Biophysical Journal - Dynamic Interreceptor Coupling: A Novel Working Mechanism of Two-Dimensional Ryanodine Receptor Array

Article Outline

Abstract
Introduction
Model layout
Results and discussion
Conclusion Remarks
Acknowledgments
References
Publication Information

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Copyright © 2007 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 92, Issue 4, 1215-1223, 15 February 2007

doi:10.1529/biophysj.106.090670

Biophysical Theory and Modeling

Dynamic Interreceptor Coupling: A Novel Working Mechanism of Two-Dimensional Ryanodine Receptor Array

Xin Liang^*, Xiao-Fang Hu^*^,^, and Jun Hu^*^,^†

^* Bio-X Life Science Research Center, College of Life Science and Biotechnology, Shanghai Jiao Tong University, Shanghai, China
^† Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai, China

Address reprint requests to Dr. Xiao-Fang Hu, Bio-X Life Science Research Center, College of Life Science and Biotechnology, Shanghai Jiao Tong University, Shanghai, China. Tel: 86-21-34204875; Fax: 86-21-34204872.

Abstract

Ryanodine receptors (RyRs) usually form two-dimensional regular array in sarcoplasmic reticulum membranes in muscle cells. The inter-RyRs coupling may be essential for the maintenance of quiescent Ca²⁺ release in resting state, as well as for the coordinated activation and rapid termination of RyR-mediated Ca²⁺ release during excitation-contraction coupling. In our previous work, we have reported that the inter-RyRs interaction is modulated by RyR channel's functional state, which inspired us to propose a novel working mechanism of RyR array: “dynamic inter-RyR coupling”. In this work, we built a simple model based on cellular automata and the Monte-Carlo method to quantitatively investigate the roles of intermolecular coupling and its modulation in regulating the signaling capabilities of RyR array. Our simulation results showed that with a suitable inter-RyR coupling strength, the combination of rest stability and high response efficiency, namely optimal signal/noise ratio, of Ca²⁺ signaling could be achieved. Moreover, we also found the continued coupling between open RyRs would delay the system termination rate. The coacquisition of robust termination of array opening relied on the proper decrease of coupling strength between activated RyRs. Obviously, such temporally asymmetric coupling would simultaneously endow the system with physiologically relevant resting stability and fast termination.

Introduction

Ryanodine receptor (RyR) is the ion channel mediating Ca²⁺ release from endoplasmic/sarcoplasmic reticulum (SR) and plays a pivotal role in intracellular Ca²⁺ signaling processes, such as excitation-contraction coupling (E-C coupling), in all muscle cell types 1,2,3,4. It is long recognized that a large amount of Ca²⁺ is released from RyRs during E-C coupling by self-amplified calcium-induced calcium release (CICR) 5,6,7. But such positive feedback is unstable, it potentially undermines the resting stability of RyRs by amplifying noise Ca²⁺, and it also potentially hinders the rapid termination of E-C coupling by regenerating Ca²⁺ release 5,6. Therefore, some design principles must be developed during evolution for RyRs to solve these problems.

Recently, electronic microscopy studies reveal that RyRs in either skeletal or cardiac muscle cells are almost exclusively found to be assembled into two-dimensional paracrystalline arrays in SR membrane 8,9,10. This organization pattern is highly conserved from crustaceans to vertebrates, suggesting that the array formation is critical to RyR-mediated Ca²⁺ release in muscle physiology 8,10,11. Some mechanism based on the RyR array may be developed to solve the problems accompanying CICR. Based on the observation of coordinated gating of neighboring RyRs in in vitro electrophysiological experiments 12,13,14, it has been proposed by Stern et al. that the allosteric interaction between neighboring resting RyRs in the array would stabilize RyRs in closed state, thus the inter-RyRs coupling provides a mechanism for the resting stability 6. However, the constant RyR-RyR coupling brings a potential design paradox into the termination process of RyR-mediated Ca²⁺ release in E-C coupling 5,15. It should be noted that in the presence of self-regenerative CICR, the rapid closure of the activated RyR channel array largely relies on the efficiency of negative feedback. Just as coupling does for resting RyRs, the continued coupling between activated RyRs will result in the stabilization of RyRs in their open state. Under such design constraints, termination mechanisms cannot efficiently transfer RyRs from open state to closed state 5,12,15. With the prolonged opening duration, both the global and local stability of SR Ca²⁺ signaling would be lost 6,16.

Intuitively, this design paradox can be ameliorated by introducing different coupling states between closed RyRs and between open RyRs. While strong coupling between closed RyRs is required to ensure the resting stability, a decoupling of RyRs accompanying their activation may remove the negative effect of inter-RyRs coupling on the termination process. This mechanism is recently hinted by our in vitro observations that the interaction between isolated RyRs decreases when the channels are activated 17. Moreover, the latest study on coupled gating of RyRs by Dulhunty et al. also reported that synchronized opening of three coupled RyRs is followed by multiple transitions between 1, 2, or 3 channels 18, which also suggested the loose coupling between activated RyRs in closing reaction. Obviously, such dynamic coupling would have profound impacts on the RyR array operation and function.

In this work, we applied a typical SR Ca²⁺ release model to quantitatively examine the impact of such dynamic coupling of RyRs on the resting stability and Ca²⁺ release duration of the two-dimensional (2-D) channel array. We demonstrated that the strong coupling between resting RyRs could increase the stability of array under rest, and an optimal coupling strength could be found for RyR array to achieve the combination of the low noise and high response efficiency, namely optimal signal/noise ratio (SNR). Moreover, the coacquisition of the timely closure of the array relied on a proper decrease of the coupling strength between activated RyRs. Our results clearly showed that such state-dependent coupling between neighboring receptors would provide a simple and efficient way to improve signaling performance of the system. In addition, the normal operation of RyR array under SR Ca²⁺ release could be dramatically damaged by biased regulation of inter-RyRs coupling, for instance in some pathological states, which would also be discussed in this paper.

Model layout

Gating scheme of single ryanodine receptor

We considered both activation and inactivation of Ca²⁺ on the activity of RyRs 2,4. Then given the law of conservation of mass and energy, a four-state scheme was built to describe RyR gating (Figure 1B), in which there are three closed states (C₁,C₂,C₃) and one open state (O). The transition probability of RyRs between different states was determined by the kinetic parameters in Table 1. The values of these parameters were set mainly according to current knowledge of single RyR gating 4, but because the RyR gating scheme in vivo is not very clear now, two additional points need to be particularly described:

1. It is still not very clear how many Ca²⁺ ions could activate the RyR in CICR 5. In our model, Ca²⁺ activation was set to be dependent on the second power of the Ca²⁺, in an effort to be consistent with data demonstrating that the rate of Ca²⁺ sparks is dependent on the square of [Ca²⁺]_i5.
2. The inherent mechanism by which clustered RyRs are closed during SR calcium release is a fundamental question not yet answered 15. In our model, accelerated Ca²⁺-dependent inactivation (with K_d ≈ 10μM; see Table 1) 6 was used as negative feedback mechanism in the gating scheme of single RyR, which is mainly based on the following considerations: 1), Some inactivation mechanisms do exist in SR Ca²⁺ signaling, in the absence of which Ca²⁺ release could not be terminated timely 4,15,19. 2), Although the Ca²⁺-dependent inactivation observed in in vitro single channel recordings (with K_d ≈ 1mM) 4 is too slow to close the RyR array in vivo, the feasibility of accelerated in vivo Ca²⁺ inactivation was already proposed to be coupled to the activity of Ca²⁺-dependent molecules, e.g., calmodulin (CaM) 20. Because it is widely accepted that the gating scheme of RyRs should be reshaped by numerous in vivo factors, we think this modification for the modeling requirement should be acceptable.

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Figure 1

Modeling layout and basic simulations. (A) Three-dimensional spatial layout of model elements. (B) Four-state gating scheme of a single ryanodine receptor (RyR) with three closed states and one open state. (C) Instantaneous pictures of coupled (e=0.25, α=1.0) and uncoupled RyR array (inset, e=0, α=1.0). (D) Basic simulations of our model (e=0.4, α=1.0): curves of SR Ca²⁺ current and Ca²⁺ content in JSR (inset) during once array opening. The red pulse curve represented the triggering Ca²⁺ signal from one L-type Ca²⁺ channel (LTCC), and three LTCCs were included in the simulations with 25 RyRs. The recovery Ca²⁺ current in JSR could be well fitted to the first-order exponential curve with the time constant of ∼30ms.

Table 1 Kinetic constants of RyRs’ gating scheme


Name	Meaning	Value
K_a	On-rate of Ca²⁺ activation	10.0 (μM⁻²×s⁻¹)
K_i	On-rate of Ca²⁺ inactivation	2 (μM⁻¹×s⁻¹)
K_ar	Off-rate of Ca²⁺ activation	400.0 (s⁻¹)
K_ir	Off-rate of Ca²⁺ inactivation	20 (s⁻¹)

Two-dimensional tetragonal lattice model of RyRs

Electronic microscope studies revealed that the geometry of the lattice and the number of RyRs in the lattice (usually 10∼100) varies with the species and muscle types 8. In this model, the typical square lattice formed of 25 (5×5) receptors (Figure 1C) was adopted. Here, it should be pointed out that we also run the simulations with RyR arrays in various sizes, and the main conclusions of our manuscript are not be affected by the size of RyR array. We consulted the arithmetic proposed by Stern et al. 6, by which the intermolecular coupling was introduced into the kinetics of arrayed RyRs (Eq. (1)):

(1)

where K_ij is the kinetic parameter of coupled RyRs; k_ij is the kinetic parameter of isolated RyRs; E_ij is the interaction energy between two contacting RyRs; i/j is current/next state of central RyR; s_m is the state of neighboring RyR; n represented the number of physical contacting neighbors of the central RyR.

Here, to present a more intuitive impression of the operation of the RyR array in our model, instantaneous cartoon pictures of uncoupled and coupled RyRs array were captured during array operation. In an uncoupled system, RyRs opened individually (inset of Figure 1C). Small patches of open channels stochastically appeared, but rarely. However, in a coupled system, opening events could be found in large patches of activated RyRs (Figure 1C). Furthermore, a coefficient matrix was constructed to modify the interaction energy (e) between neighboring RyRs in state-dependent manner (Eq. (2)). The standard principles to evaluate the coefficients, “1” and “α”, in the matrix were described as following:

(2)

Principle 1. To realize intermolecular cooperativity, the neighbors in the same state with central receptor stabilize the central one, while the neighbors in alternative states impel the central one away from its current state. Given the form of Eq. (1), the interaction energy between contacting RyRs in the same/different states should be positive/negative, respectively.
Principle 2. We have reported that the interaction between RyRs would decrease when RyRs were activated by Ca²⁺. To briefly but reasonably introduce this new information, we assumed that when two neighboring RyRs both bound to the activating Ca²⁺, the interaction strength between them would change. In the above matrix, “α” is the coefficient for the interaction energy between two RyRs bound to activating Ca²⁺ (states O or C₂). This parameter could be adjusted to simulate state-dependent coupling of arrayed RyRs in our model.

In situ SR calcium release model

The in situ SR calcium release model adopted here was modified from Sobie's model for Ca²⁺ sparks 16, and the following described the model details. Figure 1A showed the spatial configuration of the SR Ca²⁺ release unit. It consisted of several different spatial regions, including the T-Tubule membrane (TTM) with L-type Ca²⁺ channels (LTCCs), SR membrane (SRM) with a regular array of RyRs, the nanoscale space between sarcolemma (SL) and SRM subspace, the cytoplasm, the space in junctional SR (JSR), and the extensive space of network SR (NSR).

First, the volume of subspace (V_ss) was calculated as follows with the shape of the subspace simplified to be a column, and the RyR array, a regular square:

(3)

where D_ss was the distance between SRM and TTM, ∼10nm; 30nm was the dimension of a RyR 9; n_ryr was the number of RyRs in the square array.

The concentration of Ca²⁺ in the subspace was determined by the Ca²⁺ influx through LTCCs (J_LTCCs) and RyRs (J_RyRs), the contribution of several Ca²⁺ buffers (J_buf) and the Ca²⁺ efflux (J_efflux) to the cytoplasm through diffusion:

(4)

The startup of RyR array opening was normally stimulated by the inward Ca²⁺ current through the LTCCs in the TTM, and the number of LTCCs was determined according to the 7.3 RyRs/LTCCs (25 RyRs / 3 LTCCs) reported earlier 21. To emphasize our main point, a highly simplified L-type Ca²⁺ current (I_LTCC=0.5 pA, t_duration=2ms) was used to replace detailed gating behavior of LTCCs. In formula 3, I_LTCCs represented the average Ca²⁺ current of opening LTCCs, F is the Faraday's constant, and V_ss is the volume of subspace.

(5)

(6)

Ca²⁺ flux through the RyR was proportional to the Ca²⁺ concentration gradient between two sides of the SRM, and also correlated to the Ca²⁺ diffusion through the channel (D_RyR).

(7)

The total Ca²⁺ efflux (J_RyRs) through all the RyRs in the array was described as

(8)

In the subspace, Ca²⁺ could bind to calmodulin (CaM) and Ca²⁺ buffers in SRM and SL, the equation for these reactions could be written in a general form as following:

(9)

So, the contribution of total J_buf should be calculated as

(10)

Ca²⁺ efflux to global cytoplasm through diffusion was determined by the Ca²⁺ concentration gradient and the velocity of Ca²⁺ diffusion. The [Ca²⁺]_cyo was fixed at 100nM and the τ_efflux was the time constant for Ca²⁺ diffusing from subspace to cytoplasm.

(11)

Similar with the situation in subspace, three factors were mainly responsible for the calcium kinetics in JSR: outward Ca²⁺ current through RyRs, calcium flux from extensive NSR to JSR and the calcium buffer (calsequestrin) in JSR. Here, [Ca²⁺]_NSR was fixed at 10³μM and Ca²⁺ buffering in JSR by calsequestrin was treated as a rapid buffering process. Notably, the value of τ_refill was changed from 10ms in Sobie's model to 4ms here. We made this modification to satisfy the recovery time constant (∼30ms) of free [Ca²⁺] in JSR (Figure 1D), which was reported by the latest work of Brochet et al. 19.

(12)

(13)

(14)

The definitions and value of all the coefficients in our model were presented in Table 2. The basic simulations of our model under typical condition (e=0.4, α=1.0) is shown in Figure 1D. Note that the opening of RyR array could be induced by a brief inward Ca²⁺ current through L-type Ca²⁺ channels to produce a large SR Ca²⁺ release (∼5–6 pA), reflecting the high gain of this system (Figure 1D). And there was an obvious reduction of Ca²⁺ content in JSR (inset in Figure 1D). Obviously, the sample curves derived from our model exhibited similar shape and quantitative features of SR Ca²⁺ release events observed experimentally 3,16,22,23,24, thereby demonstrating the workability of our model.

Table 2 SR Ca²⁺ release model parameters


Name	Definition	Value
N	Number of RyRs in the array	25 (5×5)
D	Ca²⁺ diffusion coefficient through an open RyR	4000s⁻¹
F	Faraday's constant	96,480C mol⁻¹
τ_ss	Time constant of Ca²⁺ diffusion into cytoplasm	0.7μs
[Ca²⁺]_L	Ca²⁺ concentration in JSR	(initial) 1.0mM
V_ss	Volume of subspace	0.35×10^−12μL
CaM	Total calmodulin	24.0μM
K_on,CaM	CaM Ca²⁺ on-rate constant	100μM⁻¹ s⁻¹
K_off,CaM	CaM Ca²⁺ off-rate constant	38s⁻¹
SR	Total SR buffer	47.0μM
K_on,SR	SR buffer on-rate constant	115.0μM⁻¹ s⁻¹
K_off,SR	SR buffer off-rate constant	100s⁻¹
SL	Total SL buffer	1124.0μM
K_on,SL	SL buffer on-rate constant	115.0μM⁻¹ s⁻¹
K_off,SL	SL buffer off-rate constant	1000.0s⁻¹
V_JSR	Volume of JSR	1×10^−11μL
τ_tr	Time constant of Ca²⁺ refill from NSR	4×10⁻³ s
[Ca²⁺]_NSR	Ca²⁺ concentration in NSR	1.0mM
[CSQ]_tot	Total CSQ in JSR	10mM
KCSQ	CSQ Ca²⁺ dissociation constant	0.8mM
[Ca²⁺]_cyt	Ca²⁺ concentration in cytoplasm	0.1μM

Parameters and calculation

Signal/noise ratio

Signal/noise ratio (SNR) was defined as the ratio between a signal (effective output) and the background (noise). It has been used as an evolutionary standard to determine the optimal coupling strength between bacterial chemotactic receptors by Shimizu et al. 25. In our model, the array's response to background Ca²⁺ and L-type Ca²⁺ current were treated as noise and signal, respectively. The SNR was calculated as following (Eq. (15)):

(15)

Here, signal was represented as the average amplitude of array response to input L-type Ca²⁺ current, and noise was the average SR Ca²⁺ current from arrayed RyRs under resting state by continuously running the program for 3×10⁷ time steps (biological time=3s, much longer than the operation cycle of array opening).

Computation

The operation of the RyR lattice array during SR calcium release was run based on cellular automata and the Monte-Carlo method with the time step of 10⁻⁷ s. All programs were allowed to run a period of time (usually 2×10⁶ time steps/biological time=200ms) for stabilization before the beginning of experimental simulations. And all the codes for this model were written in Fortran and operated on a Dell workstation for scientific computation.

Results and discussion

Resting stability and high response efficiency of coupled RyR array

The impact of inter-RyR coupling on the RyR array's resting stability and response efficiency to input triggering signal was quantitatively investigated.

For an uncoupled system, all the receptors in the array behaved individually. Though the open probability of solitary RyRs under steady state (0.1μM Ca²⁺) is very low (P_o<0.01), the activation of the entire RyR array would be largely maintained due to the positive feedback of Ca²⁺-mediated regeneration (Figure 2Aa). Such frequent spontaneous activation of uncoupled RyR array greatly increased the resting noise of system. If we simulated the DHPR-generated Ca²⁺ current and input such triggering Ca²⁺ signals to the RyR array (arrows in Figure 3Aa), it was found that this signal was submerged in a sea of Ca²⁺ noise, and the RyR array could not respond efficiently to the triggering signal (Figure 3Aa). The questions arose that how could RyR array keep resting stability and response efficiency in vivo?

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Figure 2

Effects of inter-RyR coupling strength on the resting stability of a RyR array. (A) The typical activity curves of RyR array with different coupling strength under rest: (a) e=0; (b) e=0.35; (c) e=0.6. (B) E-dependent curve of spontaneous opening of RyR array, in which the data dots are the average results of 10 simulations for 10⁷ time steps (biological time=1s). For clarity, all the results shown here are obtained when “α” was 1.0.

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Figure 3

Effects of inter-RyR coupling strength on response amplitude and SNR of a RyR array. (A) The typical response of RyR array with different coupling strength to input Ca²⁺ signal: (a) e=0; (b) e=0.35; (c) e=0.6. The arrows indicate the time of triggering L-type Ca²⁺ current. (B) Regulation of coupling energy (e) on response efficiency and SNR of system. The data dots of amplitude curve are averaged from 100 simulations of SR Ca²⁺ release events, while SNR is calculated based on the formula described in the Model Layout section. For clarity, all the results shown here were obtained when “α” was 1.0.

The functional cooperation between neighboring RyRs under resting conditions provides a possible way to solve the problems. Running simulations with different cooperativity between RyRs allowed us to investigate the detailed roles of coupling in array operation. When the RyR-RyR interaction energy (e) was increased to 0.35 (Figure 2Ab), the continued opening of system changed into individual opening events. If further increasing the interaction energy to 0.6 (e) (Figure 2Ac), the system exhibited low level of noise. Statistically analysis showed that spontaneous Ca²⁺ current through RyR array decreased monotonically with the increase of inter-RyR interaction energy, and the array could keep quite stable when interaction energy (e) was more than 0.4 (Figure 2B).

Correspondingly, we also examined the impact of coupling strength on the RyR array's response to triggering signal. Suitably strengthening the inter-RyR coupling (e=0.35) could efficiently stand out the response of RyR array to triggering Ca²⁺ signal (Figure 3Ab). On the other hand, we also saw that too much strong interaction between RyR would make the array “blind” to the input signal (Figure 3Ac). As shown in Figure 3B, the mean amplitude of system response showed biphasic dependence on coupling strength (e). The e-dependent amplitude curve rose in the region of 0∼0.4 (e), but in the region of 0.5∼0.7 (e), the response amplitude decreased with the increase of interaction energy. The maximum response gain could be observed at 0.4∼0.5 (e).

Thus, the range of interaction energy suitable to maintain both RyR array resting stability and response efficiency is in the region of 0.4∼0.5 (e). In further searching for the optimal interaction energy, we determined the system SNR in response to a Ca²⁺ stimulus (mildly above activation threshold). The SNR showed bell-shaped interaction energy (e) dependence, with the maximum SNR at 0.45–0.6 interaction energy (Figure 3B). Comprehensively considering the optimal SNR and high response efficiency of RyR array, it was expected that when coupling strength (e) was in 0.45∼0.5, the resting stability and response efficiency of the system were best integrated.

Effects of “α” on resting stability and response efficiency

All the results mentioned above were obtained for a RyR array with constant coupling strength (α=1.0). To investigate the effect of the coupling strength between activated RyRs on the resting stability and response efficiency of RyR array, we ran the simulations with various “α”. As shown in Figure 4AB, the decrease of α from 1.0 to 0, with the interval of 0.2, had little effect on the e-dependent curves for both resting stability and response amplitude of RyR array. Obviously, the coupling strength between neighboring activated RyRs, represented as “α”, is relatively independent with the system behavior in resting state and activation stage.

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Figure 4

The e-dependent curves of system resting stability and response amplitude under various “α”. (A) The effect of “α” on the resting stability of RyR array. (B) The effect of “α” on the response amplitude of RyR array.

Based on these results, we noted that two factors should be responsible for the resting stability and high response efficiency of RyR array. First is the coupling between two neighboring resting RyRs, and second is the inhibitory effect from resting RyRs to their opening neighbors. These two factors keep the RyR array stable enough under rest through increasing the activation threshold for RyR array. It should be noted that the high response efficiency depends on the resting stability. The maximum amplitude appeared only when the coupling strength (e) between RyRs is strong enough to stabilize the system under rest (Figure 3BB and Figure 4BB). Therefore, the coupling between resting RyRs and their (closed/opening) neighbors play pivotal roles in keeping the system stability and response efficiency, while the coupling between activated RyRs contributes little on the performance of RyR array in this stage.

Effect of “α” on opening duration of RyR array

Coacquisition of rapid termination in a coupled system

From the evolutionary point of view, systems with both optimal SNR and high response efficiency should be favored 25 in cellular signaling. For an array of channels such as the RyR array in SR, rapid closure of the system is also physiologically required. Because the duration of array opening cannot be directly measured in vivo at present, what little knowledge we have of the process has been obtained from the analysis of temporal characteristics of elementary SR Ca²⁺ events, e.g., Ca²⁺ sparks and Ca²⁺ blinks. First, Soeller and Cannell reconstructed the SR Ca²⁺ flux underlying Ca²⁺ sparks peaked in ∼5ms and decayed with halftime of ∼5ms 22, which suggested that total duration of RyR array should be longer than 10ms. More recently, Brochet et al. reported the time to nadir of Ca²⁺ blinks was ∼22ms, longer than the time to peak of Ca²⁺ sparks (in rat ventricular myocytes ∼10ms) 19. In principle, the Ca²⁺ in JSR would not decrease further after the complete closure of RyRs, therefore the temporal characteristics of Ca²⁺ blinks suggested that the opening of clustered RyRs should be at least 22ms. Therefore, 22ms might be the proximal value that reflected the actual duration of clustered RyRs underlying the Ca²⁺ sparks at the present.

To favor the optimal SNR of system, the interaction energy was selected to be 0.45. We first tested an iteration of the operation of the RyR array with constant coupling between neighboring RyRs, regardless of their functional state. Under such control conditions (e=0.45, α=1.0), the high response and SNR were appropriately achieved, but the array opening lasted more than 70ms (Figure 5Aa). Then, it was found that the average opening duration under this condition was ∼50ms, obviously longer than the physiologically expected 22ms. Here, the question arose: how could a coupled system with high gain and optimal SNR be manipulated to realize fast termination?

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Figure 5

The effects of “α” on the opening duration of RyR array. (Aa) Array operation under control condition (e=0.45, α=1.0). (Ab) Array opening duration when “α” is reduced (e=0.45, α=0.8). (Ac) Array opening duration when “α” is further reduced (e=0.45, α=0.5). (B) The mean opening duration of RyR array could be modulated through adjusting the values of “α”, and the “α”-dependent curve could be well fitted to the exponential function (solid curve). The dashed line represents the expectation value of array opening duration (22ms) according to the analysis of Ca²⁺ sparks and Ca²⁺ blinks.

The decoupling of activated RyRs provided an easy and efficient way. The decrease of “α” in our model resulted in the rapid closure of RyR array. As shown in Figure 5B, the decrease of “α” from 1.0 to 0.8 reduced the opening duration of RyR array from 50ms to ∼30ms (Figure 5Ab). Further decrease “α” to 0.5 could shorten the duration of RyR array more to 15ms (Figure 5Ac).

To systematically investigate the effects of the decoupling of activated RyRs on the array's duration of opening, we analyzed the termination behavior of RyR array under different “α” (Figure 5B). When e was 0.45, we showed that the decrease of “α” induced an obvious reduction of opening duration. When “α” was set from 1 to 0 with the interval of 0.2, the opening duration of RyR array decreased quickly (Figure 5B), and the histogram could be well fitted with an exponential decay curve (Figure 5B, solid line). We noted that the system could be closed timely (∼20ms, indicated by dashed line in Figure 5B) when “α” was around 0.6. Compared with the coupling strength between resting RyRs (1.0×e), the decreased coupling strength between activated RyRs (0.6×e) is essential for RyR array to coachieve the rapid termination during Ca²⁺ release processes.

From the theoretical viewpoint, strong coupling between opening RyRs would delay the termination process by building high energy barrier to prevent the transition of RyRs from open state to closed state. The decoupling of activated RyRs would facilitate the rapid closure of RyR array. In addition, it should be clarified that “the decoupling of activated RyRs” itself is not a mechanism to trigger the termination process. The role of this regulatory mechanism within RyR array is to make the inherent termination mechanism (e.g., Ca²⁺ inactivation, local SR depletion, etc.) work more efficiently.

A whole picture of “dynamic coupling” mechanism

We have shown that the optimal signal/noise ratio of RyR array can be achieved by suitable inter-RyRs coupling between resting RyRs and their neighbors, while decoupling of activated RyRs could facilitate the rapid termination of the system. The operation of the RyR array under one typical “optimal” condition (e=0.45, α=0.6) was simulated and shown in Fig. 6. This coupled system kept highly stable under rest and responded efficiently to the input Ca²⁺ signal, namely acquiring high SNR. Meanwhile, the mean array opening duration of RyR array was ∼22ms and the decay constant of SR Ca²⁺ flux of ∼5ms, which seemed to approximate the experimental and numerical estimation value in the work of Soeller and Cannell and Brochet et al. 19,22. Therefore, resting stability, high response efficiency, and fast termination could be all satisfied through suitable regulation of inter-RyRs coupling, which could not be realized in either a completely uncoupled or a continued coupled system.

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Figure 6

The simulating opening course of RyR array under representative optimal conditions (e=0.45, α=0.6). Thin solid curve is one simulation. Thick solid curve is the average of 100 simulations.

In an uncoupled system, the behavior of the array is completely controlled by Ca²⁺. In this case, system gain cannot be enhanced without amplifying the array's resting noise. For a completely coupled system, although both resting stability and high gain can be acquired, the continued coupling between activated RyRs exponentially prolongs the duration of system response. Loosening the coupling between activated RyRs would make opening RyRs behave more independently, benefiting the efficiency of negative feedback to close the system. Temporally asymmetric regulation of coupling, therefore, can control the system decay rate in the termination of Ca²⁺ release, while simultaneously maintaining an optimal SNR in the system response to the input Ca²⁺ signal.

Generally speaking, it is believed that the final goal of the biological system's evolution is to optimize the system performance in its working environment. Our simulation predicts that the coupling between arrayed RyRs only occurs when necessary, the extent of which is finely controlled to satisfy physiological requirements. Within the limitations of our system calculations, suitable coupling between RyRs ensures system stability, gain and SNR, while timely and partial decoupling of activated RyRs maintains the temporal order required of physiologically relevant system activity.

Biased coupling between RyRs and abnormal SR Ca²⁺ release

Our calculations demonstrate that the extent of coupling strength always had an optimal value either in the initiation or the termination process of SR Ca²⁺ release. This implies that coupling is a significant regulatory point in SR Ca²⁺ signaling. By extension, it also suggests the potential relationship between biased inter-RyRs coupling and abnormal SR Ca²⁺ release.

For example, when the coupling between resting RyRs was weakened to a great extent (e = 0.2, α=0.6), the signal response curve exhibited high baseline and low response efficiency (Figure 7A). Loose coupling between resting RyRs would destabilize RyRs. Frequent spontaneous Ca²⁺ release from the leaky channels would potentially lead to the rise of resting Ca²⁺ in cytoplasm. Meanwhile, weakly coupled RyRs were also incapable of achieving the full activation of system, and only responded faintly to triggering Ca²⁺. Totally, the system SNR in this situation becomes sufficiently low, which could result in low efficiency in SR Ca²⁺ handling.

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Figure 7

Biased coupling causes abnormal opening of RyR array. (A) Abnormity I: inter-RyR coupling is reduced. The baseline of array's activity under rest is high, but the system gain is low. (B) Abnormity II: the interaction between activated RyRs is enhanced, rather than reduced. Once the system is activated, it will be quite difficult to recover the static state (mean opening duration>100ms).

A second scenario occurs when the coupling between activated RyRs was not decreased, but even increased. As shown in Figure 7B (e=0.45, α=1.5), once the system is activated, it will be quite difficult to recover to static state. The mean opening duration of RyR array in this situation was longer than 100ms, much longer than that acceptable in vivo. In principle, such prolonged opening of RyRs might be related to delayed termination of SR Ca²⁺ release events (Ca²⁺ spark, wave or transient), which might induce severe dysfunction of local or global SR Ca²⁺ handling system.

Here, we showed the potential relevance of biased inter-RyRs coupling, defined in our model, to the abnormal SR Ca²⁺ release. Because the Ca²⁺ release from SR is so important in muscle E-C coupling, the biased regulation of inter-RyR coupling might also be potentially involved in the dysfunction of muscle cells, especially in the pathological states.

Compared with “conformational spread” model

Another paradigm of interreceptor coupling exists in the two-dimensional array of bacterial chemotactic receptors 26. The model of “conformation spread” is proposed by Bray et al. to describe the cooperative behavior of chemotactic receptors 27. It was known that “conformational spread” conferred the chemotactic receptor array with several remarkable qualities, for instance, its ultrasensitivity, broad response spectrum (∼5 orders of magnitude chemosensing capability), etc. 27,28,29.

It should be noted that the modulation of interreceptor coupling is the communal characteristic of “conformational spread” model and our “dynamic coupling” model. In “conformational spread” model, the interreceptor coupling should be modulated at different concentration of chemoattractants to harmonize the apparently antithetical requirements for both high sensitivity and a broad response spectrum 29. In our “dynamic coupling” model, the coupling strength is modulated at different channel functional states to coachieve the optimal signal/noise ratio and fast termination of Ca²⁺ release. Obviously, the modulation of interreceptor coupling could endow the 2-D receptor array with improved performance in cellular signal transduction.

Conclusion Remarks

We have proposed a novel design principle, temporally asymmetric coupling between neighboring RyRs, for RyR array to achieve the physiologically relevant resting stability and fast termination, which cannot be simultaneously acquired by either a completely uncoupled system or completely coupled system. Obviously, this is a simple and efficient way for RyR array to improve the signaling performance. Because the clustering of functional molecules commonly exists in biological systems, such design principle may be favored by other clustered receptors to achieve both rapid “on” and “off” response.

Acknowledgments

We thank Professor Pei-Hong Zhu, Hai-Ping Fang, Mu-Ming Zhang, Jianjie Ma, and Heping (peace) Cheng for fruitful discussion, and Dr. Jerome Parness for the modification of our manuscript.

This work was supported by a grant from the National Nature Science Foundation of China (NSFC30670495).

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Received: June 7, 2006
Accepted: October 20, 2006

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Dynamic Interreceptor Coupling: A Novel Working Mechanism of Two-Dimensional Ryanodine Receptor Array

Abstract

Introduction

Model layout

Gating scheme of single ryanodine receptor

Two-dimensional tetragonal lattice model of RyRs

In situ SR calcium release model

Parameters and calculation

Signal/noise ratio

Computation

Results and discussion

Resting stability and high response efficiency of coupled RyR array

Effects of “α” on resting stability and response efficiency

Effect of “α” on opening duration of RyR array

Coacquisition of rapid termination in a coupled system

A whole picture of “dynamic coupling” mechanism

Biased coupling between RyRs and abnormal SR Ca2+ release

Compared with “conformational spread” model

Conclusion Remarks

Acknowledgments

References

Publication Information

Article Outline

Biased coupling between RyRs and abnormal SR Ca²⁺ release