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Mathematical model

Biochemical model

Ach simulation

ACH flow was simulated by using forward finite difference for time and central finite difference for space to find an explicit solution to two-dimensional heat flow. The diffusion coefficient of ACH was calculated from [link] .

[ A C H ] t = α [ A C H ] x x + [ A C H ] y y α = 0 . 4 n m 2 n s - 1

It was found experimentally that using a time step of 0.25 ns for explicit solution gave similar results as using a Gaussian lowpass filter matrix to represent a rotational symmetric point surface distribution. Since the Gaussian filter matrix, B, is built into MATLAB, it is computed faster, and thus was used for the simulation. In equation 2, B is the 3 by 3 matrix from a Gaussian lowpass filter matrix with σ = 0 . 4 . In MATLAB, B = f s p e c i a l ( ' g a u s s i a n ' , 3 , 0 . 4 ) . Time is scaled to increments of 0.25 ns.

[ A C H ] x , y t + 1 = i = - 1 1 j = - 1 1 β i , j [ A C H ] x + i , y + j t β i , j = B ( 2 + i , 2 + j )

The synaptic cleft was represented by a two-dimensional grid of length 100 nm. The width was varied from 20 to 60 nm to represent the changing distance in edematous conditions. Each cell (x, y) on the grid was 1 nm × 1 nm, and contained the concentration for that cell at a given time (t), represented in equation 2 as [ A C H ] x , y t . The initial concentration at each cell was 0 μ M , except at the neuron terminal, where the ACH was released by exocytosis at concentrations of 1000 μ M in the row of cells from [ A C H ] 45 , 2 to [ A C H ] 54 , 2 . Homogeneous Dirichlet boundary conditions were used at all boundaries except at the release site and receptor site, which were made impenetrable. These impenetrable boundaries were the row of cells from [ A C H ] 36 , 1 to [ A C H ] 63 , 1 (neuron release site) and [ A C H ] 36 , ζ to [ A C H ] 63 , ζ (membrane receptor site), where ζ was the synaptic cleft width.

Calcium control

ACH receptors influence IP3 levels, which affect cell potential, potassium channel probability, SR calcium, and intracellular calcium. The IP3 model was adapted from [link] and the remaining differential equations were adapted from [link] .

d [ I P 3 ] d t = [ A C H ] - ϵ [ I P 3 ] - V M 4 [ I P 3 ] u [ I P 3 ] u + K 4 u + P M V ( 1 - [ E ] r 2 ) K V r 2 + [ E ] r 2 d [ C a S R 2 + ] d t = B [ C a 2 + ] 2 [ C a 2 + ] 2 + C b 2 - C [ C a S R 2 + ] 2 [ C a 2 + ] 4 ( [ C a S R 2 + ] 2 + s c 2 ) ( [ C a 2 + ] 4 + c c 4 ) - L [ C a S R 2 + ] d [ W ] d t = λ ( [ C a 2 + ] + c W ) 2 ( [ C a 2 + ] + c W ) 2 + β e - ( [ E ] - v C a 3 R K ) - [ W ] d [ E ] d t = γ ( - F N a K - G C l ( [ E ] - v C l ) - 2 G C a ( [ E ] - v C a 1 ) 1 + e - [ E ] - v C a 2 R C a - G N C X [ C a 2 + ] ( [ E ] - v N C X ) [ C a 2 + ] + c N C X - G K [ W ] ( [ E ] - v K ) ) d [ C a 2 + ] d t = F [ I P 3 ] 2 [ I P 3 ] 2 + K r 2 - G C a ( [ E ] - v C a 1 1 + e - ( [ E ] - v C a 2 R C a ) + G N C X [ C a 2 + ] ( [ E ] - v N C X ) [ C a 2 + ] + c N C X - B [ C a 2 + ] 2 [ C a 2 + ] 2 + C b 2 + C [ C a S R 2 + ] 2 [ C a 2 + ] 4 ( [ C a S R 2 + ] 2 + s c 2 ) ( [ C a 2 + ] 4 + c c 4 ) - D [ C a 2 + ] ( 1 + ( [ E ] - v d ) R d + L [ C a S R 2 + ]
[ A C H ] 0 = 0 . 001 μ M Initial ACH concentration
[ I P 3 ] 0 = 0 . 49 μ M Initial IP3 concentration
[ C a S R 2 + ] 0 = 1 . 1 μ M Initial sarcoplasmic calcium concentration
[ W ] 0 = 0 . 02 Initial potassium channel probability concentration
[ E ] 0 = - 42 m V Initial cell potential
[ C a 2 + ] 0 = 0 . 17 μ M Initial intracellular calcium concentration
β = 0 . 13 μ M 2 translation factor [link]
γ = 197 m V μ M - 1 scaling factor [link]
ϵ = 0 . 015 s - 1 rate constant for linear IP3 [link]
λ = 45 channel constant [link]
B = 2 . 025 μ M s - 1 SR uptake rate constant [link]
C = 55 μ M s - 1 CICR rate constant [link]
D = 0 . 24 s - 1 Ca extrusion by ATPase constant [link]
F = 0 . 23 μ M s - 1 maximal influx rate [link]
L = 0 . 025 s - 1 leak from SR rate constant [link]
C b = 1 μ M half point SR ATPase activation [link]
c c = 0 . 9 μ M half point CICR activation [link]
c N C X = 0 . 5 μ M half point Na Ca exchange activation [link]
c W = 0 . 0 μ M translation factor [link]
F N a K = 0 . 0432 μ M s - 1 net whole cell flux [link]
G C a = 0 . 00129 μ M m V - 1 s - 1 whole cell conductance for VOCCs [link]
G C l = 0 . 00134 μ M m V - 1 s - 1 whole cell conductance Cl [link]
G K = 0 . 00446 μ M m V - 1 s - 1 whole cell conductance K [link]
G N C X = 0 . 00316 μ M m V - 1 s - 1 whole cell conductance for Na Ca exchange [link]
K 4 = 0 . 5 μ M half saturation constant IP3 degradation [link]
K r = 1 μ M half saturation constant Ca entry [link]
K V = - 58 m V half saturation constant IP3 voltage synthesis [link]
P M V = 0 . 01333 μ M s - 1 max rate voltage IP3 synthesis [link]
R 2 = 8 hill coefficient [link]
R C a = 8 . 5 m V maximum slope of VOCC activation [link]
R d = 250 . 0 m V slope of voltage dependence [link]
R K = 12 . 0 m V maximum slope Ca activation [link]
s c = 2 μ M half point CICR efflux [link]
u = 4 hill coefficient [link]
v C a 1 = 100 . 0 m V reversal potential VOCCs [link]
v C a 2 = - 24 m V half point VOCC activation [link]
v C a 3 = - 27 m V half point Ca channel activation [link]
v C l = - 25 m V reversal potential Cl [link]
v d = - 100 . 0 m V intercept voltage dependence [link]
v K = - 104 . 0 m V reversal potential K [link]
V M 4 = 0 . 0333 μ M s - 1 max nonlinear IP degradation [link]
v N C X = - 40 . 0 m V reversal potential Na Ca exchange [link]

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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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