Active transport of Ca2+ ions against chemical and electrostatic gradients, assisted by ATP hydrolisis I have a homework question that I am not able to solve. No solution is available, so I was hoping someone here could help. The question is this:


  Calculate the maximal concentration ratio that can be obtained for
  transport of Ca$^{2+}$ ions if two ions are transported per ATP hydrolyzed
  at 310 K, and they are transported to a potential that is 70 mV higher
  at the receiving side. Assume that the $\Delta G$ for hydrolysis of ATP = -45 > kJ\mol.


Relevant formulas:

\begin{align}
\Delta G_{\text{electrostatic gradient}} &amp;= ZF\Delta \Psi\\
\Delta G_{\text{chemical gradient}} &amp;= RT\ln\frac{C_\mathrm{in}}{C_\mathrm{out}}\\
\Delta G_{\text{hydrolysis of ATP}} &amp;= -45000~\mathrm{J/mol}\\
\end{align}

Moving two positive $\ce{Ca^2+}$ ions against both a concentration gradient and an electrostatic gradient implies that $G_{\text{chemical gradient}}$ and $\Delta G_{\text{electrostatic gradient}} &gt; 1$. The negative value for $\Delta G_{\text{hydrolysis of ATP}}$ means that hydrolysis of ATP is a spontaneous process, or a "work-yielding" process. The energy released from hydrolysing ATP is essentially what drives the transport process. Since we are transporting "4 charges" per ATP molecule, $\Delta G_{\text{chemical gradient}}$ is multiplied by 2. Also (I presume) the concentration of $\ce{Ca^2+}$ ions is larger on the inside of the cell than on the outside, implying that the concentration ratio is larger than 1.

The combined expression for $\Delta G$ for the unassisted transport is

$$\Delta G = RT\ln\frac{C_\mathrm{in}}{C_\mathrm{out}} + 2ZF\Delta \Psi$$

Rearranging this yields

$$\frac{C_\mathrm{in}}{C_\mathrm{out}} = \exp\left\{\frac{\Delta G - 2ZF\Delta\Psi}{RT}\right\}$$

I am not sure what to plug in for $\Delta G$, but it needs to account for the push from ATP. I believe this is done by summing the two free energies affecting the transport, ergo the sum of $\Delta G_{\text{electrostatic gradient}}$ and $\Delta G_{\text{hydrolysis of ATP}}$. However, this gives me a concentration ratio much smaller than 1.

Can anyone see where I am incorrect?
At equilibrium, the entire process (transport of $2~\ce{Ca^2+}$ ions and hydrolysis of $1~\text{ATP}$) involves no change in G.  

\begin{align}
0 &amp;=\Delta{G_\mathrm{total}}\\[2ex]
0 &amp;=\Delta{G_\mathrm{ATP}} + 2\Delta{G_{\ce{Ca^2+}}}\\[2ex]
0 &amp;=\Delta{G_\mathrm{ATP}} + 2\Delta{G_{\ce{Ca^2+},\text{ concentration gradient}}} + 2\Delta{G_{\ce{Ca^2+},\text{ electrostatic}}}\\[2ex]
0 &amp;=\Delta{G_\mathrm{ATP}} + 2RT\ln\frac{C_\mathrm{in}}{C_\mathrm{out}} + 2ZF\Delta \Psi\\
\end{align}

So basically I'm saying you made two mistakes, one a sign error and one a factor of 2 error.