General

Computational Biology, Part 20

Stochastic Modeling / Neuronal Modeling

Computational Biology, Part 20

Stochastic Modeling / Neuronal Modeling

Arvind Rao, Robert F. Murphy, Arvind Rao, Robert F. Murphy, Shann-Ching Chen, Justin Shann-Ching Chen, Justin

NewbergNewbergCopyright Copyright 2004-2009. 2004-2009.All rights reserved.All rights reserved.

Stochastic Modeling in Biology

Stochastic Modeling in Biology

Stochastic Modeling in BiologyStochastic Modeling in Biology

Why? A: Better Why? A: Better resolutionresolution in in species amountsspecies amounts

When? A: Biochemical When? A: Biochemical kineticskinetics, gene , gene expression stochasticity in cellsexpression stochasticity in cells

How? A: How? A: SSASSA Case studies:Case studies:

Chemical master equationChemical master equation Biochemical kineticsBiochemical kinetics Gene networksGene networks

Why?Why? Recall Chemical kinetics examplesRecall Chemical kinetics examples

1.1.In differential/difference In differential/difference equations, we examined regimes equations, we examined regimes where number of molecules of where number of molecules of reactants were always large reactants were always large enough to have followed mass enough to have followed mass action kinetics.action kinetics.

2.2.Think “Law of Large Numbers”Think “Law of Large Numbers”

Why Are Stochastic Models Needed?Why Are Stochastic Models Needed?• • Much of the mathematical modeling of Much of the mathematical modeling of biochemical/gene networks represents gene biochemical/gene networks represents gene expression deterministicallyexpression deterministically

• • Deterministic models describe Deterministic models describe macroscopic behavior; but many cellular macroscopic behavior; but many cellular constituents are present in small numbersconstituents are present in small numbers

• • Considerable experimental evidence Considerable experimental evidence indicates that significant stochastic indicates that significant stochastic fluctuations are presentfluctuations are present

• • There are many examples when There are many examples when deterministic models are deterministic models are not adequatenot adequate

Stochastic Chemical KineticsStochastic Chemical Kinetics

The Chemical Master EquationThe Chemical Master Equation

Challenges in the solution of CMEChallenges in the solution of CME

Exploiting Underlying BiologyExploiting Underlying Biology

Monte Carlo Simulations: Stochastic Simulation Algorithm

Monte Carlo Simulations: Stochastic Simulation Algorithm

Gillespie AlgorithmGillespie Algorithm Gillespie algorithm allows a discrete and Gillespie algorithm allows a discrete and stochastic simulation of a system with few stochastic simulation of a system with few reactants because every reaction is explicitly reactants because every reaction is explicitly simulated. simulated.

a Gillespie realization represents a random walk a Gillespie realization represents a random walk that exactly represents the distribution of the that exactly represents the distribution of the Master equation.Master equation.

The physical basis of the algorithm is the The physical basis of the algorithm is the collision of molecules within a reaction vessel collision of molecules within a reaction vessel (well mixed). (well mixed).

all reactions within the Gillespie framework must all reactions within the Gillespie framework must involve at most two molecules. Reactions involving involve at most two molecules. Reactions involving three molecules are assumed to be extremely rare three molecules are assumed to be extremely rare and are modeled as a sequence of binary reactions. and are modeled as a sequence of binary reactions.

1.1. InitializationInitialization: Initialize the number of : Initialize the number of molecules in the system, reactions constants, molecules in the system, reactions constants, and random number generators.and random number generators.

2.2. Monte Carlo StepMonte Carlo Step: Generate random numbers to : Generate random numbers to determine the next reaction to occur as well as determine the next reaction to occur as well as the time interval. The probability of a given the time interval. The probability of a given reaction to be chosen is proportional to the reaction to be chosen is proportional to the number of substrate molecules.number of substrate molecules.

3.3. UpdateUpdate: Increase the time step by the randomly : Increase the time step by the randomly generated time in Step 1. Update the molecule generated time in Step 1. Update the molecule count based on the reaction that occurred.count based on the reaction that occurred.

4.4. IterateIterate: Go back to Step 1 unless the number of : Go back to Step 1 unless the number of reactants is zero or the simulation time has reactants is zero or the simulation time has been exceeded.been exceeded.

Algorithm SummaryAlgorithm Summary

SSASSA

AdvantagesAdvantages Low memory requirementLow memory requirement Computation is not O(exp(N))Computation is not O(exp(N))

DisadvantagesDisadvantages Convergence is slowConvergence is slow Little insightLittle insight

ReferencesReferences Daniel T. Gillespie (1977). "Exact Stochastic Daniel T. Gillespie (1977). "Exact Stochastic

Simulation of Coupled Chemical Reactions". Simulation of Coupled Chemical Reactions". The The Journal of Physical ChemistryJournal of Physical Chemistry 8181 (25): 2340-2361. (25): 2340-2361. doi::10.1021/j100540a008..

Daniel T. Gillespie (2007). “Stochastic Simulation Daniel T. Gillespie (2007). “Stochastic Simulation of Chemical Kinetics". of Chemical Kinetics". Annu. Rev. Phys. Chem. Annu. Rev. Phys. Chem. 2007.58:35-55. 2007.58:35-55. 10.1146/annurev.physchem.58.032806.10463710.1146/annurev.physchem.58.032806.104637

D.Wilkinson (2009), “Stochastic modelling for D.Wilkinson (2009), “Stochastic modelling for quantitative description of heterogeneous biological quantitative description of heterogeneous biological systems.” Nature Reviews Genet. Feb 2009; systems.” Nature Reviews Genet. Feb 2009; 10(2):122-33. 10(2):122-33.

Slides adapted from: Slides adapted from: http://www.cds.caltech.edu/~murray/wiki/images/d/d9/Khammash_master-15aug06.pdf

Gillespie: Gillespie: http://en.wikipedia.org/wiki/Gillespie_algorithm

Neuronal Modeling: The Hodgkin Huxley Equations Neuronal Modeling: The Hodgkin Huxley Equations

Basic neurophysiologyBasic neurophysiology

An imbalance of charge across a An imbalance of charge across a membrane is called a membrane is called a membrane membrane potentialpotential

The major contribution to membrane The major contribution to membrane potential in animal cells comes from potential in animal cells comes from imbalances in small ions (e.g., Na, imbalances in small ions (e.g., Na, K)K)

The maintenance of this imbalance is The maintenance of this imbalance is an an activeactive process carried out by ion process carried out by ion pumpspumps


Ion pumps Ion pumps require energy (ATP) require energy (ATP) to carry ions across a membrane to carry ions across a membrane upup a concentration gradient a concentration gradient (they (they generate generate a potential)a potential)

Ion channels Ion channels allow ions to flow allow ions to flow across a membrane across a membrane downdown a a concentration gradient (they concentration gradient (they dissipatedissipate a potential) a potential)

Basic neurophysiologyBasic neurophysiology Example electrochemical Example electrochemical gradients (left)gradients (left)

Example ion channel (right)Example ion channel (right)Johnston & Wu, Johnston & Wu, Foundations of Foundations of Cellular NeurophysiologyCellular Neurophysiology, 5, 5thth ed. ed.


The cytoplasm of most cells The cytoplasm of most cells (including neurons) has an excess (including neurons) has an excess of negative ions over positive ions of negative ions over positive ions (due to active pumping of sodium (due to active pumping of sodium ions out of the cell)ions out of the cell)

By convention this is referred to By convention this is referred to as a as a negative membrane potential negative membrane potential (inside minus outside)(inside minus outside)

Typical Typical resting potential resting potential is -50 mVis -50 mV

Basic neuro-physiology

Basic neuro-physiology

An idealized An idealized neuronneuron consists of consists of somasoma or or cell bodycell body

contains nucleus and performs metabolic contains nucleus and performs metabolic functionsfunctions

dendritesdendrites receive signals from other neurons receive signals from other neurons through through synapsessynapses

axonaxon propagates signal away from somapropagates signal away from soma

terminal branchesterminal branches form form synapsessynapses with other neurons with other neurons


The junction between the soma and The junction between the soma and the axon is called the the axon is called the axonaxon hillockhillock

The soma sums (“integrates”) The soma sums (“integrates”) currents (“inputs”) from the currents (“inputs”) from the dendritesdendrites

When the received currents result in When the received currents result in a sufficient change in the membrane a sufficient change in the membrane potential, a rapid depolarization is potential, a rapid depolarization is initiated in the axon hillockinitiated in the axon hillock

Basic neuro-physiologyBasic neuro-physiology

Electrical signals regulate local Electrical signals regulate local calciumcalcium concentrations concentrations

Synaptic vesiclesSynaptic vesicles fuse with the axon fuse with the axon membrane, and membrane, and neurotransmittersneurotransmitters are are released into the space between axon released into the space between axon and dendrite in a process mediated by and dendrite in a process mediated by calcium ionscalcium ions

Binding of neurotransmitters to Binding of neurotransmitters to dendrite causes influx of sodium ions dendrite causes influx of sodium ions that diffuse into soma that diffuse into soma

Basic electrophysiologyBasic electrophysiology A cell is said to be electrically A cell is said to be electrically polarizedpolarized when it has a non-zero when it has a non-zero membrane potentialmembrane potential

A dissipation (partial or total) A dissipation (partial or total) of the membrane potential is of the membrane potential is referred to as a referred to as a depolarizationdepolarization, , while restoration of the resting while restoration of the resting potential is termed potential is termed repolarizationrepolarization

Action potential in neurons Action potential in neurons http://highered.mcgraw-hill.com/olc/dl/http://highered.mcgraw-hill.com/olc/dl/

120107/bio_d.swf120107/bio_d.swf http://bcs.whfreeman.com/thelifewire/http://bcs.whfreeman.com/thelifewire/

content/chp44/4402002.htmlcontent/chp44/4402002.html

Sodium (Na+) (Na+)

Potassium (K+)K+)insideinside

outsideoutside

Cell MembraneCell Membrane

timetime

Voltage difference (inside – outside)

Voltage difference (inside – outside)

Action potential is linked to ion channel conductances

Action potential is linked to ion channel conductances

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(m

S/c

m2) G(Na)

G(K)

0

50

100

150

Sti

mulu

s (u

A)

G is channel G is channel conductance.conductance.

High High conductance conductance allows for allows for ions to pass ions to pass through through channel channel easier.easier.

The Hodgkin-Huxley modelThe Hodgkin-Huxley model

Based on Based on electrophysiologicaelectrophysiological measurements of l measurements of giant squid axongiant squid axon

Empirical model Empirical model that predicts that predicts experimental data experimental data with very high with very high degree of accuracydegree of accuracy

Provides insight Provides insight into mechanism of into mechanism of action potentialaction potential

http://www.mun.ca/biology/desmid/http://www.mun.ca/biology/desmid/brian/BIOL2060/BIOL2060-13/1310.jpgbrian/BIOL2060/BIOL2060-13/1310.jpg

The Hodgkin-Huxley modelThe Hodgkin-Huxley model DefineDefine

v(t) v(t) voltage across the membrane at time voltage across the membrane at time tt

q(t) q(t) net charge inside the neuron at net charge inside the neuron at tt I(t) I(t) current of positive ions into neuron current of positive ions into neuron at at tt

g(v) g(v) conductance of membrane at voltage conductance of membrane at voltage vv CC capacitance of the membranecapacitance of the membrane Subscripts Na, K and L used to denote Subscripts Na, K and L used to denote specific currents or conductances specific currents or conductances (L=“other”)(L=“other”)

(I(INaNa , I , IKK , I , ILL ) )(g(gNaNa , g , gKK , g , gLL ) )

The Hodgkin-Huxley modelThe Hodgkin-Huxley model

Note:Note:

Conductance Conductance is is 1/R1/R, , where where RR is is resistanceresistance

EE indicates indicates membrane membrane potential, potential, EExx are are equilibrium equilibrium potentialspotentials

Experiments Experiments show only show only ggNaNa and and ggKK vary with vary with time when time when stimulus is stimulus is appliedapplied

The Hodgkin-Huxley modelThe Hodgkin-Huxley model Start with equation for Start with equation for capacitorcapacitor

v(t ) q( t )

C

The Hodgkin-Huxley modelThe Hodgkin-Huxley model Consider each ion separately Consider each ion separately and sum currents to get rate and sum currents to get rate of change in charge and hence of change in charge and hence voltagevoltage

)())(())((1

)(

)(

)(

)(

LLKKNaNa

LLL

KKK

NaNaNa

LKNa

vvgvvvgvvvgCdt

dv

vvgI

vvgI

vvgI

IIIdt

dq

The Hodgkin-Huxley modelThe Hodgkin-Huxley model Central concept of model: Central concept of model: Define three state variables Define three state variables that represent (or “control”) that represent (or “control”) the opening and closing of the opening and closing of ion channelsion channels mm controls Na channel opening controls Na channel opening hh controls Na channel closing controls Na channel closing nn controls K channel opening controls K channel opening

The Hodgkin-Huxley modelThe Hodgkin-Huxley model Define relationship of state Define relationship of state variables to conductances of variables to conductances of Na and KNa and Kg Na g Na m 3h

gK g K n 4

0 m, n, h 1

mm nnhh

Q: How Q: How were the were the powers powers determinedetermined? d?

A: Smart A: Smart guessingguessing

The Hodgkin-Huxley modelThe Hodgkin-Huxley model Define empirical differential Define empirical differential equations to model behavior equations to model behavior of each gateof each gate

dn

dtn (v)(1 n) n (v)n

n (v)0.01(v 10)

(e(v10)/10 1)

n (v)0.125ev / 80


dm

dtm (v)(1 m) m (v)m

m (v)0.1(v 25)

(e(v25)/10 1)

m (v)4ev /18


dh

dth (v)(1 h) h (v)h

h (v)0.07ev / 20

h (v) 1(e(v30)/10 1)

The Hodgkin-Huxley modelThe Hodgkin-Huxley model Gives set of four coupled, Gives set of four coupled, non-linear, ordinary non-linear, ordinary differential equationsdifferential equations

Must be integrated Must be integrated numericallynumerically

Constants (Constants (g g in mmho/cmin mmho/cm22 and and vv in mV) in mV)

gNa 120

gK 36gL 0.3

vNa 115vK 12vL 10.5989

Hodgkin-Huxley gatesHodgkin-Huxley gates

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Gate

para

m

valu

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m gate (Na)

h gate (Na)

n gate (K)

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G(K)

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mulu

s (u

A)

Interactive demonstrationInteractive demonstration (Integration of Hodgkin-(Integration of Hodgkin-Huxley equations using Maple)Huxley equations using Maple)

Interactive demonstrationInteractive demonstration

> Ena:=55: Ek:=-82: El:= -59: gkbar:=24.34: gnabar:=70.7: > Ena:=55: Ek:=-82: El:= -59: gkbar:=24.34: gnabar:=70.7: > gl:=0.3: vrest:=-69: cm:=0.001:> gl:=0.3: vrest:=-69: cm:=0.001:> alphan:=v-> 0.01*(10-(v-vrest))/(exp(0.1*(10-(v-vrest)))-> alphan:=v-> 0.01*(10-(v-vrest))/(exp(0.1*(10-(v-vrest)))-

1):1):> betan:=v-> 0.125*exp(-(v-vrest)/80):> betan:=v-> 0.125*exp(-(v-vrest)/80):> alpham:=v-> 0.1*(25-(v-vrest))/(exp(0.1*(25-(v-vrest)))-1):> alpham:=v-> 0.1*(25-(v-vrest))/(exp(0.1*(25-(v-vrest)))-1):> betam:=v-> 4*exp(-(v-vrest)/18):> betam:=v-> 4*exp(-(v-vrest)/18):> alphah:=v->0.07*exp(-0.05*(v-vrest)):> alphah:=v->0.07*exp(-0.05*(v-vrest)):> betah:=v->1/(exp(0.1*(30-(v-vrest)))+1):> betah:=v->1/(exp(0.1*(30-(v-vrest)))+1):> pulse:=t->-20*(Heaviside(t-.001)-Heaviside(t-.002)):> pulse:=t->-20*(Heaviside(t-.001)-Heaviside(t-.002)):> rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) +> rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) +> > gkbar*n^4*(V-Ek) + gl*(V-gkbar*n^4*(V-Ek) + gl*(V-

El)+ pulse(t))/cm:El)+ pulse(t))/cm:> rhsn:=(t,V,n,m,h)-> 1000*(alphan(V)*(1-n) - betan(V)*n):> rhsn:=(t,V,n,m,h)-> 1000*(alphan(V)*(1-n) - betan(V)*n):> rhsm:=(t,V,n,m,h)-> 1000*(alpham(V)*(1-m) - betam(V)*m):> rhsm:=(t,V,n,m,h)-> 1000*(alpham(V)*(1-m) - betam(V)*m):> rhsh:=(t,V,n,m,h)-> 1000*(alphah(V)*(1-h) - betah(V)*h):> rhsh:=(t,V,n,m,h)-> 1000*(alphah(V)*(1-h) - betah(V)*h):


> inits:=V(0)=vrest,n(0)=0.315,m(0)=0.042, h(0)=0.608;> inits:=V(0)=vrest,n(0)=0.315,m(0)=0.042, h(0)=0.608;> sol:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)),> sol:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)),diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)),diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits},diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric, {V(t),n(t),m(t),h(t)},type=numeric,

output=listprocedure);output=listprocedure);> Vs:=subs(sol,V(t));> Vs:=subs(sol,V(t));> plot(Vs,0..0.02);> plot(Vs,0..0.02);> sol20:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)),> sol20:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)), diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)),diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)), diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)),diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)), diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits},diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits}, {V(t),n(t),m(t),h(t)},type=numeric);{V(t),n(t),m(t),h(t)},type=numeric);> with(plots):> with(plots):


> > J:=odeplot(sol20,[V(t),n(t)],0..0.02):J:=odeplot(sol20,[V(t),n(t)],0..0.02):

> display({J});> display({J});

> pulse:=t->-2*(Heaviside(t-.001)-Heaviside(t-.002)):> pulse:=t->-2*(Heaviside(t-.001)-Heaviside(t-.002)):

> rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) +> rhsV:=(t,V,n,m,h)->-(gnabar*m^3*h*(V-Ena) +

gkbar*n^4*(V-Ek) + gl*(V-El)+ pulse(t))/cm:gkbar*n^4*(V-Ek) + gl*(V-El)+ pulse(t))/cm:

> sol2:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)),> sol2:=dsolve({diff(V(t),t)=rhsV(t,V(t),n(t),m(t),h(t)),

diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)),diff(n(t),t)=rhsn(t,V(t),n(t),m(t),h(t)),

diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)),diff(m(t),t)=rhsm(t,V(t),n(t),m(t),h(t)),

diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits},diff(h(t),t)=rhsh(t,V(t),n(t),m(t),h(t)),inits},

{V(t),n(t),m(t),h(t)},type=numeric);{V(t),n(t),m(t),h(t)},type=numeric);

> K:=odeplot(sol2,[V(t),n(t)],0..0.02,color=green):> K:=odeplot(sol2,[V(t),n(t)],0..0.02,color=green):

> display({J,K});> display({J,K});


> L:=odeplot(sol20,[V(t),n(t)],0..0.02,numpoints=400,> L:=odeplot(sol20,[V(t),n(t)],0..0.02,numpoints=400, color=blue):color=blue):> display({J,L});> display({J,L});> odeplot(sol20,[V(t),m(t)],0..0.02,numpoints=400);> odeplot(sol20,[V(t),m(t)],0..0.02,numpoints=400);> odeplot(sol20,[V(t),h(t)],0..0.02,numpoints=400);> odeplot(sol20,[V(t),h(t)],0..0.02,numpoints=400);> odeplot(sol20,[m(t),h(t)],0..0.02,numpoints=400);> odeplot(sol20,[m(t),h(t)],0..0.02,numpoints=400);> a:=0.7; b:=0.8; c:=0.08;> a:=0.7; b:=0.8; c:=0.08;> rhsx:=(t,x,y)->x-x^3/3-y;> rhsx:=(t,x,y)->x-x^3/3-y;> rhsy:=(t,x,y)->c*(x+a-b*y);> rhsy:=(t,x,y)->c*(x+a-b*y);> sol2:=dsolve({diff(x(t),t)=rhsx(t,x(t),y(t)),> sol2:=dsolve({diff(x(t),t)=rhsx(t,x(t),y(t)), diff(y(t),t)=rhsy(t,x(t),y(t)),x(0)=0,y(0)=-1},diff(y(t),t)=rhsy(t,x(t),y(t)),x(0)=0,y(0)=-1}, {x(t),y(t)},type=numeric, output=listprocedure);{x(t),y(t)},type=numeric, output=listprocedure);> xs:=subs(sol2,x(t)); ys:=subs(sol2,y(t));> xs:=subs(sol2,x(t)); ys:=subs(sol2,y(t));> K:=plot([xs,ys,0..200],x=-3..3,y=-2..2,color=blue):> K:=plot([xs,ys,0..200],x=-3..3,y=-2..2,color=blue):> J:=plot({[V,(V+a)/b,V=-2.5..1.5],[V,V-V^3/3,V=-2.5..2.2]}):> J:=plot({[V,(V+a)/b,V=-2.5..1.5],[V,V-V^3/3,V=-2.5..2.2]}):> plots[display]({J,K});> plots[display]({J,K});

Virtual Cell - Hodgkin-HuxleyVirtual Cell - Hodgkin-Huxley Versions of the models in Versions of the models in “Computational cell biology” by Fall “Computational cell biology” by Fall et al have been implemented in et al have been implemented in Virtual CellVirtual Cell

These are available as Public modelsThese are available as Public models The The Hodgkin-Huxley ModelHodgkin-Huxley Model is a is a scientific model that describes how scientific model that describes how action potentials in neurons are action potentials in neurons are initiated and propagatedinitiated and propagated

Within Virtual Cell, use Within Virtual Cell, use File/Open/BiomodelFile/Open/Biomodel

Then open Shared Then open Shared Models/CompCell/Hodgkin-HuxleyModels/CompCell/Hodgkin-Huxley

Biochemical Species• Potassium• Sodium• Potassium Channel Inactivation Gate-closed "n_c" • Potassium Channel Inactivation Gate-open "n_o" • Sodium Channel Inactivation Gate-closed "h_c" • Sodium Channel Inactivation Gate-open "h_o" • Sodium Channel Activation Gate-open "m_o" • Sodium Channel Activation Gate-closed "m_c"

Compartments• Extracellular• Plasma Membrane• Cytosol

m_o: : Sodium Channel Activation Gate-open

Technology

General