Multi-Scale Modeling of Neuro-Muscular Junction (NMJ)

 

1.     Objective

 

In this project, we want to construct multi-scale models for the neuro-muscular junction (NMJ) system, and calculate the electrostatic potential and diffusion rate constant by solving corresponding partial differential equations (Possion-Boltzmann equation and Smoluchowski equation).

 

http://fig.cox.miami.edu/~cmallery/150/neuro/neuromuscular-sml.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

(download tetra mesh)

(2) Membrane

(download tetra mesh)

(3) Exterior

(download tetra mesh)

 

 

 

3. Simulation

 

3.1 Possion-Boltzmann Equation to calculate the electrostatic potential

 

The Possion-Boltzmann equation (PBE) determines a dimensionless potential u(x) = ecФ(x)/(kBT) 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 Poisson-Boltzmann equation I: algorithms and examples. J. Comput. Chem. 21, 1319-1342, 2000.

2.      Baker N, Holst M, Wang F. Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II: refinement schemes based on solvent accessible surfaces. J. Comput. Chem. 21, 1343-1352, 2000.

 

3.2 Smoluchowski Equation to calculate the diffusion-influenced 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 steady-state 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 Steady-state Smoluchowski Equation for Rate Constant Calculations. Biophysical Journal, 86(4):2017-2029, 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):1558-1566, 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 Steady-State Smoluchowski Equation Using Finite Element Methods. Biophysical Journal 88(3):1659-1665, 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