## Dynamics of dendritic cells

On 23.04 I heard a talk by Michel Nussenzweig from Rockefeller University. His subject was dendritic cells, a class of white blood cells. For the casual observer, such as myself, it might seem that the class is homogeneous. It was emphasized by Nussenzweig that this is far from being the case. According to him, dendritic cells come in various types and the problem of classification may be as complicated as it is in the case of their more prominent relatives, the T-cells. As mentioned in previous posts, for the latter distinctions are made between CD4+, CD8+, Th1, Th2, Th17 etc. In the meantime various classes of dendritic cells have been recognized and it seems that not distinguishing between them sufficiently carefully has been an obstacle to progress. There are, for instance, conventional dendritic cells (cDC) and plasmacytoid dendritic cells (pDC). A related issue is insufficient precision in the use of language. As an aside: a ‘follicular dendritic cell’ is something quite different from a dendritic cell in the strict sense, sharing no more with it than its name and its morphology when seen under the microscope.

Among the themes of the talk were the questions of which cells occur in the course of development of dendritic cells from hematopoietic stem cells, where the different steps of this development take place (in bone marrow, blood, lymphatic organs or other tissues) and the relations between these precursor cells and the corresponding stages in the development of other classes of cells such as macrophages. One tool which can be used to investigate these things is parabiosis, where the bloodstreams of two mice are joined together.

At one point in the talk the speaker said, ‘Here we use a bit of mathematics’. He did not say what kind of mathematics. I sometimes have the impression in talks by biologists for biologists that the feeling is that it would be impolite to the audience, if not indecent, to show any mathematics. In this case I decided to look into what the mathematical content really is. For this I looked at the paper ‘Origin of dendritic cells in peripheral lymphoid organs of mice’ (Nature Immunology 8, 578) where Nussenzweig is one of the authors. The mathematics is not in the main text – to see it is necessary to go to the “Supplementary Methods” available as a separate file in the online version of the paper. From the point of view of a mathematician the presentation of the mathematical formulae is not very convenient. For instance, an ODE like $\frac{df}{dt}=g$ would be written as $df=g^*dt$. For me the first step in understanding the mathematical part was to convert the equations into TeX. This having been done, equations are:
$\frac{dN}{dt}=\left(\frac{V}{C}-D+P\right)N$
$\frac{dM}{dt}=\left(\frac{2V}{C}+PQ\right)N-\left(\frac{V}{C}+D\right)M$
$\frac{dL}{dt}=\frac{3PN}{10}+\left(\frac{V}{C}-D\right)L$
Here $N$, $M$ and $L$ are functions of time while the other quantities denoted by capital letters are constant parameters. The first equation is decoupled from the others and in fact plays a different role. $N$ is the number of cDC in the spleen and is taken to be $10^6$. Thus the left hand side of the first equation is assumed to vanish and this gives the relation $D=\frac{V}{C}+P$. Taking account of this, there remains a system of two linear ODE, which can easily be solved explicitly. The resulting solution can be used to determine the parameters in terms of experimental data. A question of interest is how many of the cells have divided in a certain time interval. This can be investigated experimentally by adding the nucleoside analogue bromodeoxyuridine (BrdU). When a cell divides this compound is incorporated into the DNA of the daughter cells. Thus an assay of BrdU can be used to measure the proportion of cells which have divided. At this point it is appropriate to explain the meaning of the variables $M$ and $N$.The variable $M$ is the proportion of cDC which are labelled with BrdU at a given time. $L$ is the proportion of cDC in the mouse under consideration which come from the other mouse joined to it by parabiosis.

The experimental data is as follows. The proportion of cDC in the spleen which are dividing is 5 per cent. This is the quantity $V$ in the equations. At the initial time there are no cells labelled by BrdU ($M(0)=0$) and no cells coming from the other mouse ($L(0)=0$).
After 2.2 days 50% of the cells have taken up BrdU. If ‘days’ is taken as the unit of time then this gives $M(2.2)=(0.5)N$. On long time scales this proportion approaches 91%, so that $\lim_{t\to\infty}M(t)=(0.91)N$. After 13 days the proportion of cDC from the other mouse is 22% ($L(13)=(0.22)N$) while on long time scales it is 30% ($\lim_{t\to\infty}L(\infty)=(0.3) N$). Putting this information into the explicit solution of the equations and doing some elementary algebra allows all parameters to be computed. In particular $P=0.101673$. This is the proportion of the cDC in the spleen which enter from the blood in unit time.This can be converted to the number of cDC which enter the spleen each hour by multiplying by $10^6\times (1/24)$. The result is 4236 cells per hour, a figure which, rounded up to ‘nearly 4300′, is quoted in the abstract of the paper.

The modest goal of this discussion has been to obtain an answer to the question, ‘What is the mathematical content involved in this work’.

### One Response to “Dynamics of dendritic cells”

1. Conference on modelling the immune system in Dresden « Hydrobates Says:

[…] sensing mechanism. In particular this involved the technique of parabiosis which I mentioned in a previous post. In fact he went further than standard parabiosis and joined three mice together rather than just […]