This tutorial shows how to programmatically construct a
ReactionSystem corresponding to the chemistry underlying the Smoluchowski coagulation model using ModelingToolkit/Catalyst. A jump process version of the model is then constructed from the
ReactionSystem, and compared to the model's analytical solution obtained by the method of Scott (see also 3).
The Smoluchowski coagulation equation describes a system of reactions in which monomers may collide to form dimers, monomers and dimers may collide to form trimers, and so on. This models a variety of chemical/physical processes, including polymerization and flocculation.
We begin by importing some necessary packages.
using ModelingToolkit, Catalyst, LinearAlgebra using DiffEqBase, DiffEqJump using Plots, SpecialFunctions
Suppose the maximum cluster size is
N. We assume an initial concentration of monomers,
Nₒ, and let
uₒ denote the initial number of monomers in the system. We have
nr total reactions, and label by
V the bulk volume of the system (which plays an important role in the calculation of rate laws since we have bimolecular reactions). Our basic parameters are then
## Parameter N = 10 # maximum cluster size Vₒ = (4π/3)*(10e-06*100)^3 # volume of a monomers in cm³ Nₒ = 1e-06/Vₒ # initial conc. = (No. of init. monomers) / bulk volume uₒ = 10000 # No. of monomers initially V = uₒ/Nₒ # Bulk volume of system in cm³ integ(x) = Int(floor(x)) n = integ(N/2) nr = N%2 == 0 ? (n*(n + 1) - n) : (n*(n + 1)) # No. of forward reactions
The Smoluchowski coagulation equation Wikipedia page illustrates the set of possible reactions that can occur. We can easily enumerate the
pairs of multimer reactants that can combine when allowing a maximal cluster size of
N monomers. We initialize the volumes of the reactant multimers as
# possible pairs of reactant multimers pair =  for i = 2:N push!(pair,[1:integ(i/2) i .- (1:integ(i/2))]) end pair = vcat(pair...) vᵢ = @view pair[:,1] # Reactant 1 indices vⱼ = @view pair[:,2] # Reactant 2 indices volᵢ = Vₒ*vᵢ # cm⁻³ volⱼ = Vₒ*vⱼ # cm⁻³ sum_vᵢvⱼ = @. vᵢ + vⱼ # Product index
We next specify the rates (i.e. kernel) at which reactants collide to form products. For simplicity, we allow a user-selected additive kernel or constant kernel. The constants(
C) are adopted from Scott's paper 2
# set i to 1 for additive kernel, 2 for constant i = 1 if i==1 B = 1.53e03 # s⁻¹ kv = @. B*(volᵢ + volⱼ)/V # dividing by volume as its a bi-molecular reaction chain elseif i==2 C = 1.84e-04 # cm³ s⁻¹ kv = fill(C/V, nr) end
We'll store the reaction rates in
Pairs, and set the initial condition that only monomers are present at $t=0$ in
# state variables are X, pars stores rate parameters for each rx @parameters t @variables k[1:nr] X[collect(1:N)](t) pars = Pair.(k, kv) # time-span if i == 1 tspan = (0. ,2000.) elseif i == 2 tspan = (0. ,350.) end # initial condition of monomers u₀ = zeros(Int64, N) u₀ = uₒ u₀map = Pair.(X, u₀) # map variable to its initial value
Here we generate the reactions programmatically. We systematically create Catalyst
Reactions for each possible reaction shown in the figure on Wikipedia. When
vᵢ[n] == vⱼ[n], we set the stoichiometric coefficient of the reactant multimer to two.
# vector to store the Reactions in rx =  for n = 1:nr # for clusters of the same size, double the rate if (vᵢ[n] == vⱼ[n]) push!(rx, Reaction(k[n], [X[vᵢ[n]]], [X[sum_vᵢvⱼ[n]]], , )) else push!(rx, Reaction(k[n], [X[vᵢ[n]], X[vⱼ[n]]], [X[sum_vᵢvⱼ[n]]], [1, 1], )) end end rs = ReactionSystem(rx, t, X, k)
We now convert the
ReactionSystem into a
JumpSystem, and solve it using Gillespie's direct method. For details on other possible solvers (SSAs), see the DifferentialEquations.jl documentation
# solving the system jumpsys = convert(JumpSystem, rs) dprob = DiscreteProblem(jumpsys, u₀map, tspan, pars) jprob = JumpProblem(jumpsys, dprob, Direct(), save_positions=(false,false)) jsol = solve(jprob, SSAStepper(), saveat = tspan/30)
Lets check the results for the first three polymers/cluster sizes. We compare to the analytical solution for this system:
# Results for first three polymers...i.e. monomers, dimers and trimers v_res = [1;2;3] # comparison with analytical solution # we only plot the stochastic solution at a small number of points # to ease distinguishing it from the exact solution t = jsol.t sol = zeros(length(v_res), length(t)) if i == 1 ϕ = @. 1 - exp(-B*Nₒ*Vₒ*t) for j in v_res sol[j,:] = @. Nₒ*(1 - ϕ)*(((j*ϕ)^(j-1))/gamma(j+1))*exp(-j*ϕ) end elseif i == 2 ϕ = @. (C*Nₒ*t) for j in v_res sol[j,:] = @. 4Nₒ*((ϕ^(j-1))/((ϕ + 2)^(j+1))) end end # plotting normalised concentration vs analytical solution default(lw=2, xlabel="Time (sec)") scatter(ϕ, jsol(t)[1,:]/uₒ, label="X1 (monomers)", markercolor=:blue) plot!(ϕ, sol[1,:]/Nₒ, line = (:dot,4,:blue), label="Analytical sol--X1") scatter!(ϕ, jsol(t)[2,:]/uₒ, label="X2 (dimers)", markercolor=:orange) plot!(ϕ, sol[2,:]/Nₒ, line = (:dot,4,:orange), label="Analytical sol--X2") scatter!(ϕ, jsol(t)[3,:]/uₒ, label="X3 (trimers)", markercolor=:purple) plot!(ϕ, sol[3,:]/Nₒ, line = (:dot,4,:purple), label="Analytical sol--X3", ylabel = "Normalized Concentration")
For the additive kernel we find
- Scott, W. T. (1968). Analytic Studies of Cloud Droplet Coalescence I, Journal of Atmospheric Sciences, 25(1), 54-65. Retrieved Feb 18, 2021, from https://journals.ametsoc.org/view/journals/atsc/25/1/1520-046919680250054asocdc20co2.xml
- Ian J. Laurenzi, John D. Bartels, Scott L. Diamond, A General Algorithm for Exact Simulation of Multicomponent Aggregation Processes, Journal of Computational Physics, Volume 177, Issue 2, 2002, Pages 418-449, ISSN 0021-9991, https://doi.org/10.1006/jcph.2002.7017.