30) $$ \frac\rm d \varrho_x\rm d t = – 2 \mu \nu x + 2 \mu c + 2 \alpha c \sqrt\fracx\varrho_x2 , $$ (5.31)with similar equations for \(y,\varrho_y\). Transforming to total concentrations and relative chiralities by selleck products way of $$ x = \click here displaystyle\frac12 z (1+\theta) , \quad y = \displaystyle\frac12 z (1-\theta) , \quad \varrho_x = \displaystyle\frac12 R (1+\zeta) , \quad \varrho_y = \displaystyle\frac12 R (1-\zeta) , $$ (5.32)we find $$ \frac\rm d c\rm d t = \mu \nu z – 2 \mu c – \frac\alpha c \sqrtz R2\sqrt2 \left[ \sqrt(1+\theta)(1+\zeta) + \sqrt(1-\theta)(1-\zeta)

\right] , \\ $$ (5.33) $$ \beginarrayrll \frac\rm d z\rm d t & = & 2\mu c – \mu \nu z – \alpha c z

– \frac12 \xi z^2 (1+\theta^2) \\ && + \frac\beta \sqrtzR2\sqrt2 \left[ \sqrt(1+\theta)(1+\zeta) + \sqrt(1-\theta)(1-\zeta) \right] \\ && – \frac\xi z^3/2 R^1/24\sqrt2 MAPK inhibitor \left[ (1+\theta)^3/2 (1+\zeta)^1/2 + (1-\theta)^3/2 (1-\zeta)^1/2 \right] \\ && – \frac\beta z^3/2 \sqrt2R \left[ \frac(1+\theta)^3/2(1+\zeta)^1/2 + \frac(1-\theta)^3/2(1-\zeta)^1/2 \right] , \\ \endarray $$ (5.34) $$ \frac\rm d R\rm d t = – 2\mu\nu z + 4 \mu c + \frac12 \alpha c \sqrt2zR \left[ \sqrt(1+\theta)(1+\zeta) + \sqrt(1-\theta)(1-\zeta) \right] , \\ $$ (5.35)together with the Eqs. 5.38 and 5.39 for the relative chiralities θ and ζ, which will be analysed later. Since the equations for d R/ddt and dc/dt are essentially the same, we obtain a third piece of information from the requirement that the total mass in the system is unchanged from the initial data, hence the new middle equation above. Solving these we find \(c=\frac12 (\varrho-R)\) and use this in place of the equation for c. In the symmetric case (θ = ζ = 0) we obtain the steady-state conditions $$ 0 = 2\mu\nu z – 4\mu c – \alpha c \sqrt2zR ZD1839 datasheet , \qquad\qquad \varrho \; = \; R + 2 c , \\ $$ (5.36) $$ 0 = 2\mu c – \mu \nu z – \alpha c z – \frac12 \xi z^2 + \frac12 \beta \sqrt2zR

– \beta z \sqrt\frac2zR – \frac\xi z2 \sqrt\fraczR2 . $$ (5.37)For small θ, ζ, the equations for the chiralities can be approximated by $$ \beginarrayrll \frac\rm d \theta\rm d t & = & – \left( \frac2\mu cz + \frac12 \xi z + \frac12 \beta \sqrt\fracR2z + \frac12 \beta \sqrt\frac2zR + \frac14 \xi \sqrt\fraczR2 \right) \theta \\ && + \left( \frac\beta(R+2z)2\sqrt2zR – \frac\xi4 \sqrt\fracRz2 \right) \zeta , \\ \endarray $$ (5.38) $$ \frac\rm d \zeta\rm d t = \left( \frac2\mu\nu zR – \alpha c \sqrt\fraczR2 \right) \theta – \left( \frac2\mu\nu zR – \frac4\mu cR \right) \zeta , $$ (5.