7), namely $$\displaystyle\frac\rm d c_2\rm d t = – 2\mu c_2 + \mu\nu (x_2+y_2) -\alpha c_2(x_2+y_2) , $$ (3.1) $$\displaystyle\frac\rm d x_2\rm d t = \mu c_2 – \mu\nu x_2 – \alpha c_2 x_2 – 2 \xi x_2^2

+ 2 \beta x_4 , $$ (3.2) $$\displaystyle\frac\rm d y_2\rm d t = \mu c_2 – \mu\nu y_2 – \alpha c_2 selleck inhibitor y_2 – 2 \xi y_2^2 + 2 \beta y_4 , $$ (3.3) $$\displaystyle\frac\rm d x_4\rm d t = \alpha x_2 c_2 + \xi x_2^2 – \beta x_4 , $$ (3.4) $$\displaystyle\frac\rm d y_4\rm d t = \alpha y_2 c_2 + \xi y_2^2 – \beta y_4 . $$ (3.5) Fig. 7 Simplest possible reaction scheme which might exhibit chiral symmetry-breaking We investigate the symmetry-breaking by transforming the variables x 2, x 4, y 2, y 4 according to $$ x_2 = \Stattic datasheet frac12 z (1+\theta) , \quad y_2 = \frac12

z (1-\theta) , $$ (3.6) $$ Selleckchem Vactosertib x_4 = \frac12 w (1+\phi) , y_4 = \frac12 w (1-\phi) , $$ (3.7)where z = x 2 + y 2 is the total concentration of chiral dimers, w = x 4 + y 4 is the total tetramer concentration, θ = (x 2 − y 2)/z is the relative chirality of the dimers, ϕ = (x 4 − y 4)/w is the relative chirality of tetramers. Hence $$ \frac\rm d c_2\rm d t = – 2\mu c_2 + \mu\nu z – \alpha c_2 z , $$ (3.8) $$ \frac\rm d z\rm d t = 2 \mu c_2 – \mu\nu z – \alpha c_2 z – \xi z^2 (1+\theta^2) + 2 \beta w , $$ (3.9) $$ \frac\rm d w\rm d t = \alpha z c_2 + \frac12 \xi z^2 (1+\theta^2) – \beta w , $$ (3.10) $$ \frac\rm d \theta\rm d t = – \theta \left( \frac]# cz + \frac2\beta wz+ \xi z (1-\theta^2) \right) + \frac2\beta w\phiz , $$ (3.11) $$ \frac\rm d \phi\rm d t = \theta \fraczw ( \alpha c + \xi z ) – \left( \alpha c + \frac12 \xi z (1+\theta^2) \right) \fraczw \phi . $$ (3.12)The stability of the evolving symmetric-state (θ = ϕ = 0) is given by the eigenvalues (q) of the matrix $$ \left( \beginarraycc

– \left( \displaystyle\frac2\mu cz + \displaystyle\frac2\beta wz + \xi z \right) & \displaystyle\frac2\beta wz \\ (\alpha c + \xi z) \displaystyle\fraczw & – (\alpha c + \displaystyle\frac12 \xi z) \displaystyle\fraczw \endarray \right) , $$ (3.13)which are given by $$ \beginarraylll &&\quad q^2 + q \left( \frac\alpha c zw + \frac\xi z^2w + \frac2\mu cz + \xi z + \frac2\beta wz \right) + \\ && \frac1w \left( 2\mu \alpha c^2 + \mu c \xi z + \alpha c \xi z^2 + \frac12 \xi^2 z^3 – \beta \xi z w \right) =0 . \endarray $$ (3.14)Hence there is an instability if $$ \beta \xi z w > 2\mu \alpha c^2 + \mu c \xi z + \alpha c \xi z^2 + \frac12 \xi^2 z^3 , $$ (3.15)using the steady-state result that 2βw = z(2αc + ξz) and factorising (2αc + ξz) out of the result, reduces the instability Eq. 3.15 to the contradictory ξz 2 > ξz 2 + 2μc.