What is the energy required for storing and performing computations in synthetic computational biological cells?

The reason you aren't finding this information in the literature is that this metric doesn't match the way that most genetic circuits work.

You appear to be thinking about operations in genetic circuits as being like instructions on an electronic processor. In that world, an instruction starts, executes, and stops, and you can measure the amount of energy consumed between the start and the stop.

Most genetic circuits, however, are more like wiring up electronic circuits like NOT and NOR gates. The computation is thus not discrete but continuous in time. One can certainly still consider the operating burden of each device on the cell, but (1) the load is strongly dependent on the particular state of the device (e.g., high output vs. low output), and (2) ATP availability is typically not the limiting factor.

Thus, your question is simply not one that most practitioners would find meaningful to answer. Even in the Sarpeshkar paper that you cite, by the time he gets to discussing actual experimental results, the numbers are expressed in terms of fluorescent protein concentration rather than ATP.

If you want to understand the burden of genetic circuits on cells, a much more relevant starting point is the recent paper "The Effect of Loads in Molecular Communications" from McBride, Shah and Del Vecchio

This addresses, rather than answers, the question; but is, I think, preferable to continuing to discuss the problem in comments.

As I understand it, the question deals with the use of genetic circuitry in synthetic biology to perform mathematical calculations in an analogous way to that done in computer systems. By genetic circuitry is meant the induction and feed-back regulation of transcription of certain engineered genes in the DNA of living (presumably bacterial) cells by protein molecules (together perhaps with small-molecule inducers). Such systems are modelled on natural bacterial operons.

While some of the users of this site may be familiar with synthetic biology of this type to produce bacteria with practical applications based on sensing environmental molecules, I imagine that very few will be familiar with their abstract use for mathematical operations. Certainly, although I subscribe to Nature, I had missed the 2013 article on this topic by Daniel et al. (Nature 497, 619–624), which it would take me some effort to master. Therefore the poster should not be surprised if his question has few takers.

Those who are interested or involved in synthetic biology are unlikely to have been concerned with the energetics of these circuits. Certainly RNA transcription involves the hydrolysis of phosphodiester bonds of nucleoside triphosphates, but one takes this as read, and assumes the bacterium will be in an environment where energy sources are available for growth.

Out of interest I glanced at the introduction to another paper involving Sarpeshkar — Philosophical Transaction of The Royal Society A (2014) 372 1–22 — and read the following:

Every living cell within us is a hybrid analog–digital supercomputer that implements highly computationally intensive nonlinear, stochastic, differential equations with 30000 gene–protein state variables that interact via complex feedback loops. The average 10μm human cell performs these amazing computations with 0.34 nm self-aligned nanoscale DNA–protein devices, with 20 kT per molecular operation (1 ATP molecule hydrolysed), approximately 0.8 pW of power consumption (10 M ATP s−1) and with noisy, unreliable devices that collectively interact to perform reliable hybrid analog–digital computation.

So it would appear that Sarpeshkar or others are citing the energy used per molecular operation taken apparently as 1 ATP equivalent needed for each NTP addition. (This seems to me incorrect, as the cleavage of NTP to NMP + PPi means 2 phosphodiester bond equivalents of energy are used.)

As far as I can see — and I may well be wrong — the answer to the poster’s question would seem to lie in learning how many molecular operations are involved in performing the particular mathematical operations in the systems described in these papers. I suggest he has more chance of discovering that by reading the papers and calculating how much RNA is synthesized in each operation — or even contacting their authors — than by asking on a general biology forum, like this.