1. Interplay of Ryanodine Receptor Distribution and Calcium Dynamics
Leighton T. Izu, Shawn A. Means, John N. Shadid, Ye Chen-Izu, C. William Balke 
Biophysical Journal 
Volume 91, Issue 1, Pages (July 2006)
DOI: /biophysj
Copyright © 2006 The Biophysical Society Terms and Conditions

2. Figure 1
Three-dimensional model geometry of the sarcomere and finite element mesh geometry and domain decomposition. (A) Dark blue disks are the Z-disks into which the CRUs (blue bricks) are embedded. The Z-disks lie in the xy plane and the longitudinal axis of the myocyte is along the z axis. The distance between the Z-disks is Lz. The intercalated CRUs (cyan bricks) are only on the cell periphery, forming a ring that is located midway between two Z-disks. (B) Detail of a single Z-disk. CRUs are circularly arranged with spacing Λ indicated by the white lines. See text for meaning of arrows. (C) A composite view of the unstructured tetrahedral FE mesh of the cardiac cell with two z-planes and a set of intercalated CRUs is shown. The upper left image presents the surface mesh of the cell section and the lower left image shows a close-up view of the unstructured mesh used to resolve the individual CRUs. The upper right image shows a projection of the mesh on an xz-slice plane through the domain. The lower right image shows the projection of the mesh on an xy-slice plane. The individual CRUs in these two images have been colored to make them more visible. (D) Composite view of the subdomain-to-processor assignment used to partition the mesh on 16 processors is shown. The left image shows subdomain-to-processor assignments by color. The partitioning assigns FE nodes to subdomains and then maps these subdomains to processors. The gray intersubdomain region indicates boundaries between processors for which communication of information takes place in the parallel FE code. The image on the right is a partial exploded view with the intersubdomains removed.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

3. Figure 2
Qualitative difference in Ca2+ dynamics resulting from a small change in CRU spacing. These snapshots are of two simulations that used identical parameters and initial conditions, except that the CRU spacing is Λ=0.8μm in A and 1.0μm in B. A Ca2+ wave is initiated when Λ=0.8μm but not when Λ=1.0μm. In this and subsequent figures, each image is divided as follows. The large image on the upper left is an oblique view of the sarcomere, which includes a 15-μM Ca2+ isoconcentration surface. The smaller image near the bottom right shows the Ca2+ concentration over the smaller range of 0.1–2μM, and shows the isoconcentration surface at 2μM. The small image near the middle of the right-hand side is a side view of the sarcomere. A suite of isoconcentration surfaces at 1, 2, 5, 10, and 15μM are shown. The side view is especially helpful for understanding how CRUs on different Z-disks interact. The color bar ranges from 0.1 to 50μM for the large image and from 0.1 to 2μM for the lower right image. Simulation times are 0.92, 45.92, 65.92, and 85.92ms (left to right). Simulation parameters: ISR=13 pA, Lz=1.6μm; other parameters are given in Appendix A.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

4. Figure 3
Effect of CRU spacing, number of firing CRUs, and current on Ca2+ concentration in 2-D. (A) Geometry of the 2-D lattice. The firing CRUs (black dots) are on the line x=0; they are spaced λy apart. The observation point O (gray dot), where the Ca2+ concentration is measured, is at x=λx, y=0. The Ca2+ concentrations at the other observation points are used in a later figure. (B) Ca2+ concentration when the observation point is λx=0.8 or 1μm from the line of seven firing CRUs, which are spaced λy=0.8μm apart. The modest 20% reduction in λx results in a fourfold increase in peak Ca2+ concentration at O. (C) Maximum Ca2+ concentration at O as a function of the number of firing CRUs. The square gives the Ca2+ concentration that would be seen if no Ca2+ buffers were present and a single CRU produced the concentration shown. (D) n-fold change in peak Ca2+ concentrations at O as a function of Ca2+ current. The squares show the n-fold change in Ca2+ concentration that would be seen if Ca2+ buffers were absent.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

5. Figure 4
Understanding why a wave is initiated when ISR=16 pA but not when ISR=13.4 pA. (A) Probability density function ϕ(t,N=7) for ISR=13.4 pA (dashed line) and 16 pA (solid line). ϕ(t,N) is not sufficiently sensitive to explain why a wave is initiated with ISR=16 pA but not when ISR=13.4 pA. (B) Synchronization curves show that five CRUs will fire within 11ms of each other when ISR=16 pA (circles) but it takes ∼50ms for five CRUs to fire when ISR=13.4 pA (squares). This long delay reduces the likelihood of wave initiation. The accuracy of the firing times in the synchronization curve depends on the accuracy of estimating the integral of ϕ(t,n) over infinite time. Since our simulation of the canonical Ca2+ distribution is limited to 100ms, ϕ(t,n) is known only to 100ms. When the Ca2+ concentration is high, the integral converges quickly to 1 and the firing time is accurate. When the integral is <0.8 the computed firing time is only a crude lower bound. Firing times that are not known accurately (integral <0.8) are marked with asterisks.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

6. Figure 5
Effect of CRU spacing and current in 3-D. (A) Ca2+ concentration measured at the central CRU on layer 1 (see Methods for definition) when eight contiguous CRUs fire simultaneously on layer 0. The peak Ca2+ concentration is 12-fold higher when Λ=0.8μm than when Λ=1.0μm. (B) n-fold increase in peak Ca2+ concentration at central CRU for different ISR. The dashed line shows the increase that would be seen in an unbuffered system. (C) Synchronization curves for ISR=20 pA (squares) and 32 pA (circles). Asterisks have the same meaning as in Fig. 4.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

7. Figure 6
Cooperative Z-disk interactions. CRUs on Z-disk 2 (on the right) are inactive (Pmax=0) in A and active (Pmax=0.3/ms) in B. Otherwise, the simulations are identical. Because Ca2+ released from CRUs on one Z-disk can increase the probability of firing of a CRU on the other Z-disk, this regenerative feedback reinforces the nascent wave and a stable wave propagates across the Z-disks in B. In A, this cooperative interaction is absent and no wave forms. (C) The CRUs on Z-disk 2 are inactive but a wave forms because the current has been increased from ISR=21 pA (in A and B) to 25 pA. All simulations are started with the canonical pulse. The color bar ranges from 0.1 to 50μM for the large image and from 0.1 to 2μM for the lower right image. Snapshots are taken at 0.92, 30.92, 50.92, 70.92, and 90.92ms.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

8. Figure 7
Effect of intercalated CRUs (i-CRUs). i-CRUs are absent in the simulations of A. When the current is ISR=20 pA (A), a wave is not initiated. In the absence of i-CRUs the minimum current needed to initiate a wave is ∼21 pA (not shown). The current needed to initiate a wave is significantly reduced (to ∼15 pA) when i-CRUs are present (B). The color bar ranges from 0.1 to 50μM for the large image and from 0.1 to 2μM for the lower right image. Snapshots are taken at 0.92, 20.92, 40.92, 60.92, and 80.92ms. Simulation parameters: Lz= 2.0μm, Λ=1.0μm, others standard.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

9. Figure 8
Intercalated CRUs act as relay stations. (A) The canonical pulse ends at 5.92ms (left panel), and 1ms later (at 6.92ms, center panel), two intercalated CRUs fire. Their firing is seen more clearly in the side view of the sarcomere (center panel, inset, right-hand side of image). The first CRU on Z-disk 2 fires at 9.92ms (right panel). (B) Synchronization curves for the i-CRUs (circles) and for the CRUs on Z-disk 1 (squares). Although the CRUs on layer 1 of Z-disk 1 and the i-CRUs are the same distance from the triggered CRUs (1μm), the expected times for the i-CRUs to fire are substantially shorter than for the layer-1 CRUs (for example, 4.8ms vs. 21.7ms for the first CRU to fire). The reason for this difference is shown in the curves of C, which show the Ca2+ concentration seen by the central i-CRU (solid line) and the central CRU on layer 1 (dashed line). Simulation parameters: Lz=2.0μm, Λ=1.0μm, ISR=15 pA, others standard.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

10. Figure 9
A small decrease in sarcomere length greatly increases Z-disk interaction. (A) Synchronization curves as functions of sarcomere spacing (Lz) for Lz=2.0μm (dotted line), Lz= 1.8μm (dashed line), and Lz=1.6μm (solid line). Simulation parameters: ISR=20 pA, Λ=1.0μm, others standard. (B) These three simulations used identical parameters and initial conditions, except that the Z-disk spacing was 2.0μm (upper), 1.8μm (middle), and 1.6μm (lower). Closer Z-disk spacing allows greater regenerative interaction of CRUs on adjacent Z-disks, thereby increasing the likelihood of Ca2+ wave generation. The color bar ranges from 0.1 to 50μM for the large image and 0.1 to 2μM for the lower right image. Snapshots are taken at 0.92, 20.92, 60.92, and 80.92ms. Simulation parameters: ISR=20 pA, Λ=1.0μm, others standard. (C) Currents needed to initiate a Ca2+ wave as a function of Z-disk spacing. All simulations were started with the canonical pulse (eight contiguous CRUs on layer 0 firing at t=0). Currents were varied until a Ca2+ wave was initiated within 100ms of the start of the pulse. For a given Z-disk separation, if ISR is below the curve the probability of wave initiation is low (e.g., point a) and the system is stable. If ISR remains constant but the Z-disk separation shortens to the extent that ISR moves above the curve then wave initiation is likely (e.g., point b) and the system becomes unstable. Simulation parameters: Λ=1.0μm, the rest are standard.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions

11. Figure 10
Initiation versus support of Ca2+ waves. In these 2-D simulations the initial conditions were chosen to mimic either an action-potential-triggered CRU firing (A) or a well-formed Ca2+ wave (B). White dots mark the positions of the CRUs. Simulation parameters:ℓx=ℓy=1μm (see Fig. 6 A for definition of spacing parameters), ISR=11 pA, others standard.
Biophysical Journal  , DOI: ( /biophysj )
Copyright © 2006 The Biophysical Society Terms and Conditions