).

erties in both the absence or presence of Ca2þ influx (Eq. 27). For example, the relaxation rates governing the dynamics of Eq. 27 are given by the pairwise sums of the eigenvalues of Hss (by properties of Kronecker sums and Eq. 31 (42)), and these are a function of bN T in both the presence and absence of Ca2þ influx. Fig. 4 A shows Monte Carlo simulations of domain [Ca2þ], c(t), fluctuating around the steady-state css that were generated by numerical integration of Eq. 21 using the EulerMaruyama method. In the absence of buffer, the fluctuations dc(t) ¼ c(t) – css are typically small and long-lasting (Fig. 4 Aa). Similar dynamics are observed in the presence of slow buffers (Fig. 4 Ab). Conversely, in the presence of rapid buffer, free [Ca2þ] fluctuations and subsequent relaxation to steady state occurs on a fast timescale (Fig. 4 Ac). One characterization of the influence of buffer on the time evolution of [Ca2þ] fluctuations is given by the 33 steadystate correlation function F ¼ (fi,j) with i,j ˛ {c,b,cb}, FðtÞh dðtÞdT ð0Þ ; where F(0) ¼ Sss. In the presence of buffer the correlation function for the domain [Ca2þ] fluctuation dc is given by fc,c(t), where N < t < N and FðtÞ ¼ expð jtjHss ÞSss ; f0c;c ðtÞ

(45) s0c expðkc jtjÞ

whereas in the absence of buffer ¼ (26). Fig. 4 B plots fc,c(t) using parameters that correspond to the three previously shown stochastic trajectories (Fig. 4 Aa–c). In the absence of buffer, fc,c(t) decays by 50% when t z 5 ms, consistent with the time constant for


A

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B

C

Ca2þ exchange (kc ¼ 0.2 ms1, solid line). In the presence of slow buffer, fc,c(t) is nearly identical (open circles), but in the presence of fast buffer the 50% decay occurs at a much shorter time (t z 0.1 ms, dashed line). The time evolution of the domain [Ca2þ] fluctuation dc(t) can also be characterized by its (colored) power spectrum, that is, the 33 matrix S(u) ¼ (si,j), 1 1 SðuÞ ¼ ðiuI þ Hss Þ Gss iuI þ HssT (46) that reduces to s0c;c ðuÞ ¼ 2kc css

. U u2 þ kc2

in the absence of buffer (26). Fig. 4 C shows the power spectrum sc,c(u) in the absence of buffer and in the presence of slow buffer (solid line, open circles). In both cases, the spectrum is low-pass, but for fast buffer, sc,c(u) drops off at higher frequency (smaller t, dashed line), consistent with the higher frequency fluctuations observed in Fig. 4 A. In addition to clarifying the frequency content of domain [Ca2þ] fluctuations, this power spectrum analysis illustrates how buffers influence fluctuations even in the absence of Ca2þ influx. Further analysis of the influence of buffers on the timescale of domain [Ca2þ] fluctuations is provided in Fig. S4 and Fig. S5.

FIGURE 4 Influence of buffer on the time evolution of [Ca2þ] fluctuations. (A) Monte Carlo simulation of domain [Ca2þ], c(t), for different buffer parameters. (B) Autocorrelation function (fc,c (t), Eq. 45) and (C) power spectrum (sc,c(u), Eq. 46) in the presence of slow (open circles) and fast (dashed line) buffer and in the absence of buffer (solid line). Autocorrelation function curves are normalized by the steady-state variance such that fc,c (0) ¼ 1. Parameters: css ¼ 0.1 mM, cN ¼ 0.1 mM, kc ¼ 0.2 ms1, and U ¼ 1017 L. Buffer parameters: 1 bN ms1, kþ ¼ 103 T mM, k ¼ 0.2 mM, kb ¼ 10 1 1 (slow), and 1 (fast) mM ms .

(16–18,20,47–49), as well as stochastic aspects of Ca2þinduced Ca2þ release, Ca2þ oscillations, and propagating Ca2þ waves (34,50–53). A subset of these studies utilized a Langevin formulation for the dynamics of intracellular Ca2þ channels (54–56). Models of Ca2þ-regulated Ca2þ channels and stochastic Ca2þ release usually incorporate buffering into the domain description and, more rarely, consider the fluctuations in free [Ca2þ] that result from the small volume of subcellular compartments (e.g., the cardiac dyadic subspace) (30,33– 35). To our knowledge, this is the first study that formulates Langevin equations for the buffered dynamics of intracellular Ca2þ and, subsequently, characterizes the influence of Ca2þ buffers on [Ca2þ] fluctuations that arise from the association and dissociation of Ca2þ and buffer. The analytical results pertaining to the fluctuating rapid buffer approximation (RBA) and the influence of buffers’ properties on fluctuation amplitude is, to our knowledge, entirely novel. Previous studies have, of course, utilized Langevin equations to simulate concentration fluctuations in other biochemical systems (30,33,57–60) and the stochastic gating of plasma membrane and intercellular ion channels (54,55,61–64), and used the fluctuation-dissipation theorem to characterize concentration fluctuations in biochemical reaction networks, e.g., in models of gene networks and Michaelis-Menten enzyme reactions (65–68). However, to our knowledge, this is the first mathematical analysis of domain [Ca2þ] fluctuations in the presence of Ca2þ buffers.

DISCUSSION Relation to prior work 2þ

Summary of findings 2þ

The influence of Ca buffers on Ca diffusion, oscillations, propagating Ca2þ waves, and spatially localized Ca2þ elevations has received considerable attention in recent years (for reviews focused on local signaling, see Berridge (2,43), Smith et al. (45), and Augustine et al. (46)). Previous studies have presented theoretical analysis of buffered Ca2þ diffusion and localized Ca2þ elevations

The fluctuating RBA derived above shows how the Ca2þ and buffer exchange rates (kc and kb), total buffer concentration 2þ (bN T ), and dissociation constant (k) influence [Ca ] fluctuations. Typically, fast buffer binding and high exchange 2 rates increase the size of [Ca2þ] fluctuations (sss c ¼ hdc i). The fluctuating RBA derived here shows that the relative increase in fluctuation size due to buffers, i.e., the ratio of Biophysical Journal 106(12) 2693–2709

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coefficients of variation (Eq. 40), is an increasing function of the steady-state domain [Ca2þ] (css) and thus an increasing function of Ca2þ influx rate (jin). For fixed steady-state domain [Ca2þ], css > cN, Fig. 2 and Eq. 40 show that the relative fluctuation size c0v /c0v is a biphasic N function of the total buffer concentration bN T . At low bT , the fluctuation amplitude approaches that observed in the absence of buffer (c0v z c0v , wss / 0 in Eq. 37). At high 0 0 bN T , the ratio cv /cv z 1 because increased buffering capacity attenuates the buffer-mediated increase in free [Ca2þ] fluctuations (bss / 0 in Eq. 37). The total buffer concentration that maximizes c0v /c0v is given by Eq. 42. For a fixed Ca2þ influx rate jin, css is a decreasing sigmoidal function of bN T that approaches cN (Eq. 20). In this case, sss c is a decreasing sigmoidal function that approaches cN/U. However, the buffer-mediated change in fluctuation size as measured by . pffiffiffiffiffiffiffiffiffiffiffiffi 0 cv c0v ¼ 1 þ cR1; remains a biphasic function of bN T . For fixed css and for fixed jin, Ca2þ buffers increase the amplitude of domain [Ca2þ] fluctuation. We explore the validity of the RBA for realistic buffer parameters and microdomain geometries and find the fluctuating RBA is a good approximation when buffer kinetics are rapid compared to the Ca2þ and buffer exchange rates. Exchange rates on domain volume depends on the presumed relationship between escape target size and linear dimension of the domain. However, the fluctuating RBA agrees with the full calculation whenever the physical domain volume is sufficiently large and/or the escape passage is sufficiently narrow, because both situations lead to exchange rate constants (kc and kb) that are small compared to the buffer equilibration rate (an increasing function of both kþ and k). As one would expect based on these considerations, the fluctuating RBA is a better approximation to the full equations for domain [Ca2þ] fluctuations for mushroom-shaped (as opposed to thin or stubby) dendritic spine geometries (Fig. 3 D). Our analysis shows that a buffer-mediated increase in intrinsic [Ca2þ] fluctuations requires a nonequilibrium steady state, that is, a gradient between domain and bulk Ca2þ (css > cN), which implies disequilibrium between Ca2þ and buffer within the domain. However, our primary result that buffers do not suppress intrinsic [Ca2þ] fluctuations—due to the buffer’s contribution to these fluctuations—is true in the absence, as well as the presence, of elevated domain [Ca2þ]. Although buffers do not influence the steady-state variance of domain [Ca2þ] fluctuations in the absence of Ca2þ influx, power spectrum analysis shows that buffers do alter the temporal dynamics of domain [Ca2þ] fluctuations. For fixed k ¼ k/kþ, a faster association rate constant (kþ) leads to higher frequency free [Ca2þ] fluctuations and shorter autocorrelation times. Biophysical Journal 106(12) 2693–2709

Weinberg and Smith

Buffers suppress extrinsic (not intrinsic) domain [Ca2D] fluctuations As discussed in the Introduction, the effective volume that arises in the deterministic equations for the buffered diffusion of intracellular Ca2þ is Ueff ¼ U bðcÞ; where Ueff R U because 0 < b(c) % 1 (24). Because the relative size of concentration fluctuations decreases as system size increases, one might expect that domain [Ca2þ] fluctuations would decrease as total buffer concentration increases. In marked contrast to this conjecture, the analysis of domain [Ca2þ] fluctuations presented here demonstrates that buffers increase the size of [Ca2þ] fluctuations (during periods of Ca2þ influx) or have no influence (when there is no influx). That is, buffers typically lead to domain [Ca2þ] fluctuations with variance sss c that is consistent with a domain that lacks buffers but has smaller physical volume. Some clarity with regard to this counterintuitive result can be obtained by revisiting the fast/slow system of stochastic ODEs for buffered Ca2þ dynamics (Eq. 34) under the restriction that Ca2þ-free or Ca2þ-bound buffer do not exchange with bulk (kb ¼ 0), _ ¼ ½kþ ðcss þbss Þþk þ kc dc þ ðkþ css þ k ÞdcT þ xss ðtÞ; dc c (47a) d_cT ¼ kc dc þ xss cT ðtÞ:

(47b)

Here the fluctuation is free [Ca2þ]. The value (dc) is the fast variable and dcT is the only slow variable because the domain total buffer concentration is a conserved quantity given by initial conditions (bT ¼ b(0) þ cb(0)) and does not fluctuate (dbT ¼ 0). Setting kb ¼ 0 in Eq. 37, we see that under this restriction c ¼ 0, and the fluctuating RBA result is identical to the case without buffers, .pffiffiffiffiffiffiffiffiffi css U; c0v ¼ c0v ¼ 1 because immobile buffers (kb ¼ 0) do not change the steadystate variance of domain [Ca2þ] fluctuations. Note that when kb ¼ 0, the differential fraction of free to total calcium (bss) appears in the quasistatic approximation, hdci zbss dcT and the slow equation, d_cT ¼ bss kc dcT þ xss cT ðtÞ;

(48)

precisely as one would expect. Using the fluctuation-dissipation equation (Eq. 30), we calculate the steady-state covariance of the total calcium concentration,


2 dcT ¼

gss 2kc css css c ¼ ; ¼ 2bss kc 2bss kc U bss U

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(49)

and gss r þ b2ss dc2T 2½kþ ðcss þ bss Þ þ k ¼ ð1 bss Þ

css css css þ bss ¼ ; U U U

ss gss r þ gin þ b2ss dc2T 2½kþ ðcss þ bss Þ þ k

ss ss css css jext css jext ¼ ð1 bss Þ þ bss ¼ ; 1þ 1þ bss U U 2kc css U 2kc css

hdc2 i ¼

as well as the covariances that involve the fast variable dc. In the RBA limit with kb ¼ 0, the latter are given by css (50a) hdc dcT i ¼ bss dc2T ¼ U

2 dc ¼

and

(50b)

(50c)

where for the second equality we have used bsswss ¼ 1 – bss and the definition of gss r (Eq. 23). Equation 49 shows that the covariance of the total [Ca2þ] fluctuation (hdc2T i) responds to the elementary processes of Ca2þ influx and exchange with the bulk gss c ¼ ½jin þ kc ðcss þ cN Þ U ¼ 2kc css U: The covariance of the free [Ca2þ] is also dependent on 2 2 influx and exchange (gss c ) through the term bss hdcT i. The 2 value hdc i is influenced by the elementary processes of Ca2þ and buffer association and dissociation gss r ¼ ½kþ css bss þ k cbss U; whereas hdc2T i is not. Increasing buffer concentration (bN T ) decreases bss and increases hdc2T i. The value bN also scales T 2 gss r in such a way that hdc i is unchanged. Stationary buffers do not influence the steady-state covariance of domain [Ca2þ] fluctuations (kb ¼ 0), because the intrinsic nature ss of the fluctuating force xss c (t) causes gr and bss to be related 2 in a manner that makes hdc i ¼ css/U invariant to the buffer parameter bN T and k. However, if the influx rate were modulated so that there was an extrinsic component to the fluctuations, jin ðtÞ ¼ jin0 þ jin1 ðtÞ where

0 jin0 ¼ kc ðcss cN Þ and jin1 ðtÞ jin1 ðt0 Þ ¼ gss in d t t ;

ss where gss in ¼ jext /U, then the fluctuating terms in both Eqs. 47a and 47b would have greater variance and Eqs. 49 and 50b become

ss ss 2 gss css jext c þ gin dcT ¼ ¼ 1þ 2bss kc bss U 2kc css

where we have used the fact that gss in /kþ / 0 in the RBA limit. From the last equality we conclude that the variance of the external drive gss in increases the steady-state variance of domain [Ca2þ] fluctuations. The location of the ss coefficient bss in the term ð1 þ bss jext =2kc css Þ shows that stationary buffers are able to suppress such extrinsic variability of Ca2þ influx/efflux rate, despite the fact that the buffers do not suppress intrinsic fluctuations (compare to Eq. 50b).

Analysis of the fluctuating RBA and intrinsic fluctuations Analysis of the fluctuating RBA also provides insight into intrinsic (buffer-driven) fluctuations when both Ca2þ and buffer exchange with bulk (kb s 0). The quasistatic approximation for the average value of the [Ca2þ] fluctuation in the fluctuating RBA (Eq. 35) shows that the free (dc) and total (dcT) calcium fluctuations are related by the factor bss as expected. Also, Eq. 36a shows that total buffer concentration bN T slows the relaxation of total calcium fluctuations (dcT) through the factor bss. However, the relaxation of total buffer (dbT) fluctuations are not slowed in this manner (Eq. 36b) and, importantly, the enss tries of the covariance matrix for the random terms xss c , x cT , ss and xbT , are all proportional to the total buffer concentration bN T (Eq. S18 and Eq. S20 in the Supporting Material). To understand the interplay between the effect of bN T on the random terms (fluctuation) and the relaxation rate (dissipation) in the case of mobile buffers (kb > 0), we analyze the Langevin domain model when the exchange rates for Ca2þ and buffer are identical (kc ¼ kb ¼ k). Under this restriction, the covariances of the slow variables are given by

css þ cbss ; U cbss þ cbN ; hdcT dbT i ¼ 2U 2 bN dbT ¼ T ; U dc2T

¼

(51)

where the css þ cbss value in the numerator of hdc2T i is the steady-state total calcium concentration in the domain. The variance of the fast variable hdc2i can be written in terms of the covariances of the slow variables, Biophysical Journal 106(12) 2693–2709

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h c 2 i ss dc ¼ bss wss þ bss dc2T 2nss hdcT dbT iþn2ss db2T ; U (52) and it can be shown that hdcdbTi is negative whereas hdcdcTi and hdc2i are positive. Upon substitution of Eq. 51, we have 2 bss dc ¼ wss css þ bss css þ cbss nss ðcbss þ cbN Þ U (53a) þ n2ss bN ; T ¼

1 ð1bss Þcss þ b2ss css þcbss nss ðcbss þ cbNÞþ n2ss bN : T U (53b)

In this case, the fluctuating RBA simplifies to css 0 ð1 þ cÞ; sc ¼ U wss css cN c ¼ : 2 k þ cN ð1 þ wss Þ

(54)

This analysis of the fluctuating RBA under the restriction kc ¼ kb shows that an increase in total buffer bN T increases the absolute value of the slow covariances (Eq. 51). These covariances (the terms within square brackets in Eq. 53) combine to create a net positive impact on hdc2i that is attenuated to some extent by a decrease in the bss that scales these terms. The bss/U value outside the curly brackets in Eq. 53a might be interpreted as an effective volume that attenuates all of these contributions to hdc2i, but this is misleading because bsswsscss ¼ (1 bss)css is an increasing function of bN T (compare to Eq. 53b). Furthermore, Eq. 54 shows that the variance of the free [Ca2þ] fluctuations in the presence of buffer bN T > 0 is never less than what would occur in the absence of buffer (bN T ¼ 0). All of the expressions in this section have counterparts when the fluctuating RBA is derived in full generality (kc s kb, see Appendix B in the Supporting Material). The derivation of the fluctuating RBA when kc ¼ kb is shown in Appendix C in the Supporting Material. Physiological implications Ca2þ is an important signaling molecule in many cell types. Intracellular Ca2þ regulates cellular responses through Ca2þ-regulated ion channels, Ca2þ-dependent kinases, and phosphotases, and key processes in excitable cells, including muscle contraction and neurotransmitter release (1). Many of these signaling pathways occur in small spatially restricted volumes (microdomains) and, consequently, fluctuations in [Ca2þ] may be physiologically significant. Our results are broadly applicable to subcellular Ca2þ domains, including dendritic spines, the dyadic cleft, Biophysical Journal 106(12) 2693–2709

and many other situations where localized Ca2þ elevations occur in spatially restricted subcellular compartments, such as the primary cilium, mitochondria-ER junctions, and endoplasmic reticulum/plasma membrane (ER/PM) junctional regions (Fig. 1). As a concrete example of an application of the theory, we found—using physiological values for dendritic spine shape—that the exogenous Ca2þ buffer BAPTA does not suppress intrinsic domain [Ca2þ] fluctuations, and may increase the coefficient of variation (cv) of domain [Ca2þ] fluctuations by a factor of 2 or more (Fig. 3 D). Spatially localized subcellular dynamics can greatly alter whole-cell or even tissue-level physiological processes. For example, in the cardiac dyadic subspace, the regenerative (i.e., positive feedback) nature of Ca2þ-induced Ca2þrelease creates a subcellular environment where the spontaneous opening of only a few Ca2þ channels can lead to opening of nearly all of the channels in a dyad (essentially an all-or-none response) (69). Ca2þ release that spontaneously initiates within one dyad may activate neighboring dyads resulting in propagating Ca2þ waves. In pathophysiological conditions this may lead to ectopic heart beats that are potentially arrhythmogenic. This is one of many examples where the spatially localized stochastic dynamics of a few Ca2þ channels may influence the global Ca2þ response and, for this reason, the regulation of Ca2þ channels by buffer-modulated [Ca2þ] fluctuations may be physiologically significant. In Weinberg and Smith (35), we developed and analyzed a minimal model of a Ca2þ microdomain that included a stochastically gating Ca2þ-regulated Ca2þ channel and accounted for the small number of Ca2þ ions present in the domain at any given time. This prior work demonstrated that [Ca2þ] fluctuations due to small domain volumes could alter the open probability of Ca2þ channels, typically reducing the open probability compared with larger volumes. The fluctuating RBA derived here shows that Ca2þ buffers typically enhance domain [Ca2þ] fluctuations and, when combined with previous work, this suggests that buffers may influence the stochastic dynamics of Ca2þ-regulated Ca2þ channels (the frequency and duration of channel openings) and potentially influence the stochastic properties of local Ca2þ release events (sparks and puffs). Our analysis of the fluctuating RBA raises the intriguing question of whether or not domain [Ca2þ] fluctuations are physiologically significant aspects of local (and perhaps global) Ca2þ signaling. This topic requires further study, in part because our mathematical analysis indicates that Ca2þ chelators and indictor dyes enhance intrinsic domain [Ca2þ] fluctuations. The physiological relevance of domain [Ca2þ] fluctuations on the dynamics of signaling complexes will presumably depend on the timescale of Ca2þ influx, the geometry of the microdomain, and the kinetics of downstream Ca2þ-dependent signaling.


Limitations of this analysis 2þ

We have shown that [Ca ] fluctuations in spatially restricted subcellular compartments cannot be suppressed by increasing the total concentration of Ca2þ buffer (bN T ). Of course, this conclusion that Ca2þ buffers typically enhance domain [Ca2þ] fluctuations pertains only to buffers whose dynamics are well represented by the bimolecular association reaction that was our starting point (Eq. 1). It is possible that Ca2þ buffers with more complicated and realistic Ca2þ-binding kinetics (e.g., cooperative binding) may suppress free [Ca2þ] fluctuations. This question could be addressed rigorously using chemical reaction network theory. Although we consider Ca2þ and buffer transport via exchange between domain and bulk, this work does not use continuum modeling (partial differential equations) to explicitly represent spatial aspects of the buffered diffusion of intracellular Ca2þ. Spatial correlations in fluctuations are of interest, but beyond the scope of this study, in part because the mathematical description of diffusion-mediated concentration fluctuations is less accessible than the analysis of stochastic differential equations presented here. This article has focused on intrinsic Ca2þ and buffer concentration fluctuations around nonequilibrium steady states in the presence of Ca2þ influx (jin > 0) and equilibrium steady states in the absence of Ca2þ influx (jin ¼ 0). Our analytical work is relevant and easily applied whenever a separation of timescales exists between the concentration fluctuations per se (that arise from finite size effects) and temporal variations in Ca2þ influx rate. For example, during the cardiac cycle, triggered release of junctional sarcoplasmic reticulum (SR) Ca2þ elevates dyadic subspace [Ca2þ] for durations of ~300 ms, and the low [Ca2þ] of the diastolic interval persists for ~500 ms. Because domain [Ca2þ] fluctuations occur on a faster timescale (1–10 ms), it is straightforward to apply our theory to the cardiac dyad under the assumption that domain concentrations (expected values) are in quasistatic equilibrium with the time-varying influx rate, i.e., css(t) ¼ css(jin(t)) through Eq. 20 and similarly for bss(t) and cbss(t). Even when no such timescale separation exists, domain [Ca2þ] fluctuations in the context of time-varying Ca2þ influx can be calculated by numerical integration of Eq. 27, with Hss(t) and Gss(t) evaluated at the time-varying expected values of the domain Ca2þ and buffer concentrations (26). Such quasistatic and nonstationary calculations agree with numerical simulations of domain [Ca2þ] fluctuations during triggered release (see Fig. S6). Perhaps most importantly, such calculations confirm that buffers increase intrinsic domain [Ca2þ] fluctuations during systole, and buffers do not suppress [Ca2þ] fluctuations during diastole, just as one would expect from the nonequilibrium steady-state analysis that has been our focus. In other physiological settings, domain [Ca2þ] is elevated for longer or shorter durations. In pancreatic b-cells, for

2707

example, insulin secretion is regulated by minute-long duration [Ca2þ] elevations in nuclear (8) and subplasma membrane (70) Ca2þ microdomains. In pituitary cells, minute-long spontaneous [Ca2þ] oscillations have been observed in mitochondrial microdomains (71). In such cases, it is straightforward to apply our nonequilibrium steady-state analysis (e.g., the fluctuating RBA). This article’s conclusions are also directly applicable to the case of pulsatile Ca2þ influx with characteristic time shorter than the domain equilibration time (for buffers and escape). When the timescale for pulsatile influx is comparable to the domain time constant, quantitative studies will require simulation of the full system of stochastic ODEs. In numerical simulation of a stochastically gated influx with mean open time of 1 ms and mean closed time of 9 ms, we find that buffers do not decrease intrinsic domain [Ca2þ] fluctuations (see Fig. S7), consistent with our analysis of constant and slowly changing Ca2þ influx. It would be interesting to understand the impact of intrinsic fluctuations (due to the finite number of Ca2þ ions in a domain) on Ca2þ-mediated effectors, including Ca2þactivated or inactivated channels, initiation of Ca2þ sparks, and so on. In complementary projects we have made some progress in that direction (35). It is important to understand that even when [Ca2þ] fluctuations are much faster than the timescale of a downstream Ca2þ-mediated process (e.g., opening of a Ca2þ-activated channel), these fluctuations may (34) or may not (35) average-out over the slower timescale. Further studies could address such issues by including downstream signaling events in model formulations, but this article’s primary conclusion—that Ca2þ buffers do not suppress intrinsic [Ca2þ] fluctuations—is an important first step in our developing understanding of the physiological significance of domain [Ca2þ] fluctuations. SUPPORTING MATERIAL Supporting Results, with Validation of the Linear Noise Approximation, [Ca2þ] Fluctuations in the Presence of Multiple Buffers, Influence of the Buffer Dissociation Constant k on [Ca2þ] Fluctuations, Timescale of Domain [Ca2þ] Fluctuations, [Ca2þ] Fluctuations during a Time-Varying Ca2þ Influx, Appendix A: The Chemical Langevin Equation and Linear Noise Approximation, Appendix B: Derivation of the Rapid Buffer Approximation, Appendix C: Analysis of the Fluctuating RBA and Intrinsic Fluctuations, and seven figures are available at http://www.biophysj.org/ biophysj/supplemental/S0006-3495(14)00464-0. The authors thank the referees for helpful and constructive comments. The work was supported by National Science Foundation grant No. DMS 1121606, the Biomathematics Initiative at The College of William & Mary, and a 2013 Plumeri Award for Faculty Excellence to G.D.S.

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Supporting material for “The influence of Ca2+ bu↵ers on free [Ca2+] fluctuations and the e↵ective volume of Ca2+ microdomains”

Seth H. Weinberg and Gregory D. Smith

Department of Applied Science, The College of William & Mary, Williamsburg, VA 23187, USA

31

SUPPORTING RESULTS Validation of the linear noise approximation To validate our analytical analysis, we performed Monte Carlo simulation of Ca2+ and bu↵er dynamics in a microdomain using Gillespie’s stochastic simulation algorithm, an exact simulation algorithm that accounts for the small system size (1). We found close agreement between the coefficient of variation of the domain [Ca2+ ] fluctuations calculated from the linear noise approximation and the Monte Carlo simulations (Figure S1), in particular reproducing the bell-shaped dependence on total bu↵er concentration b1 T . kb = 10−2 2 ms −1 k = 10 b

c

k = 10

ccss ==1 1µMµM

0.6

c v v

kb = 10−11 ms −1

1

ms

b

1

ms

ss

0.4 0.2 0 c = 10 µM ss= 10 µM css

c

c v v

0.4 0.3 0.2 0.1 0

−2

10

0

10

2

10 bT (µM) bT (µM)

4

10

6

−2

10

10

0

10

2

4

10 10 bT (µM) bT (µM)

6

10

Figure S1: Comparison of the relative magnitude of domain [Ca2+ ] fluctuations, the coefficient of variation cv , calculated using the linear noise approximation (solid gray lines), computed numerically using Eq. 32, and the exact value, determined from the ensemble average of 1000 Gillespie-type stochastic simulations (black circles). Parameters as in Figure 2.

[Ca2+ ] fluctuations in the presence of multiple bu↵ers The fluctuating RBA (Eq. 37) is readily generalizable to multiple bu↵ers. For N rapid bu↵ers with parameters bnT , n , and kbn for n 2 {1, . . . , N }, the variance of the domain [Ca2+ ] fluctuations is given by

0 c

=

css · (1 + ) ⌦

=

ss

·

N n n X wss kb c + n n=1 1

kc +

N X n=1

32

n n wss kb

· (css

c1 )

(S1)

0

20

bb1T1 (µM) (µM)

40

t

60

80

100

0.54 0.52

Buffer 1

c

v

0.5 0.48

Weighted Average

0.46 0.44

Both

0.42 0.4 100

80

Buffer 2

b22 (µM)

60

bTt (µM)

40

20

0

Figure S2: Combined influence of two rapid bu↵ers on the coefficient of variation of domain [Ca2+ ] fluctuations (cv ) as the total concentration of bu↵er 1 (b1T ) and bu↵er 2 (b2T ) are varied with constant total amount of bu↵er (b1T + b2T = 100 µM). The fluctuating RBA (solid black line, Eq. S1) agrees with numerical solution of the full model (Eq. 30, open circles). The cv in the presence of either bu↵er 1 or 2 in isolation (dashed lines) and a weighted average based on total bu↵er concentration (solid gray line) are also shown. Parameters: css = 1 µM, c1 = 0.1 µM, kc = 0.2 ms 1 , ⌦ = 10 17 . Bu↵er 1: k+ = 102 µM 1 ms 1 ,  = 0.1 µM, kb = 10 1 ms 1 . Bu↵er 2: k+ = 102 µM 1 ms 1 ,  = 1 µM, kb = 10 3 ms 1 . where

ss

n and wss are

ss

=

1+

N X n=1

n wss

!

1

and

n wss

bnT n = , (css + n )2

(S2)

and for readability we have dropped the superscripted 1 in the notation for total bu↵er ,n concentration (bnT ⌘ b1 ). T Figure S2 shows the cv for domain [Ca2+ ] fluctuations in the presence of two rapid bu↵ers with di↵erent dissociation constants (2 > 1 ) and exchange rates (kb2 > kb1 ). Moving left to right, the total concentration of bu↵er 1 (b1T ) increases and the total concentration of bu↵er 2 (b2T ) decreases, while the sum is held constant (b1T + b2T = 100 µM). The fluctuating RBA (Eq. S1, solid line) is in agreement with full calculations based on the appropriate generalization (Eq. 30, open circles), that is, the numerical solution of a Lyapunov equation composed of 5 ⇥ 5 known matrices Hss and ss , and the symmetric 5 ⇥ 5 matrix ⌃ss , the steady-state covariance matrix for the fluctuating concentrations in the presence of two bu↵ers, representing 10 unknown covariances. For comparison, Figure S2 shows the cv obtained when bu↵er 1 or, alternatively, bu↵er 2 is included (dashed lines), as well as a weighted average of these values utilizing the total bu↵er concentrations (biT /(b1T + b2T )). As expected, this naive weighted average does not agree with the full calculation (circles), because the correct weighting is as implied by the fluctuating RBA (solid line, Eq. S1).

33

10 (µM)

10

bT

3

10

100 µM

2

10 µM

10

1

cv /c0v

B

4

4 3 2 1 0

bT = 10 µM

cv /c0v

A

4 3 2 1 0

bT = 100 µM

css = 1 µM

0

10

10

cv /c0v

cv /c0v

15

5 0 −2 −1 10 10

0

10

1

10 (µM)

2

10

3

10

bT = 1000 µM 4 3 2 1 0 −2 −1 0 1 2 10 10 10 10 10 (µM)

3

10

Figure S3: Dependence of [Ca2+ ] fluctuations on dissociation constant . (A, top) The total 2+ bu↵er concentration that maximizes [Ca2+ ] fluctuations (b1 T ⇤ ) for a fixed steady-state [Ca ], 0 css (Eq. 42). (A, bottom) The enhancement of [Ca2+ ] fluctuations, c0⇤ v /cv , that occurs with optimal total bu↵er concentration (b1 T ⇤ ) as a function of the bu↵er dissociation constant  (Eq. 43). (B) The enhancement of [Ca2+ ] fluctuations, c0v /c0v , for di↵erent values of css and 1 b1 T (Eq. 40) plotted as functions of . Parameters: c1 = 0.1 µM, kc = kb = 0.2 ms .

Influence of the bu↵er dissociation constant  on [Ca2+ ] fluctuations The bu↵er dissociation constant  influences bu↵er-mediated increase in [Ca2+ ] fluctuations in a complex manner. For example, the variance of free [Ca2+ ] fluctuations ( c0 , Eqs. 37 and 38) depends on the relative (as opposed to absolute) concentrations of domain (css /) and bulk (c1 /) Ca2+ and total bu↵er (b1 T /) via the dimensionless quantities ss and wss , 2 because the later can expressed as wss = (b1 T /)/(1+css /) and, similarly, the second factor of can be written as (css / c1 /)/(1 + c1 /). However, an increase or decrease in  does not simply shift the bell-shaped curves describing cv vs. b1 T (as in Figure 2) to the left or right. For a given steady-state [Ca2+ ], css , the total bu↵er concentration with maximal e↵ect (b1 T ⇤ , Eq. 42) is a u-shaped function of the bu↵er dissociation constant  (Figure S3A, top). The value of  that minimizes this optimal total bu↵er concentration is found as the  that zeros @b1 T ⇤ /@ and is given by  = css . This suggests that as css increases, weaker affinity (i.e., larger ) bu↵ers are more e↵ective at enhancing domain [Ca2+ ] fluctuations. However, for a given value of css , a bu↵er with dissociation constant  = css does not necessarily enhance domain [Ca2+ ] fluctuations to the greatest extent possible, as the fluctuation enhancement 0 is a decreasing function of  (given by c0⇤ v /cv , Eq. 43, Figure S3A, bottom). 34

In order to determine an optimal dissociation constant ⇤ that maximizes domain [Ca2+ ] fluctuations, it is instructive to plot the fluctuation enhancement, c0v /c0v (Eq. 40) for a given fixed css and total bu↵er concentration b1 T (Figure S3B). These curves are bell-shaped functions of : for both small  (high affinity bu↵er) and large  (low affinity bu↵er), steady-state bu↵ering capacity wss ! 0 (Eq. 38) and, consequently, c0v /c0v ⇡ 1. For a given css and b1 T , the optimal dissociation constant ⇤ (corresponding to the peak of the bell-shaped curve) is found as the  that zeros @ /@. This optimal dissociation constant is the positive solution of the quartic equation, 4⇤ + a3⇤ + b2⇤ + c⇤ + d = 0, (S3) where a = 12 (c1 + 4css ) 1 b = 12 [css (b1 T + 2css + c1 ) + bT (css c = 12 css c1 (b1 css ) T d = 12 c1 c3ss .

c1 )]

Figure S3 shows that for fixed b1 T , ⇤ is an increasing function of css , that is, when css is large, fluctuations are enhanced to the largest extent by low affinity bu↵ers. For a given 2+ css , ⇤ is a decreasing function of b1 T . Figure S3 shows that domain [Ca ] fluctuations may be enhanced several-fold for physiological values of b1 T and , especially for larger values of css = 10 and 100 µM.

Time scale of domain [Ca2+ ] fluctuations Figure S4 characterizes the time scale of [Ca2+ ] fluctuations by plotting the domain [Ca2+ ] auto-correlation function, denoted by c,c (⌧ ) in Eq. 45, for a range of model parameters. In brief, our calculations suggest that in the presence of rapid, high concentration bu↵er, the time scale of domain [Ca2+ ] fluctuations may range over several orders of magnitude (0.1 µs to 10 ms), depending on the steady-state domain [Ca2+ ]. For low domain [Ca2+ ] (css = 0.1 µM) and total bu↵er (b1 T ) concentrations (top row), a semi-logarithmic plot of the autocorrelation function c,c is a decreasing sigmoid with 50% decay ⇠1 ms, as in the absence of bu↵er (dashed black lines). As b1 T increases, the characteristic time scale of [Ca2+ ] fluctuations decreases ( c,c shifts to the left). For example, 4 for a slow bu↵er at total concentration of b1 T = 10 µM (blue), the time scale is ⇠0.1 ms (Figure S4A, top panel). For a fast bu↵er, the influence of b1 T is more pronounced, for 4 4 example, when b1 = 10 µM (blue), the time scale is ⇠ 10 ms (Figure S4B, top panel). T 2+ For large steady-state domain [Ca ], the autocorrelation function c,c is a decreasing (double) sigmoid with two “knees,” indicating that [Ca2+ ] fluctuations are characterized by 4 two distinct time scales. For example, for a fast bu↵er, css = 10 µM, and b1 T = 10 µM (blue), 3 the two time scales are ⇠ 10 and 10 ms (Figure S4B, third panel) that may correspond to the bu↵er kinetics and exchange rates, respectively (see Figure S5). As css increases further, the fast time scale (small ⌧ ) becomes less prominent and the slow time scale (large ⌧ ) dominates. These e↵ects are more pronounced when bu↵ers are rapid (right column). Increasing the bu↵er exchange rate kb has minimal influence on the shape of c,c , but does increase the values of b1 T that lead to dominance of the show time scale (not shown). 35

A

B

Slow buffer k+ = 10

3

µM

1

ms

1

( )

ms

1

css = 1 µM

1

( )

1

Increasing bT

0.5 0

0.5 0

css = 10 µM

1

( )

k+ = 1 µM

css = 0.1 µM

1

0.5 0

css = 100 µM

1

( )

Fast buffer

0.5 0 −6 −5 −4 −3 −2 −1 0 1 2 10 10 10 10 10 10 10 10 10 (ms)

−6

−5

−4

−3

−2

−1

10 10 10 10 10 10 (ms)

0

1

Figure S4: Dependence of the auto-correlation function c,c (⌧ ) on steady-state domain [Ca2+ ] (css ) and total bu↵er concentration (b1 T ) (Eq. 45). Parameters: c1 = 0.1 µM,  = 0.2 µM, ⌦ 17 1 2 2 = 10 L, kc = 0.2 ms , kb = 10 ms 1 , b1 (magenta), T = 0 (no bu↵er, dashed black), 10 2 4 1 (green), 10 (red) and 10 (blue) µM. The physical processes that correspond with the observed knees in the autocorrelation functions plotted in Figure S4 can be identified to some extent but not isolated. The relaxation time of fluctuations is governed by the eigenvalues of the steady-state Jacobian matrix Hss . In fact, the relevant rates are (up to a sign change) given by the six pairwise sums of the eigenvalues of the Jacobian matrix Hss (three eigenvalues with negative real parts). To see this, note that that Eq. 27, .

T ⌃ = Hss ⌃ + ⌃Hss +

ss ,

can be written as

d vec(⌃) = (Hss Hss )vec(⌃) + vec( ss ), (S4) dt where is a Kronecker sum and the vec operation creates a column vector from a matrix by stacking its column vectors. It is well-known (2) that if A 2 Rn⇥n has eigenvalues i for i = 1, . . . , n and B 2 Rm⇥m has eigenvalues µj = 1, . . . , m, then A B has mn eigenvalues, 1

+ µ1 , . . . ,

1

+ µm ,

Denoting the three eigenvalues of Hss as 2 1,

1

+

2,

2

+ µ1 , . . . ,

1, 1

+

2

and 3,

36

3,

2 2,

2

+ µm , . . . ,

n

+ µm .

it follows that Hss Hss has eigenvalues 2

+

3,

2 3.

2

10 10 10

(S5)

A

kb = 10

2

ms

1

kb = 10

css = 1 µM

6

1

ms

1

k+ = 1 µM 1 ms 1 k+ = 10 3 µM 1 ms

−11 −1 −1 −1 −1 1 −1 −1 −1 −1 1 | | (ms (ms(ms | (ms (ms | )| (ms (ms ) )(ms ) (ms ) ) |(ms ) ) ) (ms(ms ) ))

10

63

10 10 630 10 10 10

B

C

3−3

0 10 10 10 0−3 6 10 10 10

css = 10 µM

−3

63 10 10 10 63 0 10 10 10 3

0−3 10 10 10 0−3 6 10 10 10 −3 63 10 10 10

css = 100 µM

6

30 10 10 10 30−3 10 10 10

−2

0−3 10 10 10 −2 −3 10 10 −2 10

0

10

0

10 0 10

2

10

2

10 2 10

4

10

4

10 4 10

6

10

6

10 6 10

−2

10

−2

10 −2 10

bT (µM)

0

10

0

10 0 10

2

10

2

10 2 10

bT (µM)

10

4

4

10 4 10

6

10

6

10 6 10

Figure S5: [Ca2+ ] fluctuation relaxation rates (| i |, i 2 {1, 2, 3}) plotted as a function of bu↵er parameters (cf. Eq. S5 and nearby text). Horizontal dotted black line indicates rate in the absence of bu↵er (2kc , Eq. 15). Parameters: css = 1 (A), 10 (B), and 100 (C) µM, c1 = 0.1 µM,  = 0.2 µM, ⌦ = 10 17 L, kc = 0.2 ms 1 , k+ = 10 3 (solid blue) and 1 (dashed red) µM 1 ms 1 . Figure S5 shows a numerical calculation of the relaxation rates derived from the eigenvalues of Hss (| i |, i 2 {1, 2, 3}) and the dependence of these rates on Ca2+ bu↵er parameters. The rates range over several orders of magnitude (10 3 to 104 ms 1 ). As total bu↵er concentration b1 T increases, the fastest rate increases, the slowest rate remains constant, and the intermediate rate decreases (asymptotically approaching the slow rate). Increasing the rate of exchange of bu↵er between domain and bulk (kb ) increases the slow and intermediate rates, but does not change the fast rate (compare left and right panels). Increasing the bu↵er kinetics (k+ , k ) increases the fast rate, but does not influence the slow and intermediate rate (compare solid blue and dashed red lines). The fastest rate is an increasing function of the steady-state domain [Ca2+ ] (css ) (cf. A, B and C).

[Ca2+ ] fluctuations during a time-varying Ca2+ influx In the main text, we demonstrate that in the presence of a constant Ca2+ influx rate, bu↵ers enhance the size of domain [Ca2+ ] fluctuations around the non-equilibrium steady-state. Figure S6 illustrates an example of a time-varying Ca2+ influx rate, during which bu↵ers do not suppress and may enhance domain [Ca2+ ] fluctuations. In Figure S6A, Ca2+ influx rate is characteristic of triggered SR Ca2+ release, i.e., a Ca2+ spark (top panel, inset). Numerical simulations demonstrate that bu↵ers increase intrinsic domain [Ca2+ ] fluctuations during 37

1

B

Numerical

Buffer

cv

50 ms

jin

c(t) (µM)

A

Quasistatic Non-stationary

Theoretical

No Buffer

cv

cv

Buffer No Buffer

Quasistatic Non-stationary

Figure S6: [Ca2+ ] fluctuations during a time-varying Ca2+ influx. (A, top) Monte Carlo simulations of domain [Ca2+ ], c(t), in the presence of bu↵er and a time-varying Ca2+ influx, jin (t), generated by numerical integration of Eq. 21 using the Euler-Maruyama method. The time-varying jin (t) is characteristic of triggered SR Ca2+ release and given by an ↵-function, jin (t) / (exp( t/⌧2 ) exp( t/⌧1 )) (see inset). (A, bottom) The ensemble domain [Ca2+ ] coefficient of variation, cv , calculated from 1000 Monte Carlo simulations, in the presence (solid red) and absence (solid green) of bu↵er, is plotted throughout the time-varying Ca2+ influx. Theoretical calculation of cv for non-stationary domain concentrations, determined from integration of Eq. 27 in the presence (dashed black) and absence (dashed blue) agree closely with the numerical simulations. (B) Theoretical calculation of cv , for non-stationary (solid) and quasistatic (dashed) domain concentrations in the presence (top) and absence (bottom) of bu↵er. See text for details of the calculations. Note that the dashed black and dashed blue traces in (A) and (B) are the same. Parameters: b1 T = 50 µM, c1 = 0.1 µM, 1 1 1 1  = 0.2 µM, kc = 0.2 ms , kb = 0.1 ms , k+ = 1 µM ms , ⌦ = 10 17 L, ⌧1 = 10 ms, ⌧2 = 40 ms. systole, and bu↵ers do not suppress [Ca2+ ] fluctuations during diastole (A, bottom, solid red and green). A calculation of the domain [Ca2+ ] fluctuations during a time-varying Ca2+ influx rate jin (t) can be obtained by numerical integration of Eq. 27 (3), .

⌃ = H⌃ + ⌃H T + , where the Jacobian matrix H and two-time covariance matrix are evaluated at the timevarying expected value of the domain Ca2+ and bu↵er concentrations, i.e., c(t), b(t), and 38

c(t) (µM)

10 ms

jin

30

15

0 0

10

20

30

40

50

40

50

cv

2.5 2

Buffer No Buffer

1.5 1 0

10

20 30 time (ms)

Figure S7: [Ca2+ ] fluctuations during a stochastically-gated Ca2+ influx. (top) Monte Carlo simulations of domain [Ca2+ ], c(t), in the presence of bu↵er and a time-varying Ca2+ influx, jin (t), generated by numerical integration of Eq. 21 using the Euler-Maruyama method. The time-varying jin (t) is stochastically gated with a mean open time of 1 ms and mean closed time of 9 ms (see inset). (bottom) The ensemble domain [Ca2+ ] coefficient of variation, cv , calculated from 1000 Monte Carlo simulations, in the presence (red) and absence (black) of bu↵er, is plotted throughout the time-varying Ca2+ influx. Parameters: b1 T = 50 µM, 1 1 1 1 c1 = 0.1 µM,  = 0.2 µM, kc = 0.2 ms , kb = 0.1 ms , k+ = 1 µM ms , ⌦ = 10 17 L. cb(t), found by numerical integration of Eq. 19. Non-stationary calculations agree closely with numerical simulations in the presence (Figure S6A, bottom, dashed black) and absence (dashed blue) of bu↵er. Alternatively, we may assume that the domain concentrations are in quasistatic equilibrium with the time-varying influx rate, i.e., css (t) = css (jin (t)) and similarly for bss (t) and cbss (t) as determined by Eq. 20, and estimate ⌃(t) using the the fluctuation-dissipation theorem (Eq. 30, the steady-state of Eq. 27) and the quasistatic values of ss (t) and Hss (t). Figure S6B shows that this approximation deviates only slightly from the aforementioned non-stationary calculation, and thus both theoretical methods yield results in close agreement with stochastic simulation. We also show Monte Carlo simulations of [Ca2+ ] during a stochastically gated influx (Figure S7). The stochastically gated influx has mean open time of 1 ms and mean closed time of 9 ms. Throughout the time course, [Ca2+ ] fluctuations in the presence of bu↵er (red) are larger compared with in the absence of bu↵er (black).

39

SUPPORTING APPENDICES Appendix A: The chemical Langevin equation and linear noise approximation Consider a well-stirred compartment with I chemical species undergoing elementary reactions indexed by q. The q-th elementary reaction can be characterized by (a) the forward and backward reaction rates per unit volume, ⌫q+ (⇢) and ⌫q (⇢), that may depend on species concentrations ⇢(t) = (⇢1 , ⇢2 , · · · , ⇢I )T , and (b) the change in the number of molecules when the q-th reaction occurs, denoted by the column vector ! q = (!q 1 , !q 2 , · · · , !q I )T , a di↵erence of stoichiometric coefficients (products minus reactants). If N (t) = (N1 , N2 , · · · , NI )T (a random vector) is the copy number of each species at time t, the probability distribution W (n, t) = Pr{N (t) = n} solves the chemical master equation: X d W (n, t) = Vq+ (n ! q )W (n ! q , t) + Vq (n + ! q )W (n + ! q , t) dt q X [Vq+ (n) + Vq (n)]W (n, t),

(S6)

q

where Vq± (n) = ⌦ ⌫q± (n/⌦) are the forward and backward reaction rates for the q-th elementary reaction. The chemical master equation is a system of ODEs, one for each possible state. Often the high dimensionality of the chemical master equation makes its use prohibitive. Several approaches have been utilized to show that for sufficiently large system size and reaction rates, the stochastic dynamics of the species concentrations ⇢i are wellapproximated by solutions of the chemical Langevin equation (4-6), X . ⇢i = !q i (⌫q+ ⌫q ) + ⇠i (t) (S7) q

where the column vector of fluctuating forces, ⇠ = (⇠1 , ⇠2 , . . . , ⇠I )T , has mean zero h⇠i (t)i = 0,

i = 1, 2, . . . , I

(S8)

and two-time covariance h⇠⇠ (t) ⇠ T (t0 )i = (⇢) (t

t0 ).

where the general form of the two-time covariance matrix, = ( X 1 !q i (⌫q+ + ⌫q ) !q j , ij = ⌦

(S9) ij ),

is given by (3) (S10)

q

that is, the covariance ij is the sum of forward and backward reaction rates, multiplied by the change in copy number of species i and j, summed over each reaction q and scaled by the inverse of domain volume ⌦. Assuming a stable steady state ⇢ss = nss /⌦, the linear noise approximation to Eq. S7 is .

⇢ = H ss ⇢ + ⇠iss (t) 40

(S11)

where the fluctuation ⇢ = ⇢(t) ⇢ss , the matrix H ss = (hss ij ) is the Jacobian of reaction terms in the chemical Langevin equation (Eq. S7) evaluated at steady state, that is, ⇥ ⇤ P @ q !q i ⌫q+ (⇢) ⌫q (⇢) ss hij = (S12) @⇢j ss ⇢=⇢

and ⇠iss (t) has the same form as ⇠i (t) in Eq. S7 with evaluated at ⇢ss . The approximations relating the chemical master equation, chemical Langevin equation, and linear noise approximation are subtle (3-7). The validity of these approximations depends on multiple factors, including reaction rates (propensities) and the number of molecules n = ⌦c, where c is a characteristic concentration for the species being simulated. Our use of the linear noise approximation was validated through comparison of the analytical and symbolic results (Eq. S11) to Gillespie-type stochastic simulation (Figure S1) that is an exact sampling of the steady-state probability distribution solving the chemical master equation (Eq. S6). For the analysis of the fluctuations produced by Eq. S11, we consider an ensemble of identically prepared systems with ⇢(0) = ⇢ss or, equivalently, ⇢(0) = 0, and numerically or analytically calculate the I ⇥ I symmetric covariance matrix ⌃(t) = h ⇢(t) ⇢T (t)i that solves Eq. 27,

.

T ⌃ = Hss ⌃ + ⌃Hss +

ss ,

which follows from Eq. S11 and the definition of ⌃(t). We are primarily interested in the steady-state covariance matrix ⌃ss that solves the following Lyapunov equation, which relates ss and ⌃ss and is called the fluctuation-dissipation theorem, T Hss ⌃ss + ⌃ss Hss =

ss .

Appendix B: Derivation of the rapid bu↵er approximation In this Appendix we derive a general analytical expression for the variance of the free [Ca2+ ] fluctuations in the presence of rapid Ca2+ bu↵er (Eq. 37) using Eqs. 34 and 36 as our starting point, .

c =

.

cT = .

bT =

[k+ (css + bss ) + k + kc ] c + (k+ css + k ) cT ss

(kc + wss kb ) cT +

ss

(kc

kb ) ⌫ss bT + ⇠cssT

kb bT + ⇠bssT ,

k+ css bT + ⇠css (t) (S13a) (S13b) (S13c)

where c is the fast variable, cT and bT are slow variables, and we have written ⇠cT (t) = ⇠c (t)+⇠cb (t) and ⇠bT (t) = ⇠b (t)+⇠cb (t). Eq. S13b is derived using a quasistatic approximation for the average value of the [Ca2+ ] fluctuation (Eq. 35), h ci⇤ ⇡

ss

[ cT

⌫ss bT ] ,

(S14)

where ⌫ss = css /(css + ) and h·i⇤ indicates a time average (as opposed to an ensemble average). As a consequence, the reaction terms of Eq. S13b that involved the fast fluctuation 41

c have been replaced by a time average that is a function of the slow fluctuations cT and bT , that is, the slow SDEs (Eqs. S13b and S13c) are now expressed in terms of the slow fluctuations. The quasistatic approximation for c (Eq. S14) is obtained from Eq. S13a by dividing both sides by k+ ⇣ss to yield 1 . c= k+ ⇣ss

c+

ss

( cT

⌫ss

⇠css (t) bT ) + , k+ ⇣ss

where ⇣ss = css + bss +  + kc /k+ ,  = k /k+ , and we have used (css + )/⇣ss = rapid bu↵er limit (k+ , k ! 1 with  fixed) this expression becomes 0=

c+

ss

cT

ss ⌫ss

ss .

In the

bT ,

where ⇣ss ! css + bss +  and bss ! bT /(css + ) (Eq. 20). Because this is an outer solution for a fluctuating quantity, for this zeroth order approximation we write h ci⇤ ⇡

ss

( cT

⌫ss bT ) ,

where h ci⇤ is a time average, that is, an average of c over an intermediate time scale, long compared to the fluctuations in ⇠c (t), but short compared to the relaxation time for the slow variables cT and bT . The steady-state covariances of the slow subsystem (Eqs. S13b and S13c) are found by solving the 2 ⇥ 2 Lyapunov equation (Eq. 30), slow slow slow T Hss ⌃ss + ⌃slow ss (Hss ) =

slow ss ,

(S15)

where the unknown symmetric matrix is ✓ ◆ h c2T i h cT bT i slow ⌃ss = ? h b2T i and the ? indicates that h bT cT i = h cT bT i. Eq. S15 is expanded to yield three equations for three unknowns, (hslow cT cT

2 slow 2hslow cT cT h cT i + 2hcT bT h cT bT i =

+ hslow bT bT )h cT slow 2hbT cT h cT

bT i + hslow cT bT h bT i + 2hslow bT bT h

b2T i b2T i

= =

ss cT c T ss cT bT ss bT bT

(S16a) (S16b) (S16c)

slow where the elements of the relaxation matrix Hss are (cf. Eq. 36),

hslow cT cT = hslow cT bT

↵ss =

= = (kc

slow hslow kb . bT cT = 0 and hbT bT = The 2 ⇥ 2 covariance matrix relationships

slow ss

ss (kc

kb )

ss

+ kb wss ),

⌫ss = (kc

(S17a) kb ) css /⇣ss ,

(S17b)

for ⇠cssT and ⇠bssT is calculated using Eq. 23 and the

2 h⇠c2T i = h⇠c2 i + 2h⇠c ⇠cb i + h⇠cb i h⇠cT ⇠bT i = h⇠c ⇠b i + h⇠c ⇠cb i + h⇠cb ⇠b i + h⇠cb ⇠cb i 2 h⇠b2T i = h⇠b2 i + 2h⇠b ⇠cb i + h⇠cb i,

42

(S18a) (S18b) (S18c)

where we have dropped the superscripted ss for clarity. In this way we identify cssT cT = ss ss ss ss ss ss ss c + cb , cT bT = cb and bT bT = b + cb . Importantly, these expressions are independent of the rate constants k+ and k that appear in rss . Solving Eq. S16 simultaneously we find  1 1 ↵ss cbss + kb cb1 2 h cT i = · kc css + kb cbss + (kc kb ) ⌫ss · (S19a) ⌦ ↵ss ↵ss + kb 1 ↵ss cbss + kb cb1 h cT bT i = · (S19b) ⌦ ↵ss + kb b1 h b2T i = T (S19c) ⌦ where ↵ss = ss (kc + kb wss ). Finally, we note that in the ( c, cT , bT ) system, c is the fast variable and the 3 ⇥ 3 covariance matrix ⌃fssast for the fast subsystem, 0 1 h c2 i h c cT i h c bT i h c2T i h cT bT i A ⌃fssast = @ ? ? ? h b2T i satisfies

f ast f ast f ast T Hss ⌃ss + ⌃fssast (Hss ) =

f ast ss

(S20)

where h c2T i, h cT bT i and h b2T i are known. In the rapid bu↵er limit (k+ , k ! 1 with  = k /k+ fixed) this matrix equation simplifies considerably, because elements and terms f ast of Hss and fssast that do not involve k+ and k can be dropped, namely, 0 1 [k+ (bss + css ) + k ] k+ css + k k+ css f ast A 0 0 0 Hss =@ 0 0 0 and

f ast ss

0

ss r

=@ 0 0

1 0 0 0 0 A. 0 0

Expanding Eq. S20 we write the following three equations for the unknown h c2 i, h c cT i and h c bT i, f ast 2 f ast 2(hfc ast c h c i + hc cT h c cT i + hc bT h c bT i) =

hfc ast c h

hfc ast c h

c cT i +

c bT i +

hfc ast cT h

hfc ast cT h

c2T i cT

+ hfc ast bT h cT bT i + hfc ast bT h

f ast cc

=

bT i = 0 b2T i

= 0,

ss r

(S21a) (S21b) (S21c)

where h c2 i, h c cT i, and h c bT i are the unknowns. Combining these three equations, we express h c2 i as a function of the slow covariances, h c2 i =

css wss ⌦

ss

+

2 ss

h c2T i

2 2⌫ss h cT bT i + ⌫ss h b2T i .

(S22)

Using Eq. S22, algebraic manipulations allow us to express c0 = h c2 i in terms of model parameters (Eq. 37). The remaining covariances of the original system with three fast 43

variables ( c, b, cb) can be found through the relationships cT = c + cb, bT = b + cb, and h c cT i h c bT i h c2T i h cT bT i h b2T i

h h h h h

= = = = =

c2 i + h c cbi c bi + h c cbi c2 i + 2h c cbi + h cb2 i c bi + h c cbi + h b cbi + h cb2 i b2 i + 2h b cbi + h cb2 i.

From the perspective of analytical work, the two-step process of the fluctuating rapid bu↵er approximation is far easier than solving the Lyapunov equation for the covariance matrix ⌃ss for c, b and cb, because the fast and slow versions of the ss and Hss matrices of the RBA are simpler than those of the full calculation.

Appendix C: Analysis of the fluctuating RBA and intrinsic fluctuations We analyze the Langevin domain model when the exchange rates for Ca2+ and bu↵er are identical (kc = kb = k). In this case the fluctuating RBA simplifies to 0 c

=

css · (1 + ) ⌦

=

wss css c1 · . 2 (1 + wss )  + c1

(S24)

Under this p restriction, the relative increase in the coefficient of 2variation2 due to bu↵er 0 0 (cv /cv = 1 + ) remains biphasic through the factor wss /(1 + wss ) = wss ss = ss (1 ss ) that is a biphasic function of b1 . This result is more easily interpreted when k = k = k c b T because the equations for the slow variables (Eqs. 34b and 34c) simplify, .

cT = .

bT =

k cT + ⇠cssT (t) k bT + ⇠bssT (t).

Though cT and bT remain correlated because ⇠cssT (t) and ⇠cssT (t) are correlated, the Lyapunov equation satisfied by the covariances (Eq. S16) also simplifies and h c2T i, h cT bT i and h b2T i may be calculated independently by solving (cf. Eq. S16), 2kh c2T i =

(

ss c

+

ss cb )

2kh cT bT i =

ss cb

2kh b2T i =

to find

(

ss b

+

ss cb )

css + cbss cbss + cb1 b1 h cT bT i = h b2T i = T (S25) ⌦ 2⌦ ⌦ where the css + cbss in the numerator of h c2T i is the steady state total calcium concentration in the domain, and the dependence of this result on k is hidden in css through the steady-state flux balance jin = k(css c1 ) + k(cbss cb1 ). The final step in the fluctuating RBA is to calculate the covariances that involve the fast variable c. These covariances satisfy (see Appendix B), h c2T i =

f ast 2 f ast hfc ast c h c i + hc cT h c cT i + hc bT h c bT i =

44

ss r /2

where rss = (k+ css ·bss +k cbss )/⌦ can be written as rss = 2k+ css ·bss /⌦ or rss = 2k cbss /⌦, f ast hfc ast [k+ (bss + css ) + k ], hfc ast k+ css (cf. Eq. S21). Dividing c = cT = k+ css + k and hc bT = f ast through by hc c gives h c2 i

ss h

c cT i +

ss ⌫ss h

c bT i =

ss wss css /⌦

(S26)

f ast f ast f ast where we have used hfc ast = = ss ⌫ss and have written ⌫ss = ss and hc bT /hc c cT /hc c 1 css /(css + ) = cbss /bT for the steady-state fraction of Ca2+ -bound bu↵er in the domain. The right hand of Eq. S26 that originates from the covariance of ⇠css may be expressed in several ways (wss css = ⌫ss bss = css bss /(css + ) = cbss /(css + )). The covariances h c cT i and h c bT i in Eq. S26 are ⇥ ⇤ h c cT i = ss h c2T i ⌫ss h cT bT i = ss css + cbss ⌫ss 12 (cbss + cb1 ) /⌦ ⇥ ⇤ h c bT i = ss h cT bT i ⌫ss h b2T i = ss 12 (cbss + cb1 ) ⌫ss b1 T /⌦.

Thus, the variance of the fast variable h c2 i can be written in terms of the covariances of the slow variables, h c i ss 2 h c2 i = ss wss + ss h c2T i 2⌫ss h cT bT i + ⌫ss h b2T i (S27) ⌦ and it can be shown that h c bT i is negative while h c cT i and h c2 i are positive. Upon substitution of Eq. S25, we have h c2 i =

ss

wss css +

⌦ 1 = (1 ⌦

ss )css

ss

⇥ css + cbss

+

2 ss

2 1 ⌫ss (cbss + cb1 ) + ⌫ss bT

⇥ css + cbss

⇤

2 1 ⌫ss (cbss + cb1 ) + ⌫ss bT

(S28a) ⇤

.

(S28b)

This analysis of the fluctuating RBA under the restriction kc = kb shows that an increase in total bu↵er b1 T increases the absolute value of the slow covariances (Eq. S25). These covariances (the terms within square brackets in Eq. S28) combine to create a net positive impact on h c2 i that is attenuated to some extent by a decrease in the ss that scales these terms. The ss /⌦ outside the curly brackets in Eq. S28a might be interpreted as an e↵ective volume that attenuates all of these contributions to h c2 i, but this is misleading because 1 ss wss css = (1 ss )css is an increasing function of bT (cf. Eq. S28b). Furthermore, Eq. S24 2+ shows that the variance of the free [Ca ] fluctuations in the presence of bu↵er b1 T > 0 is 1 never less than what would occur in the absence of bu↵er (bT = 0).

SUPPORTING REFERENCES 1. Gillespie, D., 2007. Stochastic simulation of chemical kinetics. Annual review of physical chemistry 58:35-55. 2. Laub, A. J., 2004. Kronecker Products. In Matrix Analysis for Scientists and Engineers, Society for Industrial and Applied Mathematics. 3. Keizer, J., 1987. Statistical Thermodynamics of Nonequilibrium Processes. Springer Verlag. 45

4. Gillespie, D. T., 2000 The chemical Langevin equation. Journal of Chemical Physics 113:297-306. 5. Higham, D. J., 2008 Modeling and simulating chemical reactions. SIAM Review 50:347368. 6. Gardiner, C.W., 1997 Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer Verlag, New York. 7. Van Kampen, N. G., 1981. Stochastic Processes in Physics and Chemistry. NorthHolland Publishing Company, Amsterdam.

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