TrypanosomesAfter working on models of the evolution of minicircles in trypanosomes (tryps), I became interested in the population dynamics of tryps in the mammalian host.
Several species of trypanosome (including T. brucei) develop through several stages within the mammalian host, shown in Fig. 1. Tryps initially have a slender morphology and divide rapidly in the blood stream. On receiving a signal to differentiate from an unknown soluble factor termed Stumpy Induction Factor (SIF), produced by the tryps themselves, they halt division and differentiate into a stumpy form that is pre-adapted for life in the tsetse fly.
Tryps that don't differentiate can change their antigenic surface glycoprotein coat to evade the host's building immune response to the dominant antigenic type. This process, called antigenic variation, has been widely studied and modelled.
Some of the best data on the slender to stumpy transition had been collected by Dick Seed at the University of North Carolina. I contacted him and he kindly agreed to send me some of his data. The parasitaemias are wholly due to parasite behaviours and are not modified by the immune response because of immuno-suppression. Some of his data are shown in Fig. 2.
The simplest model one could imagine is shown schematically and quantified in Model 1. Slender cells divide, they produce SIF which induces differentiation of slender into stumpy cells which then die off. SIF also decays in the blood. There are 6 parameters in the model, including the initial conditions, This might seem a lot, but some of them may not be needed.
The model is fitted to the data by maximising the log of a likelihood function. By maximising the likelihood function we find the set of parameters that maximses the probability of the data given the model. We used the downhill simplex method to search parameter space. You can never be sure you've found the global maximum, but with trial and error and different initial parameter values you can be reasonably confident that you're pretty close.
Fig. 3 shows the best fit of Model 1 to the data. Okay, but not great.
To see if the model is a good fit to the data we did a goodness-of-fit test. If your residuals are normally distributed than you can use a χ2-test. Our residuals were not, so we used a more laborious, although more general, technique. We simulate 1,000 artificial data sets using our model with the best fit parameters. Next we fit the model to the simulated data to get 1,000 log-likelihood values. These give us the distribution of log-likelihood values if the model were true. We compare the log-likelihood value we got from the real data to this distribution of expected values. If the actual value lies within the distribution then the model fits the data well. If it doesn't lie within the distribution then we can reject the model as a bad fit.
Our log-likelihood value for the data was -80, the expected values if the model were true lie within the range -60 to -45 (Fig. 4). Therefore this model is a bad fit, which is what we expected from Fig. 3.
The problem is how to find a model that will fit the data well. This is a process of trial and error of trying out lots of ideas.
In the end we had to postulate new stages in the differentiation process in order for the model to fit the data well (see Final Model). Differentiation in this model begins when a slender cell receives a SIF signal (L1). The cell is committed to differentiation but can still divide a few more times (L2). It then exits the cell cycle, but it still has a slender morphology (L3). Over a period of time it transforms into a stumpy cell (S). In addition the model explicity included delayed progression between stages. This is much more biologically realistic; Model 1 assumed an exponentially distributed time to differentiation. As an aside, many biological processes are delayed, but most mathematical models developed to describe them often use exponentially distributed delay.
More to follow on work done with Paula MacGregor and Keith Matthews at UoE.
The equations for this model are quite horrendous, but in essence the model is similar to the first model. The model also has a couple more parameters. We again fit the model to the data:
This time the fit looks much better. The goodness-of-fit suggests an excellent fit:
This research was published in Parasitology.
Created 1 September 2006. Updated 28 July 2010.