Biology, Computation, Math Modeling

Iterative curve fitting of biological systems

As I discussed in my post on sensitive parameters, optimizing parameter values may sometimes be difficult for reasons other than a poorly defined model. Initial parameter values have a high impact on the ability of the optimizer to find a solution, unless you are using a global optimizer. The solution is iterative curve fitting.

What parameters are known?

Some parameters may be experimentally derived or heavily constrained, such as synthesis or degradation parameters. These should be fairly homogeneous and may be directly observed in separate experiments. Using additional experimental data to fit these parameters and then holding them constant will make estimating the remaining parameters easier.

Identify sensitive parameters

Some parameters control the behavior of the system than others. After eliminating known parameters, what is the subset that are sensitive? 

The rest

At this point, you have 3 sets of parameters: known parameters which are removed from the pool of estimates, sensitive parameters, and the rest of the parameters. Estimating sensitive and unsensitive parameters together may be difficult for a local solver. Estimating unsensitive parameters separately may result in very unlikely system behaviors. 

In general, I’ve found that fitting small groups together works best. That may require re-estimates for some groups, but if the groupings are good, the estimates won’t change much. 

Selecting a grouping

I choose my groupings based off both the dynamic behavior and the conceptual meaning of the parameters. I first take plenty of time to observe the behavior of the model at extreme combinations of parameters (too high or low for biological relevance). Some parameters might be sensitive in general, but have few effects when some other parameters are extreme or not varied. 

These types of relationships suggest groupings, where the sensitive parameters are not causing other parameter estimates to be skewed, although the system might be. Iterative estimates across the groups solves that issue. 

Parameters that don’t matter 

You may come across parameters whose values do not effect the system at all. In such at case, I prefer to pick a value to keep them at until the end of the estimation. I imagine it improves the solution slightly. 

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