4.3 Intracellular calcium dynamics

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1 coupled equations: dv dv 2 dv 3 dv 4 = i + I e A + g,2 (V 2 V ) = i 2 + g 2,3 (V 3 V 2 )+g 2, (V V 2 )+g 2,4 (V 4 V 2 ) = i 3 + g 3,2 (V 2 V 3 ) = i 4 + g 4,2 (V 2 V 4 ) This can be written in matrix/vector format and solved numerically: dv dv 2 dv 3 dv 4 A = i i 2 i 3 i 4 A + I e A A + g,2 g,2 g 2, (g 2,3 + g 2, + g 2,4 ) g 2,3 g 3,2 g 3,2 g 4,2 g 4,2 A General purpose simulators, such as NEURN, solve such coupled DE equations numerically. There is a large body of numerical methods dedicated to solving such problems, which will be subject to a later chapter. 4.3 Intracellular calcium dynamics Intracellular calcium dynamics play an important role in relaying electrical activity to an intracellular, biochemical machinery relevant for learning and cell survival. Intracellular calcium signals that enter the cell nucleus selectively activate gene transcription relevant cascades, where the amplitude, duration, and frequency of the calcium signal determine which cascades are activated and at what intensity. The question then becomes: how are calcium signals, that eventually enter the cell nucleus through nuclear pore complexes, shaped and how do they propagate over long distances from synapses to the nucleus? Neurons have a large set of calcium regulating components. This section will discuss a number of these to explain the spatio-temporal, intracellular calcium dynamics. The cell s plasma membrane is equipped with calcium exchangers that can bidirectionally exchange calcium ions between the intra- and extracellular space. In the intracellular space, calcium ions diffuse and react with second messenger molecules. Given these circumstances, calcium signals would be confined to a small microdomain around a given calcium source. To overcome this microdomain confinement, neurons make use of intracellular calcium stores, that can be activated through free calcium ions, and other signaling molecules. ne such calcium store is the endoplasmic reticulum (ER). Similar to the plasma membrane, neurons can bidirectionally exchange calcium across the ER membrane, thus allowing a calcium-induced calcium release mechanism to propagate calcium signals over longer distances towards the cell nucleus. In the following we will discuss these exchange mechanisms. V V 2 V 3 V 4 A 35

2 o o / / / 4.3. Inositol-3-phosphate receptors Inositol-3-phosphate receptors (IP3R) have three binding sites, one for the molecule IP3, one for activating calcium ions, and one for inactivating calcium ions. Definition 2. Let x ijk be the fraction of IP3 receptors in one of the 8 states s ijk, i, j, k 2,. Then the kinetics in the front plane between s, s, s,ands is of the following form: s a 2 [a 2+ ] b 2 s (4.24) a [IP3] b a 3 [IP3] b 3 s a 4 [a 2+ ] b 4 The front to back plane kinetics are described by: s s ij a 5 [a 2+ ] Taken together this leads to 8 differential equations: b 5 s ij o (4.25) x x x x x x x x = b 4 x + b 5 x + b x (a 4 [a 2+ ]+a 5 [a 2+ ]+a [IP3])x = a 4 [a 2+ ]x + b 5 x + b 3 x (b 4 + a 5 [a 2+ ]+a 3 [IP3])x = a 5 [a 2+ ]x + b 4 x + b x (b 5 + a 4 [a 2+ ]+a [IP3])x = a 5 [a 2+ ]x + a 4 [a 2+ ]x + b 3 x (b 5 + b 4 + a 3 [IP3])x = a [IP3]x + b 2 x + b 5 x (b + a 2 [a 2+ ]+a 5 [a 2+ ])x = a 3 [IP3]x + a 2 [a 2+ ]x + b 5 x (b 3 + b 2 + a 5 [a 2+ ])x = a [IP3]x + a 5 [a 2+ ]x + b 2 x (b + b 5 + a 2 [a 2+ ])x = a 3 [IP3]x + a 5 [a 2+ ]x + a 2 [a 2+ ]x (b 3 + b 5 + b 2 )x (4.26) In thermodynamic equilibrium the sum of reactions is zero. Therefore we get: = a 2 [a 2+ ][S k ] b 2 [S k ] ) b 2 = [a2+ ][S ik ] a 2 [S k ] 36

3 and a 3 [S k ] = b 3 [IP3][S k ] a 4 [S k ] = b 4 [a 2+ ][S k ] b = [IP3][S k] a [S k ] (4.27) This leads to Let d i := b i a i, then b b 2 a 3 a 4 = [IP3][S k][a 2+ ][S k ][S k ][S k ] a a 2 b 3 b 4 [S k ][S k ][IP3][S k ][a 2+ =. (4.28) ][S k ] p IP3R = x 3 d 2 [a 2+ 3 ][IP3] = ([a 2+ ][IP3] + d 2 [IP3] + d 3 [a 2+ ]+d d 2 )([a 2+ (4.29) ]+d 5 ) omputing d, d 2, d 3, and d 5 Using sets of experimental data, we can compute the necessary parameters. In a first step we will use K d -values at different calcium concentrations. Definition 3. The K d -value denotes the concentration of a substance at which 5 % of all binding sites are occupied. Joseph et al. (989) computed the following values for IP3 binding to IP3 receptors:. no a 2+ : K d = 45 nm. 2. µm a 2+ : K d = 542 nm. For the model this means 5% = x + x + x + x = [a 2+ ]+[IP3] + d 2 [IP3] [a 2+ ][IP3] + d 2 [IP3] + d 3 [a 2+ ]+d d 2, d 3 [a 2+ ]+d d 2 = [a 2+ ][IP3] + d 2 [IP3] Now we can plug in our experimental data. (c,p) = (µm, 45nM) and 2. (c2,p2) = (µm, 542nM). 37

4 o o ) c p + d 2 p = d 3 c + d d 2 c 2 p 2 + d 2 p 2 = d 3 c 2 + d d 2 ) d 3 = c p c 2 p 2 + d 2 (p p 2 ) c c 2 d = d 2(c p 2 c 2 p )+c c 2 (p 2 p ) d 2 (c c 2 ) (4.3) Using data from ezprozvanny et al. (99), who found that p max =.5 at concentrations [IP3] = 2µM and [a 2+ ]=ax =.25µM, we get d 2 = (p max )/3 ax ([IP3] + d 3 )(ax + d 5 ) ax [IP3] (p max )/3 ([IP3] + d )(ax + d 5 ) (4.3) and with = dx3 d[a 2+ ] follows d 5 = c2 max([ip3] + d 3 ) d 2 ([IP3] + d ) (4.32) Finally, [IP3] can be computed via a diffusion reaction equation: [IP3] t = D IP3 [IP3] + k IP3 ([IP3] [IP3] eq ) (4.33) with [IP3] eq equilibrium state concentration and k IP3 a reaction rate for free IP3. This model can then be integrated into a three-dimensional model for calcium dynamics, where the total calcium flux through a piece of membrane can be computed from the IP3-model Ryanodine receptors Keizer and Levine (996) propose the following model for ryanodine receptors (RyR): k + a [a 2+ ] n / k a k + b [a2+ ] m / 2 k b (4.34) k c k + c 2 38

5 This leads to dx dx dx 2 = k a x k + a [a 2+ ] n x = k + a [a 2+ ] n x + k b x 2 + k c x 2 (k a + k + b [a2+ ] m + k c )x = k + b [a2+ ] m x k b x 2 dx 2 = k c + x k c x 2 In equilibrium the left hand sides are zero. With x + x 2 + x + x 2 =we can solve the system for p RyR = x + x 2 = +K b [a 2+ ] m +K c +(K a [a 2+ ] n ) + K b [a 2+ ] m (4.35) with K i := k+ i k i. The values m and n are fitted using data from Keizer/Levine and K a, K b,andk c are least squares fitted with data from Gyoerke and Fill (993) SERA pumps A model for sarco-/endoplasmic reticulum a 2+ -ATPases (SERA) was proposed by Sneyd et al. (23). SERA pumps operate against concentration gradients and thus are active transporters. The model is based on a Hill-type equation: J total SERA = With data from Fink et al. (2) g total pump g SERA. gtotal SERA [a2+ ] K SERA +[a 2+ ] SERA [a 2+ ] (4.36) can be broken down to a value for a single g SERA [a 2+ ] j SERA = SERA K SERA [a 2+ ] [a 2+ (4.37) ] In resting state all ER-transmembrane fluxes need to be in equilibrium. This is achieved by adding a leakage term of the type j leak ER = g El ([a 2+ ER ] Plasma membrane processes [a2+ cyt ]) (4.38) Plasma membrane a 2+ -ATPases (PMA) and sodium/calcium exchangers (NX) can be modeled by Hill-equations [a 2+ cyt j P = P g ]2 P KP 2 +[a2+ cyt ]2 j NX = NX g NX [a 2+ cyt ] K NX +[a 2+ cyt ] 39

6 Finally, voltage-dependent a 2+ channels (VD) are modeled by org-graham (998) in a Hodgkin-Huxley-type formalism: dx where x is the open probability of the gate. = (V )( x ) (V )x (4.39) 4