Condition-Specific Parameters

As discussed in this extended tutorial, sometimes a subset of the model parameters to estimate have different values across experimental/simulation conditions. For such models, runtime can drastically improve by setting split_over_conditions=true when creating a PEtabODEProblem. This example explores this option in more detail, and it assumes that that you are familiar with condition-specific parameters in PEtab (see this tutorial) and with the gradient methods in PEtab.jl (see this page).

As a working example, we use a published signaling model referred to as the Beer model after the first author [24]. The Beer model is available in the PEtab standard format (a tutorial on importing problems in the standard format can be found here), and the PEtab files for this model can be downloaded from here. Given the problem YAML file, we can import the problem as:

using PEtab
path_yaml = joinpath(@__DIR__, "assets", "beer", "Beer_MolBioSystems2014.yaml")
model = PEtabModel(path_yaml)

Efficient Handling of Condition-Specific Parameters

The Beer problem is a small model with 4 species and 9 parameters in the ODE system, but there are 72 parameters to estimate. This is because most parameters are specific to a subset of simulation conditions. For example, cond1 has a parameter τ_cond1, and cond2 has τ_cond2, which map to the ODE model parameter τ, respectively. This can be seen by printing some model statistics:

using Catalyst
petab_prob = PEtabODEProblem(model)
println("Number of ODE model species = ", length(unknowns(model.sys)))
println("Number of ODE model parameters = ", length(parameters(model.sys)))
println("Number of parameters to estimate = ", length(petab_prob.xnames))

Number of ODE model species = 4
Number of ODE model parameters = 9
Number of parameters to estimate = 72

For small ODE models like the Beer model, the most efficient gradient method is gradient_method=:ForwardDiff, and it is often feasible to compute the Hessian using hessian_method=:ForwardDiff as well (see this page for details). Typically, with :ForwardDiff, PEtab.jl computes the gradient with a single call to ForwardDiff.gradient. However, for the Beer model, this approach is problematic because for each simulation condition, n forward passes are required to compute all derivatives, where n depends on the number of gradient parameters. Since many parameters only belong to a subset of conditions, actually only ni < n forward passes are needed for each condition. To this end, PEtab.jl provides the split_over_conditions=true keyword when building the PEtabODEProblem, which ensures that one ForwardDiff.gradient call is performed per simulation condition. Let us examine how this affects gradient runtime for the Beer model:

using Printf
petab_prob1 = PEtabODEProblem(model; split_over_conditions = true)
petab_prob2 = PEtabODEProblem(model; split_over_conditions = false)
x = get_x(petab_prob1)
g1, g2 = similar(x), similar(x)
b1 = @elapsed petab_prob1.grad!(g1, x)
b2 = @elapsed petab_prob2.grad!(g2, x)
@printf("Runtime split_over_conditions = true: %.2fs\n", b1)
@printf("Runtime split_over_conditions = false: %.2fs\n", b2)

Runtime split_over_conditions = true: 0.07s
Runtime split_over_conditions = false: 0.40s

For the Hessian, the difference in runtime is even larger:

h1, h2 = zeros(length(x), length(x)), zeros(length(x), length(x))
b1 = @elapsed petab_prob1.hess!(h1, x)
b2 = @elapsed petab_prob2.hess!(h2, x)
@printf("Runtime split_over_conditions = true: %.1fs\n", b1)
@printf("Runtime split_over_conditions = false: %.1fs\n", b2)

Runtime split_over_conditions = true: 0.6s
Runtime split_over_conditions = false: 25.7s

Given that split_over_conditions=true reduces runtime in the example above, a natural question is: why is it not the default option in PEtab.jl? This is because calling ForwardDiff.gradient for each simulation condition, instead of once for all conditions, introduces an overhead. Therefore, for models with none or very few condition-specific parameters, split_over_conditions=false is faster. Determining exactly how many condition-specific parameters are needed to make true the faster option is difficult. Currently, the default is to enable this option when the number of condition-specific parameters is at least twice the number of parameters to estimate in the ODE model. For the Beer model, this means split_over_conditions=true is set by default, but this is a rough heuristic. Therefore, for models like these, we recommend benchmarking the two configurations to determine which is fastest.

References