MultiScale Modeling of NeuroMuscular Junction (NMJ)


1.
Objective In this project, we want to construct
multiscale models for the neuromuscular junction
(NMJ) system, and calculate the electrostatic potential and diffusion rate constant
by solving corresponding partial differential equations (PossionBoltzmann
equation and Smoluchowski equation). http://fig.cox.miami.edu/~cmallery/150/neuro/neuromuscularsml.jpg 

2.
Modeling We first construct volumetric data from PDB
data or the input geometry, then use our LBIE_Mesher
software (Level Set Boundary Interior and Exterior Mesher)
to generate adaptive and quality tetrahedral meshes for each components in
the NMJ system. 2.1 Membrane 

Interior mesh (224475 vertices, 1077728 tetrahedra) 
Exterior mesh (74299 vertices, 374524 tetrahedra) 

2.2
AChBP (1I9B) – the top part of AChR 

Blobbiness
= 0.5, interior mesh (106971 vertices, 527438 tetra), exterior mesh (113528
vertices, 559670 tetra). 

Blobbiness
= 0.1, interior mesh (77110 vertices, 381280 tetra), exterior mesh (109438
vertices, 560535 tetra). (download
interior/exterior tetra meshes) 

2.3
AChR – Receptor 



Blobbiness = 0.5 



Blobbiness = 0.1 

2.4
A model of an AChR
and membrane (small local region) within a sphere 

(1) AChR 
(2) Membrane 
(3) Exterior 



3. Simulation 

3.1 PossionBoltzmann Equation – to calculate the electrostatic potential The PossionBoltzmann equation (PBE) determines a dimensionless potential u(x) = e_{c}Ф(x)/(k_{B}T) around a charged biological structure immersed in a salt solution, where Ф(x) is the electrostatic potential at , with d = 2 or d = 3.For a 1:1 electrolyte, the PBE can be written as Where
References: 1. Holst M, Baker N, Wang F. Adaptive multilevel finite
element solution of the PoissonBoltzmann equation
I: algorithms and examples. J. Comput. Chem.
21, 13191342, 2000. 2. Baker N, Holst M, Wang F.
Adaptive multilevel finite element solution of the PoissonBoltzmann equation II: refinement schemes based on
solvent accessible surfaces. J. Comput. Chem.
21, 13431352, 2000. 3.2 Smoluchowski Equation – to calculate the diffusioninfluenced biomolecular reaction rate constant The Smoluchowki equation describes the overdamped dynamics of multiple particles while neglecting interparticle interactions. For a stationary diffusion process, the Smoluchowski equation has the steadystate form of Where Lp(x) represents (dp(x, t)/dt) (t is the time), p(x) is the distribution function of the reactants, D(x) is the diffusion coefficient, β = 1/kT is the inverse Boltzmann energy, k is the Boltzmann constant, T is the temperature, and W(x) is the potential mean force (PMF) for the diffusing particle.
References: 1.
Y. Song, Y.
Zhang, T. Shen, C. Bajaj,
J. McCammon, N. Baker. Finite Element Solution of the Steadystate Smoluchowski Equation for Rate Constant Calculations.
Biophysical Journal, 86(4):20172029, 2004. 2.
Y. Song, Y.
Zhang, C. Bajaj, N. Baker. Continuum Diffusion Reaction Rate Calculations of
Wild Type and Mutant Mouse Acetylcholinesterase:
Adaptive Finite Element Analysis. Biophysical Journal
87(3):15581566, 2004. 3.
D. Zhang, J. Suen, Y. Zhang, Y. Song, Z. Radic,
P. Taylor, M. J. Holst, C. Bajaj,
N. A. Baker, J. A. McCammon. Tetrameric Mouse Acetylcholinesterase:
Continuum Diffusion Rate Calculations by Solving the SteadyState Smoluchowski Equation Using Finite Element Methods.
Biophysical Journal 88(3):16591665, 2005. 3.3 Particle Diffusion Equation  to model the diffusion of neurotransmitters across the synaptic cleft The Reaction Diffusion Equation is used to model the diffusion of particles across a domain. This is given by the following equation with boundary and initial conditions: Where: C (x,y,z,t) = concentration of the Neurotransmitters at a given time C0 = initial concentration at time t=0 n(x,y,z) = surface normal kappa = specific reactivity A = Diffusion Reaction Coefficient 


