% Differential equation for the lac operon
% RMM, 20 Jan 07
%
% Based on Yildirim et al, Chaos 2004, with time delays set to zero.
%
% x(1)	M	mRNA concentration
% x(2)  B	beta-gal concentration
% x(3)  A	allolactose concentration

function xder = lac(t, x)

% Parameters (from YSHM04; converted to uM)
mu_bar = 3.03e-2;			% dilution rate
alpha_M = 0.997;   gamma_M = 0.411;	% mRNA growth/decay rates
n = 2;					% cooperativity
K = 7200;				% basal transcription constant
K_1 = 2.52e-2;				% Hill constant for lacI binding
alpha_B = 1.66e-2; gamma_B = 8.33e-4;	% beta-gal growth/decay
alpha_A = 1.76e4;  gamma_A = 1.35e-2;	% allolactose growth/decay
beta_A = 2.15e4;
K_L = 970;	   K_A = 1950;		% allolactose Michaelis Menten terms
L = 100;				% external lactose concentration

% mRNA concentration
dM = alpha_M * ...
      ( (1 + K_1 * x(3)^n ) / ...
        (K + K_1 * x(3)^n ) ) - gamma_M * x(1);

% B-gal concentration
dB = alpha_B * x(1) - gamma_B * x(2);

% Allolactose concentration
dA = alpha_A * x(2) * L / (K_L + L) - ...
     beta_A * x(2) * x(3) / (K_A + x(3)) - gamma_A * x(3);

xder = [dM; dB; dA];