Modeling dissociation using Hill function provides misleading results

I stumbled upon a simple dissociation model using Hill function. The process in question is

[A:B] -> A + 2 B

A, B, [A:B] species [nanomole]

Now, dependent on how one formulates it mathematically, the results are completely different.

Case 1: Vmax/((EC50/[A:B])^n+1)

Vmax = 1.2 [nanomole/hour]

EC50 = 10 [nanomole]

n = 2 [-]

Simulation produces figure 1.

Case 2: Vmax/((EC50/([A:B]/comp1))^n+1)

Vmax = 1.2 [nanomole/hour]

EC50 = 10 [nanomole/liter]

n = 2 [-]

comp1 = 1 [microliter]

Simulation produces figure 2.

The question I am wandering about why in the second case the amount becomes negative.

sbproj files attached. Any comments would be appreciated. Emjey

Answer 1

[Arthur Goldsipe]

Similar questions come up periodicaly. I suggest looking at my previous responses here and here.

The bottom line is that both of these plots show the expected numerical behavior of the differential equations as you implemented them. My recommendation is to replace the [A:B] term with max(0,[A:B]).

In your case, the reactions for the two cases proceed at vastly different rates. In Case 1, EC50 = 10 nanomole. In Case 2, EC50*comp1 = 10 nanomole/liter*microliter = 1e-5 nanomole.This leads to much faster dynamics for Case 2, and eventually the solver takes a large enough time step that the concentration goes negative. And it turns out that the solver tolerances are still satisifed under these conditions, due to the form of your reaction rate. The problem here is that a "negative amount" of [A:B] of -10 nanomole results in the exact same reaction rate as +10 nanomole of [A:B]. So a slightly negative amount leads to even more negative amounts of [A:B]. I typically say that negative concentrations that are within the solver's absolute tolerance of zero should be considered as equivalent to zero. And you can actually enforce such a constraint on your equations by replacing terms involving [A:B] with max(0,[A:B]).

I made such a change in your model, and this eliminated the negative concentrations. Note however that if you make such a change and still see concentrations with large magnitude negative values (say, -10 or -100 nanomole, for this model), then you may have a modeling error that leads to non-physical behavior.

[emjey]

Thanks for the explanation! In that context I have a question about 'AbsoluteToleranceScaling'.

In the documentation you write 'When AbsoluteToleranceScaling is enabled (by default), each state has its own absolute tolerance that can increase over the course of simulation. Sometimes the automatic scaling is inadequate for models that have kinetics at largely different scales'.

Could you explain please how the scaling is performed so that each state has its own absolute tolerance? Is it based on it's initial value? In that particular example it has no bearing on the simulation results although the range of the reaction rates is significant.

[emjey]

Btw, I simulated the above using ode45 solver and get the same results (using the max(0,[A:B]) detour). Could you explain what do you do to the model to be able to handle the different scales with a non-stiff solver?

[Arthur Goldsipe]

You will find a detailed description of the algorithm for absolute tolerance scaling here.

I'm not sure I understand what you're asking about ode45. For your model diss_conc.sbproj with the max detour, I get basically the same results when simulating with ode45, ode15s, and sundials.