8.8 Exploring the biochemical and mechanical effects of intestinal (Page 4/6)

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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] .

\begin{matrix} {[A C H]}^{t} & = α ({[A C H]}_{x x} + {[A C H]}_{y y}) & α = 0.4 n m^{2} n s^{- 1} \end{matrix}

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.

\begin{matrix} {[A C H]}_{x, y}^{t + 1} & = \sum_{i = - 1}^{1} \sum_{j = - 1}^{1} β_{i, j} {[A C H]}_{x + i, y + j}^{t} & β_{i, j} = B (2 + i, 2 + j) \end{matrix}

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 $\times$ 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] .

\begin{matrix} \frac{d [I P 3]}{d t} & = [A C H] - ϵ [I P 3] - \frac{V_{M 4} {[I P 3]}^{u}}{{[I P 3]}^{u} + K_{4}^{u}} + \frac{P_{M V} (1 - {[E]}^{r_{2}})}{K_{V}^{r_{2}} + {[E]}^{r_{2}}} \\ \frac{d [C a_{S R}^{2^{+}}]}{d t} & = \frac{B {[C a^{2^{+}}]}^{2}}{{[C a^{2^{+}}]}^{2} + C_{b}^{2}} - \frac{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^{+}}] \\ \frac{d [W]}{d t} & = \frac{λ {([C a^{2^{+}}] + c_{W})}^{2}}{{([C a^{2^{+}}] + c_{W})}^{2} + β e^{- (\frac{[E] - v_{C a_{3}}}{R_{K}})}} - [W] \\ \frac{d [E]}{d t} & = γ (- F_{N a K} - G_{C l} ([E] - v_{C l}) - \frac{2 G_{C a} ([E] - v_{C a_{1}})}{1 + e^{- (\frac{[E] - v_{C a_{2}}}{R_{C a}})}} \\ - G_{N C X} \frac{[C a^{2^{+}}] ([E] - v_{N C X})}{[C a^{2^{+}}] + c_{N C X}} - G_{K} [W] ([E] - v_{K})) \\ \frac{d [C a^{2^{+}}]}{d t} & = \frac{F {[I P 3]}^{2}}{{[I P 3]}^{2} + K_{r}^{2}} - \frac{G_{C a} ([E] - v_{C a_{1}}}{1 + e^{- (\frac{[E] - v_{C a_{2}}}{R_{C a}})}} \\ + \frac{G_{N C X} [C a^{2^{+}}] ([E] - v_{N C X})}{[C a^{2^{+}}] + c_{N C X}} \\ - \frac{B {[C a^{2^{+}}]}^{2}}{{[C a^{2^{+}}]}^{2} + C_{b}^{2}} + \frac{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})} \\ - \frac{D [C a^{2^{+}}] (1 + ([E] - v_{d})}{R_{d}} + L [C a_{S R}^{2^{+}}] \end{matrix}

${[A C H]}_{0} =$	$0.001 μ M$	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]

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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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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