Achiral clusters are denoted by C r , and we allow clusters to change their morphology spontaneously according to $$ \beginarrayrclclccrclcl C_r & \AZD0156 supplier rightarrow & X_r & \quad& \rm rate = \mu_r , && X_r & \rightarrow & C_r & \quad& \rm rate = \mu_r \nu_r , \\[4pt] C_r & \rightarrow & Y_r & \quad& \rm rate = \mu_r , && Y_r & \rightarrow & C_r & \quad& \rm rate = \mu_r \nu_r . \endarray $$ (2.7)We allow clusters to grow by coalescing with clusters of similar LY2835219 handedness or an achiral cluster. In the case of the latter process, we assume that the cluster produced is chiral with the same chirality as the parent.

Thus $$ \beginarrayrclcl X_r + X_s & \rightarrow & X_r+s , && \rm rate = \xi_r,s, \\[6pt] X_r + C_s & \rightarrow & X_r+s , && \rm rate = \alpha_r,s,\\[6pt] C_r + C_s & \rightarrow & C_r+s , && \rm rate

= \delta_r,s,\\[6pt] Y_r + C_s & \rightarrow & Y_r+s , && \rm rate = \alpha_r,s,\\[6pt] Y_r + Y_s & \rightarrow & Y_r+s , && \rm rate = \xi_r,s . \endarray $$ (2.8)We do not permit clusters of opposite to chirality to merge. Finally we describe fragmentation: all clusters may fragment, producing two smaller clusters each of Copanlisib manufacturer the same chirality as the parent cluster $$ \beginarrayrclcl X_r+s & \rightarrow & X_r + X_s && \rm rate = \beta_r,s, \\[4pt] C_r+s & \rightarrow & C_r + C_s && \rm rate = \epsilon_r,s, \\[4pt] Y_r+s & \rightarrow & Y_r + Y_s &\quad& \rm rate = \beta_r,s . \endarray$$ (2.9)Setting up concentration variables for each size and each type of cluster by defining c r (t) = [C r ], x r (t) = [X r ], y r (t) = [Y r ] and applying the law of mass action, we obtain $$ \beginarrayrll \frac\rm d c_r\rm d t &=& -2\mu_r c_r + \mu_r\nu_r(x_r+y_r) – \sum\limits_k=1^\infty \alpha_k,r c_r (x_k+y_k) \\[6pt] && + \frac12 \sum\limits_k=1^r-1 \left( \delta_k,r-k c_k c_r-k – \epsilon_k,r-k

c_k c_r-k \right) – \sum\limits_k=1^\infty \left( \delta_k,r c_k c_r – \epsilon_k,r c_r+k \right) , \endarray $$ (2.10) $$ \beginarrayrll \frac\rm d x_r\rm d t &=& \mu_r c_r – \mu_r \nu_r x_r + \sum\limits_k=1^r-1 \alpha_k,r-k c_k x_r-k Thiamine-diphosphate kinase – \frac12 \sum\limits_k=1^r-1 \left( \xi_k,r-k x_k x_r-k – \beta_k,r-k x_r \right) \\[2pt] && – \sum\limits_k=1^\infty \left( \xi_k,r x_k x_r – \beta_k,r x_r+k \right) , \endarray $$ (2.11) $$ \beginarrayrll \frac\rm d y_r\rm d t &=& \mu_r c_r – \mu_r \nu_r y_r + \sum\limits_k=1^r-1 \alpha_k,r-k c_k y_r-k – \frac12 \sum\limits_k=1^r-1 \left( \xi_k,r-k y_k y_r-k – \beta_k,r-k y_r \right) \\[2pt] && – \sum\limits_k=1^\infty \left( \xi_k,r y_k y_r – \beta_k,r y_r+k \right) . \endarray $$ (2.12)The main problem with such a model is the vast number of parameters that have been introduced (α r,k , ξ r,k , β r,k , μ r , ν r , δ r,k , ϵ r,k , for all k, r).