a dynamical model for neural cell development - CiteSeerX

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Volkswagen Foundation Research Group, Humboldt University Berlin. 4. Max Delbrück Center for Molecular Medicine (MDC), Berlin-Buch. Overview. Aim of the ...
A DYNAMICAL MODEL FOR NEURAL CELL DEVELOPMENT Christoph Bandt1, Elena Beisswanger1,2, Laurenz Wiskott2,3 and Gerd Kempermann3,4 1 Department of Mathematics, Arndt University Greifswald 2 Institute for Theoretical Biology, Humboldt University Berlin 3 Volkswagen Foundation Research Group, Humboldt University Berlin 4 Max Delbr¨uck Center for Molecular Medicine (MDC), Berlin-Buch

Overview

How cells are counted

Modelling

Aim of the study. In recent years, adult hippocampal neurogenesis has become a subject of intense research [1], with the hope to cure diseases like Alzheimer in the future. However, quantitative modelling has not been done so far. Based on various experiments with mice performed in the group of Kempermann [2, 3, 5], we establish a model which can estimate the number of cells produced per day, and the time which the cells remain in various stages of development. An error analysis is included.

General. Bromodeoxyuridine (BrdU) is a marker for dividing cells which can substitute thymidine in the DNA which is synthesized during the S-phase of the cell cycle. It will label not all developing brain cells, but only those which are in S-phase at the time of injection of BrdU, and their progeny. When a cell division is completed, the daughter cells will share the BrdU molecules. After several cell divisions, the concentration of BrdU is below the limit of detection, and the progeny cells are not recognized any more. The problem is that the cells which multiply very fast are those which become invisible first!

Stationarity assumption. The (expected) number Ck of all cells of type k = 1, 2, 3 is assumed to be constant over time. Although more and more neurons of type 5 are produced, the cell numbers in intermediate stages do not increase. Thus, if Bk0 denotes the number of BrdU-marked cells of type k at time zero and τk the length of their cell cycle in hours, then

Problems of modelling. With BrdU and genetic markers, we can observe only certain parts of the cell populations. What is worse: most of the initially observed cells become invisible during the course of the experiment. In order to fit the data, a model must adapt to the experimental methods. It turns out that for the mitotic stages, a Leslie matrix model works best while for the postmitotic stages differential equations seem to be appropriate. Conclusions from dynamical systems. Certain stationarity conditions in the Leslie model provide various relations among the parameters of the cell population. This can be used to reduce the dimension of parameter space. A simple formula for the total production of postmitotic cells can be derived. Inequalities prove the influence of BrdU thinning already in early stages of the experiment.

Neuronal development Cell types. According to morphology, the following six stages of development of brain cells can be distinguished [2, 3].

T1 are the putative stem cells, with astrocyte shape, which never die. They divide rarely and only asymmetrically, into one cell T1 and one T2a. T2 cells have simpler morphology and much more proliferation activity. Together with T3 they form the progenitor cells, while T4 and T5 are the postmitotic stages where the cells do not divide and only develop their morphology. While in stage T4 there is a strong selection, and many cells are eliminated, for T5 no apoptosis seems to occur. Expression of proteins. In the figure, the marker proteins are noted which are typical for various stages: nestin for T1 and T2, GFAP for T1. Type T2b already expresses DCX, like T3 and T4. Moreover, T4 expresses Calretinin, T5 Calbindin, and both express NeuN. These markers, together with examination of the cell morphology, serve to identify the cell type, which is difficult and sometimes ambiguous.

References [1] G. Kempermann, H.G. Kuhn and F.H. Gage, More hippocampal neurons in adult mice living in an enriched environment, Nature 386, 493-495 (1997). [2] S. Jessberger, M.D. Brandt et. al., Transient calretinin expression defines early postmitotic step of neuronal differentiation in adult hippocampal neurogenesis of mice, Mol. Cell. Neurosc. 24, 603-613 (2003). [3] G. Kronenberg, K. Reuter et. al., Subpopulations of proliferating cells of the adult hippocampus respond differently to physiologic neurogenic stimuli, J. Comp. Neurol. 467, 455-463 (2003). [4] L.P. Lefkovich, The study of population growth in organisms grouped by stages, Biometrics 21, 1-18 (1965). [5] B. Steiner et. al., Differential regulation of gliogenesis in the context of adult hippocampal neurogenesis in mice, GLIA 46, 41-52 (2004).

e-mail contact: [email protected]

BrdU

BrdU

S

τk 0 Ck = Bk · . 8 Leslie matrix. We choose discrete time steps of length 8 hours. Each cell can be in state k = 1, 2, 3, 4 or in S-phase Sk for type k = 1, 2, 3. In the simplest model, the following transitions occur:

8h

G2

G1 8h

M

Antigen reactions. BrdU is detected by fluorochrome-labelled antibodies. We can only measure its presence, not the concentration. The types of BrdU-labelled cells are also determined by antibody-antigen reactions with certain marker proteins.

Data and error analysis Experimental setting. We describe only part of [3]. 20 genetical identical mice, housed under equal conditions, got a single injection of BrdU. At given times (4 hours, 1, 3 and 7 days) five mice were sacrificed, and their brain cut into 60 coronal slices. BrdU positive cells in the dentate gyrus were counted under light microscope after antibody-antigen reaction. Only one sixth of the slices were studied. Relative numbers of cell types were determined by fluorochrome-conjugated antibodies and confocal laser scanning microscopy. Resulting numbers were extrapolated to the dentate gyrus and averaged over five mice. Neglect differences between mice. A Poisson distribution models absolute numbers of BrdU cells, as they would appear in repeated experiments with one mouse. Comparing with data it is observed that fluctuations of BrdU numbers between different mice at the same time point are not larger than the model predicts for repeated measurements of one individual. Thus data from different time points can be treated as if they all came from the same mouse. Example of data. In simplified form, the essential part of the data of [3] is given in the following table. Time t (days) Type 1 Type 2=2a Type 3=2b+3 Sum

0 5 157 56 218

1 5 139 234 378

3 6 91 289 386

7 4 18 107 129

Size of errors. A probabilistic analysis of the counting method shows that the standard error of absolute BrdU numbers (last row) is 10 up to 20 % . In addition, the relative numbers of types, which are determined using only part of the BrdU cells, also have a standard error of about 20 % . This agrees with the standard deviations for groups of mice at each time point. Averaging reduces the error. Nevertheless, ±20% deviation in the table would be normal, so that precise implications are not yet possible. What we see in the table. The first column (4 hours after injection) shows the number of cells in S-phase, the other columns show the number of their progeny after 1,3 and 7 days – as far as they have not become invisible. After 1 day the first cell cycle is completed, so that twice as many cells should exist. After 3 days further cell divisions have occured, but the BrdU numbers do not increase. Some of the cells have changed to type 4, and many have become invisible, as can be seen from the last two columns. The numbers also indicate that type 2 divides asymmetrically into type 2 and type 3.

Here sk denotes the rate of cell division of type k which is connected with the average length of cell cycle: τk = (1 + s1k ) · 8h. Moreover, wk is the probability that a cell of type k = 2, 3 will grow into the next type by morphological change. vk is the probability that a cell division is symmetrical, with two successors of type k. An asymmetrical cell division results in one progeny of type k and type k + 1, and this is always the case for type k = 1. Since S-phase is assumed to last 8 hours, each cell in state Sk will return in the form of successors in the next time step. Stationary distribution. With this scheme we can calculate as with a Markov chain. For the stationary distribution, states 1 · C cells are in state k and Sk together include Ck cells. 1+s k k sk · Ck cells in state Sk. Type 1 is the input of k and Bk0 = 1+s k the system, and s1, C1 are chosen so that always 5 cells are in S-phase, as required by the data: s1C1 = 5(1 + s1). In reality, the process is not stationary in state 4, but m4 is chosen so that stationarity is formally fulfilled. Then m4C4, the output of postmitotic cells of the system per unit of time, is m4C4 = B10 + B20 + B30, the total number of BrdU cells at time zero, in our case 218 for 8 hours. This estimate changes only little when mortalities m2, m3 for type 2 and 3 are introduced. Mathematical conditions. There are two more equations for stationarity which allow to relate the parameters s2, v2, s3, v3. Since these values must be between 0 and 1, we obtain restrictions! We can also calculate the BrdU population after one day in the model, and compare with the data Bk1 . For example w2 ≥ 0

implies

m2 1 0 B2 ≥ B2 (1 + ). s2

If this relation does not hold, as in our data, this means m2 ≈ 0, w2 ≈ 0, and the BrdU-thinning is already effective within oneday-values! Using generating functions, gradual BrdU thinning is incorporated into the model and gives an excellent fit to the data, with s2 = 0.25, s3 = 0.03, w3 = 0.1, v2 and w2 near to zero, v3 near to 1.

Conclusions • Quantitative estimates for brain cell proliferation are possible with today’s experimental techniques. Precise results, however, require more data. Statistical measurement errors are still very large, while differences between individual mice are so small that they can be neglected. • Under normal conditions, several hundreds of new cells are produced each day in the hippocampus of an adult mouse. • Cells of type 2 undergo asymmetric division into type 2 and type 3. For type 3, the data indicate symmetric division and transition to type 4 mainly by morphological change. • The decrease of observed cells of type 2 already in the first days of measurement is due to BrdU thinning rather than cell death. The data from time points 0 and 1 are particularly important since they are less confounded by BrdU thinning.