Chapter 4: cell division

Each cell divides at each timestep following the initial time. We begin by defining this initial value:

n0=3;

At each subsequent timestep we double the previous count, which yields the recursive relation:

n_(i+1) = 2n_i

or:

fdiv=@(Nprev) 2*Nprev;

As we saw in the text, this corresponds to the closed-form solution:

fdivcf=@(N0,t) N0*2^t;

Thus, we expect the same output from the two functions, which we verify here for the first 4 timesteps starting from n0:

nnow=n0*[1 1];

for t=1:4, nnow=[fdiv(nnow(1)) fdivcf(n0,t)];

disp(['t:' num2str(t) '; n:[' num2str(nnow(1)) ', ' num2str(nnow(2)) ']']); end

Part of the background information for this problem stipulated that no cells die during our counting ‘experiment’. However, we can imagine how cell death might work. For example, we might assume that a proportion, , of the previous population dies during the time separating the current and previous timestep. To see the effect of deaths, we subtract the cells that die during the interval between the previous and current timestep from the original (at the end of the previous timestep) count, assuming deaths always occur after dividing. This yields a new recursive relation:

n_(t+1) = 2n_t - (alpha)n_t = (2 - alpha)n_t

We can further run this relation forward from to death-adjusted cell counts for the first few timesteps:

fdie=@(Nprev,alpha) round((2-alpha)*Nprev);

n0=1; alpha=.1; nnow=n0*[1 1]; for n=1:5, nnow=[fdiv(nnow(1)) fdie(nnow(2),alpha)], end

At which timestep do we see the effect of the cell-death term of the new equation? why?