%0 Book Section %1 statphys23_1046 %A Sokolovsky, A.I. %A Sokolovsky, Y.U.A. %A Chernaya, V.I. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K bogolyubov bogolyubov-grad boundary conditions effective functional hypothesis initial kinetics layer michaelis-menten problem statphys23 topic-2 %T Effective initial conditions in generalized Michaelis-Menten kinetics %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1046 %X The Michaelis-Menten kinetics, which describes transformation of a substrate (S) into a product (P) with the help of enzyme (E) through a complex (C) \ S+Ek_1 .1ex$\,_\stackrelto 10mm \rightarrowfill $C, Ck_-1 .1ex$\,_\stackrelto 10mm \rightarrowfill $S+E, Ck_2 .1ex$\,_\stackrelto 10mm \rightarrowfill $P+E, \ is considered ($k_1,k_-1,k_2$ are rate constant). Kinetic equations for these reactions can be written in the form $$x=-ax(z_0-u)+bu, u=ax(z_0-u)-(b+c)u,$$ equation1 equation $$y=cu, z=z_0-u$$ where $x,y,z$, and $u$ are concentrations of substrate, product, enzyme, and complex ($ak_1, bk_-1, ck_2$). We consider situation, in which $z_0x_0, y_0=0, u_0=0$, where $x_0,y_0,z_0,u_0$ are initial values of concentrations $x,y,z,u$. In dimensionless variables problem of solution of equations (1) is reduced to a singular perturbation problem. In this framework the steady-state approximation in the Michaelis-Menten kinetics can be understand. However, ideas developed by Chapman and Bogolyubov in theory of nonequilibrium processes alow to cast the steady-state theory in very simple form. According to these ideas after a transition period concentration of the complex $u(t)$ is synchronized with concentration of the substrate $x(t)$ equation2 u(t).1ex$\,_\stackrelto 8mm \rightarrowfill t\tau_0$u^(+)(t), x(t).1ex$\,_\stackrelto 8mm \rightarrowfill t\tau_0$x^(+)(t), u^(+)(t)=u(x^(+)), equation where $\tau_0$ is duration of the transition period. In fact, relations (2) is an analog of the Bogolyubov functional hypothesis and lead to closed equation for concentration of the substrate $x^(+)(t)$ and simple equation for asymptotic concentration of the product $y^(+)(t)$ equation3 x^(+)(t)=L(x^(+)(t)), y^(+)(t)=cu(x^(+)) equation at long times $t\tau_0$. Simple calculation gives introduced in (2) and (3) functions $$L(x)=-z_0cxx+k+z_0^2ck(ax+b)xa(x+k)^4+O(z_0 ^3)$$ equationequation $$u(x)=z_0xx+k+z_0^2ckxa(x+k)^4+O(z_0^3)$$ ($k(b+c)/a$ is the Michaelis constant). Evolution equation (3) for $x^(+)(t)$ and $y^(+)(t)$ must be completed by effective initial conditions $x^(+)_0, y^(+)_0$. They can be introduced by a continuation of solution of equations (3) for times $0t \tau_0$. Calculation of the effective initial conditions gives $$x^(+)_0=x_0-z_0(ax_0+b)x_0a(x_0+k)^2+O(z_0^2)$$ equation4equation $$y^(+)_0=-z_0x_0ca(x_0+k)^2+O(z_0^2)$$ This is solution for our model of the Bogolyubov-Grad initial boudary layer problem. A comparison of numerical solution of equations (1) with real initial conditions and equation (3) with effective initial condition gives an excellent agreement.

@incollection{statphys23_1046, abstract = {The Michaelis-Menten kinetics, which describes transformation of a substrate (S) into a product (P) with the help of enzyme (E) through a complex (C) \[ S+E{\stackrel {k_1} {\raisebox{.1ex}{\mbox{$\,_{\stackrel{\hbox to 10mm {\rightarrowfill}} {}}$}}}}C, \quad C{\stackrel {k_{-1}} {\raisebox{.1ex}{\mbox{$\,_{\stackrel{\hbox to 10mm {\rightarrowfill}} {}}$}}}}S+E, \quad C{\stackrel {k_2} {\raisebox{.1ex}{\mbox{$\,_{\stackrel{\hbox to 10mm {\rightarrowfill}} {}}$}}}}P+E, \] is considered ($k_1,k_{-1},k_2$ are rate constant). Kinetic equations for these reactions can be written in the form $$\dot x=-ax(z_0-u)+bu, \dot u=ax(z_0-u)-(b+c)u,$$ \begin{equation}\label{1} \end{equation} $$\dot y=cu, z=z_0-u$$ where $x,y,z$, and $u$ are concentrations of substrate, product, enzyme, and complex ($a\equiv k_1, b\equiv k_{-1}, c\equiv k_2$). We consider situation, in which $z_0\ll x_0, y_0=0, u_0=0$, where $x_0,y_0,z_0,u_0$ are initial values of concentrations $x,y,z,u$. In dimensionless variables problem of solution of equations (1) is reduced to a singular perturbation problem. In this framework the steady-state approximation in the Michaelis-Menten kinetics can be understand. However, ideas developed by Chapman and Bogolyubov in theory of nonequilibrium processes alow to cast the steady-state theory in very simple form. According to these ideas after a transition period concentration of the complex $u(t)$ is synchronized with concentration of the substrate $x(t)$ \begin{equation}\label{2} u(t){\stackrel {}{\raisebox{.1ex}{\mbox{$\,_{\stackrel{\hbox to 8mm {\rightarrowfill}} {t\gg \tau_0}}$}}}}u^{(+)}(t), x(t){\stackrel {}{\raisebox{.1ex}{\mbox{$\,_{\stackrel{\hbox to 8mm {\rightarrowfill}} {t\gg \tau_0}}$}}}}x^{(+)}(t), u^{(+)}(t)=u(x^{(+)}), \end{equation} where $\tau_0$ is duration of the transition period. In fact, relations (2) is an analog of the Bogolyubov functional hypothesis and lead to closed equation for concentration of the substrate $x^{(+)}(t)$ and simple equation for asymptotic concentration of the product $y^{(+)}(t)$ \begin{equation}\label{3} \dot x^{(+)}(t)=L(x^{(+)}(t)), \quad \dot y^{(+)}(t)=cu(x^{(+)}) \end{equation} at long times $t\gg \tau_0$. Simple calculation gives introduced in (2) and (3) functions $$L(x)=-\frac{z_0cx}{x+k}+\frac{z_0^2ck(ax+b)x}{a(x+k)^4}+O(z_0 ^3)$$ \begin{equation}\end{equation} $$u(x)=\frac{z_0x}{x+k}+\frac{z_0^2ckx}{a(x+k)^4}+O(z_0^3)$$ ($k\equiv (b+c)/a$ is the Michaelis constant). Evolution equation (3) for $x^{(+)}(t)$ and $y^{(+)}(t)$ must be completed by effective initial conditions $x^{(+)}_0, y^{(+)}_0$. They can be introduced by a continuation of solution of equations (3) for times $0\leq t \leq \tau_0$. Calculation of the effective initial conditions gives $$x^{(+)}_0=x_0-\frac{z_0(ax_0+b)x_0}{a(x_0+k)^2}+O(z_0^2)$$ \begin{equation}\label{4}\end{equation} $$y^{(+)}_0=-\frac{z_0x_0c}{a(x_0+k)^2}+O(z_0^2)$$ This is solution for our model of the Bogolyubov-Grad initial boudary layer problem. A comparison of numerical solution of equations (1) with real initial conditions and equation (3) with effective initial condition gives an excellent agreement.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Sokolovsky, A.I. and Sokolovsky, Y.U.A. and Chernaya, V.I.}, biburl = {https://www.bibsonomy.org/bibtex/27bbe7c6e2a493b7f22ce640e1a358cbd/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {f8cf83071314af227abdf773eaff3b39}, intrahash = {7bbe7c6e2a493b7f22ce640e1a358cbd}, keywords = {bogolyubov bogolyubov-grad boundary conditions effective functional hypothesis initial kinetics layer michaelis-menten problem statphys23 topic-2}, month = {9-13 July}, timestamp = {2007-06-20T10:16:37.000+0200}, title = {Effective initial conditions in generalized Michaelis-Menten kinetics}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1046}, year = 2007 }