Accurate particle-based simulation of adsorption, desorption and partial transmission

Abstract

Particle-based simulators represent molecules of interest with point-like particles that diffuse and react in continuous space. These simulators are often used to investigate spatial or stochastic aspects of biochemical systems. This paper presents new particle-based simulation algorithms for modeling interactions between molecules and surfaces; they address irreversible and reversible molecular adsorption to, desorption from and transmission through membranes. Their central elements are: (i) relationships between adsorption, desorption and transmission coefficients on the one hand, and simulator interaction probabilities on the other, and (ii) probability densities for initial placements of desorbed molecules. These algorithms, which were implemented and tested in the Smoldyn simulator, are accurate, easy to implement and computationally efficient. They allow longer time steps and better address reversible processes than an algorithm that Erban and Chapman recently presented (Physical Biology 4:16-28, 2007). This paper also presents a method for simulating unbounded diffusion in a limited spatial domain using a partially absorbing boundary, as well as new solutions to the diffusion differential equation with reversible Robin boundary conditions.

Figure 1

(A) Model system for adsorption and desorption. Molecules, shown with dots, adsorb to the grey surface with adsorption coefficient κ and desorb with rate constant k. (B) Model system for partial transmission. Molecules transmit from the front side of the surface (x > 0) to the back side (x < 0) with permeability coefficient κ_F and transmit from back to front with permeability coefficient κ_B.

Figure 2

Particle-based simulator design assumed throughout this work. The probability, P, refers to the adsorption, desorption, or partial transmission probability, as appropriate, and “rand()” is a uniformly distributed random number between 0 and 1. This article presents molecule-surface interaction probabilities (P) with which simulators following this design will adsorb, desorb, and transmit molecules at the correct rates.

Figure 3

Steady-state concentration profiles for model systems that (A) reversibly or irreversibly adsorb molecules to a surface or (B) reversibly transmit molecules through a surface. The grey line at x = 0 represents the surface.

Figure 4

Steady-state concentration profiles at different steps of the simulation process. These apply to (A) irreversible adsorption and (B) reversible adsorption. Solid lines represent the profiles at both the beginning and end of time steps, long-dashed lines represent the profiles after the simulator's Step 1, and the short-dashed line in Panel B represents the profile after the reflection and adsorption portions of Step 2, but before desorption (this situation doesn't apply to Panel A). Areas on the figure represent molecular populations, as labeled. The grey line at x = 0 represents the surface. Panel (A) results were computed with the simulation emulator (see the main text and Supplementary Information) and Panel (B) with Mathematica software [44].

Figure 5

Relationships between transition coefficients and transition probabilities, and initial separations between surfaces and desorbed molecules. (A) Relationship between κ′ and P_a for irreversible adsorption. The solid black line, calculated by the simulation emulator and approximated by Eq. 21, enables accurate adsorption simulation. Points with one standard deviation error bars represent results from stochastic simulations of a steady-state system (see Supplementary Information for details). (B) Probability densities for the initial separation between a desorbed molecule and the surface that it desorbed from. The solid line represents the density that should be used for irreversible desorption (Eq. 25) and the dashed line represents the density for reversible adsorption (Eq. 33). (C) Relationships between P_a (solid black lines), P_d (dashed red lines), κ′ (abscissa), and k′ (marked on figure) for reversible adsorption; these are also given in Eqs. 32 and 37. The k′ = 0 desorption probability line is coincident with the abscissa. (D) Relationships between P_tF (solid black lines), P_tB (dashed red lines), κ_F′ (abscissa), and κ_B′ (marked on figure) for reversible transmission; these are also given in Eqs. 47 and 48. The κ_B′ = 0 transmission probability line is coincident with the abscissa. The green bold long-dashed line presents the transmission probabilities for the condition that κ_B′ = κ_F′. Where present, the dotted blue line represents the relationship given by Eq. 1.

Figure 6

Algorithm tests for systems that started with well-mixed states. Dots represent stochastic simulation results and lines represent predictions for the corresponding model systems. (A) Number of molecules adsorbed to a surface or transmitted through the surface, as appropriate, in tests of irreversible adsorption (IA, circles), reversible adsorption (RA, triangles), or reversible transmission (RT, diamonds). Upper black points are for “fast” rates: κ = 85.9 μm/s for irreversible and reversible adsorption (P_a = 1), k = 276 s⁻¹ for reversible adsorption (P_d = 0.128), and κ_F = κ_B = 5045 μm/s for reversible transmission (P_tF = P_tB ≈ 1). Lower red points are for “slow” rates: κ = 4.23 μm/s for irreversible and reversible adsorption (P_a = 0.1), k = 28 s⁻¹ for reversible adsorption (P_d = 0.026), and κ_F = κ_B = 4.36 μm/s for reversible transmission (P_tF = P_tB = 0.1). Every third simulation data point is shown for clarity. (B) Concentration profiles at several time points for the irreversible adsorption test with the “fast” rate. Black circles represent time 0, red triangles represent time 1 ms (1 time step), green diamonds represent time 10 ms, and blue squares represent time 100 ms. See Supplementary Information for details.

Figure 7

Simulation of effective unbounded diffusion with a partially absorbing boundary. (A) a 0.5 μm thick slice, taken parallel to the x,y-plane, through the middle of a cubical simulated system. Large dots represent molecular emitters, small dots represent molecules, and the grey band across the system (5 μm long by 0.5 μm wide and thick) represents the volume in which molecules were counted. Each face of the system is assembled from a grid of 25 panels, each with an absorption coefficient from Eq. 54; panels with larger absorption coefficients are depicted with thicker boundary lines. (B) Dots represent simulated molecule concentrations, measured along the grey band from panel A, at steady-state and averaged over 1000 time steps. Dashed lines represent theoretical concentrations from each individual source, along the grey band, for the unbounded model system. The solid line represents the sum of the dashed lines, which is the total theoretical concentration for the unbounded model system. See Supplementary Information for details.