4) $$ \frac\rm d y\rm d t = k_1 r s + k_2 selleckchem r s y – k_3 x y – k_-1 y – k_-2 y^2 , $$ (1.5) $$ \frac\rm d p\rm d t = k_3 x y – k_4 p , $$ (1.6)from which we note that at steady-state we have $$ rs=\frack_0+k_-1(x+y) + k_-1(x^2+y^2)2k_1+k_2(x+y). $$ (1.7)We write the absolute enantiomeric excess as ee = x − y and the total concentration as σ = x + y; adding and subtracting

the equations for dx / dt and dy / dt, we find $$ \sigma^2 = \frac2k_0k_3 + ee^2 , $$ (1.8) $$ ee \left[ \frack_2(k_-2ee^2+k_-2\sigma^2+2k_-1\sigma+2k_0) 2(2k_1+k_2\sigma) - k_-1 - k_-2 \sigma \right] = 0 . $$ (1.9)Hence ee = 0 is always a solution, and there are other solutions with ee ≠ 0 if the rate constants k * satisfy certain conditions (these include k 3 > k  − 2 and k 0 being sufficiently large). The important issues to note here are: (i) this system is open, it requires the continual supply of fresh R, S to maintain the asymmetric steady-state. Also, the removal of products is required to avoid the input terms causing the total amount of material to increase indefinitely;   (ii) the forcing input term drives the system away from

an equilibrium solution, into a distinct steady-state solution;   (iii) the system has cross-inhibition which removes equal numbers of X and Y, amplifying any differences Dorsomorphin mw caused by random fluctuations in the initial data or in the input rates.   Saito and Hyuga (2004) discuss a sequence of toy models describing homochirality caused by nonlinear autocatalysis and recycling. Their family of models can be summarised PR-171 molecular weight by $$ \frac\rm d r\rm d t = k r^2 (1-r-s) – \lambda r , $$ (1.10) $$ \frac\rm d s\rm d t = k s^2 (1-r-s) – \lambda s , $$ (1.11)where r and s are the

concentrations of the two enantiomers. Initially they consider k r  = k s  = k and λ = 0 and find that enantiomeric exess, r − s is constant. Next the case k r  = kr, k s  = ks, λ = 0 is analysed, wherein the relative enantiomeric excess \(\fracr-sr+s\) is constant. Then the more complex case of \(k_r=k r^2\), \(k_s=k s^2\), λ = 0 is analysed, and amplification of the enantiomeric excess is obtained. This amplification persists when the case λ > 0 is see more finally analysed. This shows us strong autocatalysis may cause homochiralisation, but in any given experiment, it is not clear which form of rate coefficients (k r , k s , λ) should be used. Saito and Hyuga (2005) analyse a series of models of crystallisation which include some of features present in our more general model. They note that a model truncated at tetramers exhibits different behaviour from one truncated at hexamers. In particular, the symmetry-breaking phenomena is not present in the tetramer model, but is exhibited by the hexamer model.