Before that, however, we illustrate the behaviour of the system by briefly presenting the results of some numerical simulations. In Figs. 2 and 3 we show the results of a simulation of Eqs. 2.28–2.33. The former shows the evolution of the concentrations c 1 which rises then decays, c 2 which decays since the parameters have been chosen to reflect a cluster-dominated system. Also

plotted are the learn more numbers of clusters N x , N y and the mass of material in clusters \(\varrho_x\), \(\varrho_y\) defined by $$ \varrho_x = \sum\limits_j=2^K j x_j , \qquad \varrho_y = \sum\limits_j=2^K j y_j . $$ (2.34)Note that under this definition \(\varrho_x + \varrho_y + c_1 + 2c_2\) is conserved, and this is plotted as rho. Both the total number of clusters, N x  + N y , and total mass of material in handed clusters \(\varrho_x + \varrho_y\) appear to equilibrate by t = 102, however, at a much later time (t ∼ 104 − 105) a symmetry-breaking bifurcation occurs, and the system changes from almost racemic (that is, symmetric) to asymmetric. This is more clearly seen in Fig. 3,

where we plot the cluster size distribution at three time points. At t = 0 there are only dimers present (dashed line), and we impose a small difference in the concentrations of x 2 and y 2. At a later time, t = 112 (dotted line), there is almost no difference between the X- and Y-distributions, however by the end of the simulation Androgen Receptor signaling pathway Antagonists (t ∼ 106, solid line) one distribution clearly completely dominates the other. Fig. 2 Plot of the concentrations c 1, c 2, N x , N y , N = N x  + N y , \(\varrho_x\), \(\varrho_y\), \(\varrho_x + \varrho_y\)

and \(\varrho_x + \varrho_y + 2c_2 + c1\) against time, t on a logarithmic timescale. Bupivacaine Since model equations are in nondimensional form, the time units are arbitrary. Parameter values μ = 1.0, ν = 0.5, δ = 1, ε = 5, a = 4, b = 0.02, α = 10, ξ = 10, β = 0.03, with initial conditions c 2 = 0.49, x 4(0) = 0.004, y 4(0) = 0.006, and all other concentrations zero Fig. 3 Plot of the cluster size distribution at t = 0 (dashed line), t = 112 (dotted line) and t = 9.4 × 105. Parameters and initial conditions as in Fig. 2 Simplified Macroscopic Model To obtain the simplest model which involves three polymorphs corresponding to right-handed and left-handed chiral clusters and achiral clusters, we now aim to simplify the processes of cluster aggregation and fragmentation in Eqs. 2.28–2.33. Our aim is to retain the symmetry-breaking phenomenon but eliminate physical processes which are not necessary for it to occur. Our first simplification is to remove all clusters of odd size from the model, and just consider dimers, tetramers, hexamers, etc. This corresponds to putting a = 0, b = 0 which removes x 3 and y 3 from the system. Furthermore, we put ε = 0 and make δ large, so that the achiral monomer is rapidly and irreversibly converted to achiral dimer.