Basic Principles Underlying Cellular Processes

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$ \newcommand{\avg}[1]{\langle #1 \rangle} \newcommand{\cc}[1]{[\mathrm{#1}]^{\mathrm{cell}}} \newcommand{\cgdp}{\mathrm{C \! \cdot \! GDP}} \newcommand{\cgtp}{\mathrm{C \! \cdot \! GTP}} \newcommand{\comb}[1]{{#1}^{\mathrm{comb}}} \newcommand{\conc}[1]{[\mathrm{#1}]} \newcommand{\conceq}[1]{[\mathrm{#1}]^{\mathrm{eq}}} \newcommand{\concss}[1]{[\mathrm{#1}]^{\mathrm{ss}}} \newcommand{\conctot}[1]{[\mathrm{#1}]_{\mathrm{tot}}} \newcommand{\cu}{\conc{U}} \newcommand{\dee}{\partial} \newcommand{\dgbind}{\Delta G_0^{\mathrm{bind}}} \newcommand{\dgdp}{\mathrm{D \! \cdot \! GDP}} \newcommand{\dgtp}{\mathrm{D \! \cdot \! GTP}} \newcommand{\dmu}{\Delta \mu} \newcommand{\dphi}{\Delta \Phi} \newcommand{\dplus}[1]{\mbox{#1}^{++}} \newcommand{\eq}[1]{{#1}^{\mathrm{eq}}} \newcommand{\fidl}{F^{\mathrm{idl}}} \newcommand{\idl}[1]{{#1}^{\mathrm{idl}}} \newcommand{\inn}[1]{{#1}_{\mathrm{in}}} \newcommand{\ka}{k_a} \newcommand{\kcat}{k_{\mathrm{cat}}} \newcommand{\kf}{k_f} \newcommand{\kfc}{k_{fc}} \newcommand{\kftot}{k_f^{\mathrm{tot}}} \newcommand{\kd}{K_{\mathrm{d}}} \newcommand{\kdt}{k_{\mathrm{dt}}} \newcommand{\kdtsol}{k^{\mathrm{sol}}_{\mathrm{dt}}} \newcommand{\kgtp}{K_{\mathrm{GTP}}} \newcommand{\kij}{k_{ij}} \newcommand{\kji}{k_{ji}} \newcommand{\kkeq}{K^{\mathrm{eq}}} \newcommand{\kmmon}{\kon^{\mathrm{ES}}} \newcommand{\kmmoff}{\koff^{\mathrm{ES}}} \newcommand{\kconf}{k_{\mathrm{conf}}} \newcommand{\konf}{k^{\mathrm{on}}_{\mathrm{F}}} \newcommand{\koff}{k_{\mathrm{off}}} \newcommand{\kofff}{k^{\mathrm{off}}_{\mathrm{F}}} \newcommand{\konu}{k^{\mathrm{on}}_{\mathrm{U}}} \newcommand{\koffu}{k^{\mathrm{off}}_{\mathrm{U}}} \newcommand{\kon}{k_{\mathrm{on}}} \newcommand{\kr}{k_r} \newcommand{\ks}{k_s} \newcommand{\ku}{k_u} \newcommand{\kuc}{k_{uc}} \newcommand{\kutot}{k_u^{\mathrm{tot}}} \newcommand{\ktd}{k_{\mathrm{td}}} \newcommand{\ktdsol}{k^{\mathrm{sol}}_{\mathrm{td}}} \newcommand{\minus}[1]{\mbox{#1}^{-}} \newcommand{\na}{N_A} \newcommand{\nai}{N_A^i} \newcommand{\nao}{N_A^o} \newcommand{\nb}{N_B} \newcommand{\nbi}{N_B^i} \newcommand{\nbo}{N_B^o} \newcommand{\nc}{N_{C}} \newcommand{\nl}{N_L} \newcommand{\nltot}{N_L^{\mathrm{tot}}} \newcommand{\nr}{N_R} \newcommand{\nrl}{N_{RL}} \newcommand{\nrtot}{N_R^{\mathrm{tot}}} \newcommand{\out}[1]{{#1}_{\mathrm{out}}} \newcommand{\plus}[1]{\mbox{#1}^{+}} \newcommand{\rall}{\mathbf{r}^N} \newcommand{\rn}[1]{\mathrm{r}^N_{#1}} \newcommand{\rdotc}{R \!\! \cdot \! C} \newcommand{\rstarc}{R^* \! \! \cdot \! C} \newcommand{\rstard}{R^* \! \! \cdot \! D} \newcommand{\rstarx}{R^* \! \! \cdot \! X} \newcommand{\ss}{\mathrm{SS}} \newcommand{\totsub}[1]{{#1}_{\mathrm{tot}}} \newcommand{\totsup}[1]{{#1}^{\mathrm{tot}}} \newcommand{\ztot}{Z^{\mathrm{tot}}} % Rate notation: o = 1; w = two; r = three; f = four \newcommand{\aow}{\alpha_{f}} \newcommand{\awo}{\alpha_{u}} \newcommand{\kow}{\kf} % {\kf(12)} \newcommand{\kwo}{\ku} % {\ku(21)} \newcommand{\kor}{\conc{C} \, \konu} % \konu(13)} \newcommand{\kwf}{\conc{C} \, \konf} % \konf(24)} \newcommand{\kro}{\koffu} % {\koffu(31)} \newcommand{\kfw}{\kofff} % {\kofff(42)} \newcommand{\krf}{\kfc} % {\kfc(34)} \newcommand{\kfr}{\kuc} % {\kuc(43)} \newcommand{\denom}{ \krf \, \kfw + \kro \, \kfw + \kro \, \kfr } $
splitscreen view

Reaction Coupling

Biosynthesis is Driven by Free Energy

Synthesis Basic Eqn

Chemical syntheses in the cell, such as the formation of X-Y above, typically require an input of free energy. Free energy could be supplied directly if there were an over-abundance of reactants (compared to equilibrium) and/or a dearth of products. More commonly, there is a complex sequence of reactions required for synthesis, and free energy drives these intermediate reactions. Below, we see how the hydrolysis of ATP can be coupled to a series of two synthesis reactions. Other carriers could also supply the free energy to drive synthetic reactions.

Note that if a synthetic reaction were favorable (did not require free energy), it would only be necessary for the cell to supply a catalyst.

Reaction coupling, driven by ATP

An example of the mechanism here is the synthesis of glutamine from glutamic acid, where X-OH = glutmatic acid, Y-H = NH$_3$. Below, -P = -O-PO$_3$.

The synthesis of X-Y can be coupled to ATP hydrolysis by the following two reactions

Synthesis Mechanistic Eqns

The net result of the two reactions is the synthesis of X-Y and the hydrolysis of ATP. Graphically the process looks like this

Synthesis Schematic

Steady-state analysis of the coupled reactions

Will the two-step synthesis above really occur under typical cellular conditions, even if the direct synthesis ($X + Y \rightarrow XY$) is unfavorable? We can answer this question using a steady-state analysis of the equations above in a mass action picture. Specifically, we can calculate the rate of formation of XY (and Pi) in the second of the two reactions, which is marked with a green (*). We use a steady-state condition on the intermediate XP:

Synthesis Steady State

Because the last two terms are precisely the net rate of XY and Pi synthesis - based on Eq (*) - and the four terms together sum to zero in a steady state, we can rewrite the synthesis rate to see the dependence on ATP and ATP:

Synthesis Rate

So long as the ratio $\conc{ATP} / \conc{ADP}$ is sufficiently large, the rate of XY synthesis will be positive. And because ATP is an activated carrier that ratio is large.

Bottom line: XY is synthesized at a positive rate if ATP is sufficiently activated.

splitscreen view

Reaction Coupling

Biosynthesis is Driven by Free Energy

Synthesis Basic Eqn

Chemical syntheses in the cell, such as the formation of X-Y above, typically require an input of free energy. Free energy could be supplied directly if there were an over-abundance of reactants (compared to equilibrium) and/or a dearth of products. More commonly, there is a complex sequence of reactions required for synthesis, and free energy drives these intermediate reactions. Below, we see how the hydrolysis of ATP can be coupled to a series of two synthesis reactions. Other carriers could also supply the free energy to drive synthetic reactions.

Note that if a synthetic reaction were favorable (did not require free energy), it would only be necessary for the cell to supply a catalyst.

Reaction coupling, driven by ATP

An example of the mechanism here is the synthesis of glutamine from glutamic acid, where X-OH = glutmatic acid, Y-H = NH$_3$. Below, -P = -O-PO$_3$.

The synthesis of X-Y can be coupled to ATP hydrolysis by the following two reactions

Synthesis Mechanistic Eqns

The net result of the two reactions is the synthesis of X-Y and the hydrolysis of ATP. Graphically the process looks like this

Synthesis Schematic

Steady-state analysis of the coupled reactions

Will the two-step synthesis above really occur under typical cellular conditions, even if the direct synthesis ($X + Y \rightarrow XY$) is unfavorable? We can answer this question using a steady-state analysis of the equations above in a mass action picture. Specifically, we can calculate the rate of formation of XY (and Pi) in the second of the two reactions, which is marked with a green (*). We use a steady-state condition on the intermediate XP:

Synthesis Steady State

Because the last two terms are precisely the net rate of XY and Pi synthesis - based on Eq (*) - and the four terms together sum to zero in a steady state, we can rewrite the synthesis rate to see the dependence on ATP and ATP:

Synthesis Rate

So long as the ratio $\conc{ATP} / \conc{ADP}$ is sufficiently large, the rate of XY synthesis will be positive. And because ATP is an activated carrier that ratio is large.

Bottom line: XY is synthesized at a positive rate if ATP is sufficiently activated.