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. 2021 Jun 1;120(11):2112-2123.
doi: 10.1016/j.bpj.2021.03.027. Epub 2021 Apr 20.

Stochastic reaction-diffusion modeling of calcium dynamics in 3D dendritic spines of Purkinje cells

Victor Nicolai Friedhoff  1 , Gabriela Antunes  2 , Martin Falcke  1 , Fabio M Simões de Souza  3
Affiliations

Stochastic reaction-diffusion modeling of calcium dynamics in 3D dendritic spines of Purkinje cells

Victor Nicolai Friedhoff et al. Biophys J. .

Abstract

Calcium (Ca2+) is a second messenger assumed to control changes in synaptic strength in the form of both long-term depression and long-term potentiation at Purkinje cell dendritic spine synapses via inositol trisphosphate (IP3)-induced Ca2+ release. These Ca2+ transients happen in response to stimuli from parallel fibers (PFs) from granule cells and climbing fibers (CFs) from the inferior olivary nucleus. These events occur at low numbers of free Ca2+, requiring stochastic single-particle methods when modeling them. We use the stochastic particle simulation program MCell to simulate Ca2+ transients within a three-dimensional Purkinje cell dendritic spine. The model spine includes the endoplasmic reticulum, several Ca2+ transporters, and endogenous buffer molecules. Our simulations successfully reproduce properties of Ca2+ transients in different dynamical situations. We test two different models of the IP3 receptor (IP3R). The model with nonlinear concentration response of binding of activating Ca2+ reproduces experimental results better than the model with linear response because of the filtering of noise. Our results also suggest that Ca2+-dependent inhibition of the IP3R needs to be slow to reproduce experimental results. Simulations suggest the experimentally observed optimal timing window of CF stimuli arises from the relative timing of CF influx of Ca2+ and IP3 production sensitizing IP3R for Ca2+-induced Ca2+ release. We also model ataxia, a loss of fine motor control assumed to be the result of malfunctioning information transmission at the granule to Purkinje cell synapse, resulting in a decrease or loss of Ca2+ transients. Finally, we propose possible ways of recovering Ca2+ transients under ataxia.

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Figures

Figure 1
Figure 1
Illustration of a spine segment of a Purkinje cell showing the spine head at the top, neck in the middle, and beginning of the dendrite at the bottom. Visible are the signaling pathways of parallel and climbing fiber stimulation (1, 2, 3, 4, 5), which can trigger a cytosolic Ca2+ transient because of an opening of IP3Rs on the ER (6,7). To see this figure in color, go online.
Figure 2
Figure 2
(A) Model geometry. The endoplasmic reticulum (ER) is visible in the head and neck. Vtotal = 0.512 μm3, Vhead = 0.100 μm3, and VER = 0.020 μm3. Release sites of Ca2+ and IP3 for PF and Ca2+ for CF activation are marked by dots. Exact sizes of the geometry can be found in Table S4. (B) Interaction scheme of particle species used in the simulations from a cytosolic perspective. SERCAs, leak channels, and IP3Rs are located on the ER membrane, and PMCAs, NCXs, and more leak channels are located on the outer plasma membrane. Ca2+, IP3, and the buffers are free to diffuse in the cytosol, the volume within the plasma membrane, and outside the ER.
Figure 3
Figure 3
Overview of buffer models for (A) Pv, (B) Cb, and (C) CaM. CaM can hold up to four Ca2+. We used a 16-state model with individual binding sites. For ease of reading, we omitted the six states with two bound Ca2+. Reaction rates for all can be found in Table S3.
Figure 4
Figure 4
(A) Doi’s model: T00 represents an empty IP3R. The IP3R acts as a coincidence detector and only opens if IP3 binds before Ca2+, making T11 the open state. Otherwise, it buffers Ca2+ and becomes inactivated, preventing binding of IP3. (B) Subsection of Moraru’s model. All vertical and horizontal transitions are possible. When four IP3 and two Ca2+ (activating) are bound, T42 (light gray), there is a probability to go into the open state Topen, whence additional Ca2+ will be released. Further Ca2+ binding of the IP3R will lower the probability of opening, effectively promoting the 10 states Tx3 and Tx4 to inhibitory Ca2+ states (gray), x ∈ [0, 4]. We set the Ca2+ binding and dissociation rates of the inhibitory states to be slower than the rates of the activating states, expressed by ratio rs. Parameter values can be found in Table S1. (C) IP3R open probability at constant [IP3] = 10 μM for Doi’s (dotted) and our version of Moraru’s model with inhibitory Ca2+ binding scaling rs = {10−1, 10−2} (dashed and solid, respectively). Data points are results of stochastic computations with Copasi. The rise of the open probability at low Ca2+ causes CICR.
Figure 5
Figure 5
Snapshots of the spine head from a typical simulation with Ca2+ (red), IP3 (beige), and IP3Rs colored according to their state (Moraru’s model) as explained by the legend. At t = 0 ms, the large red dot shows the 110 Ca2+ coming from the first PF stimulus, and at t ≈ 100 ms, 1700 Ca2+ from the CF stimulus. The highly localized Ca2+ stimuli very rapidly spread by diffusion and are absorbed by buffers. Spatial averages of the Ca2+ concentration are shown in Fig. 6. To see this figure in color, go online.
Figure 6
Figure 6
(A) Moraru’s model: peak values of Ca2+ transients in the spine head for different parameter sets to PF burst and PF burst + CF (PFb + CF) stimuli, averaged over 12 simulations. Error bars show SDs. Parameters are A, N = 54 IP3Rs, PF = 90 Ca2+, and CF = 1700 Ca2+; B, N = 54 IP3Rs, PF = 150 Ca2+, and CF = 1700 Ca2+; and C, N = 56 IP3Rs, PF = 110 Ca2+, and CF = 1700 Ca2+. Set C is our control parameter set and will be used from now on. (B) Ca2+ in spine head after PF burst (dotted) and PF burst + CF (solid) stimuli with control set. The inset shows a magnification of the first 220 ms. At t = 0.1 s, the highly localized Ca2+ from the CF stimulus is visible as a spike. The injected Ca2+ gets absorbed by buffers immediately explaining the immediate return to lower [Ca2+] shortly after the stimulus. The averages with SDs (greyish areas) of 12 simulations are shown.
Figure 7
Figure 7
Peak scaling of Ca2+ transients against (A) CF Ca2+ amplitude for the PF burst + CF scenario and (B) against PF Ca2+ amplitude for the PF burst scenario for three different inhibitory Ca2+ binding timescale ratios rs = {1.0, 0.1, 0.01}; see Table S1. The slower the rate of inhibition, the larger the resulting Ca2+ transients for a given CF or PF Ca2+ amplitude. The value of rs sets the largest possible peak value when varying PF and CF Ca2+ amplitudes. Large enough transients triggered by CF coactivation with intermediate Ca2+ amplitudes are only reached when rs is on the order of 10−2. Data points are averages of 12 stochastic simulations and were fitted to Hill curves, error bars show SDs.
Figure 8
Figure 8
(A) Peak of Ca2+ transients against timing of CF stimulus for three parameter sets from Fig. 6 and an additional one with rs = 0.03. 12 stochastic simulations per data point were used, and points were fitted to Gaussians with the condition to converge against peak values of PF burst stimulation alone for very large and small tCF. Error Bars indicate SDs. The maxima of the Gaussian fits occur at 91 ms (red), 59 ms (blue), 16 ms (green), and 80 ms (purple), in qualitative agreement with experimental data (62) (+92 ± 37 ms). The halfwidths of the Gaussian fits are 257 ms (red), 294 ms (blue), 443 ms (green), and 260 ms (purple), in qualitative agreement with experimental data (62) (212 ± 85 ms). (B) Illustrating PF and CF stimulus timing by showing five Ca2+ spikes in spine head from a PF burst stimulus at 100 Hz from 0 to 40 ms and a Ca2+ spike from a CF stimulus, here tCF = 100 ms. To see this figure in color, go online.
Figure 9
Figure 9
(A) Time courses of average [IP3](t) and SDs in the spine head for different IP3 decay rates are shown. (B) Data points show averages of Ca2+ peak values and SDs for different values of IP3 decay rates kdecay against the binding rate of IP3. Each color and fit represent one decay rate. Binding rates smaller than the original value possibly represent ataxia. (C) Averages of 12 Ca2+ transients and SDs for cases with modified IP3 decay and binding rates, taken from (B), that approximately peak around the original peak value of 360 Ca2+ (dotted red line in (B)) are shown. To see this figure in color, go online.
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