Kinetic Proofreading and the limits of thermodynamic uncertainty: a review. Jenny Poulton

This paper attempts to link together two important concepts in theoretical biophysics: kinetic proofreading and the thermodynamic uncertainty relation. It analyses both in the context of copying sequence information in a polymer, but feels like it confuses more than it clarifies. The paper does make some clear statements about the predictability of travelling through a copying network. However, it fails to link this quantity to either speed or accuracy.  It further fails to make the case for the intrinsic use of this predictability, in systems such of this.

It is worth taking a moment to briefly discuss the ideas of kinetic proofreading and the thermodynamic uncertainty relation separately, before we attempt to link the together. Kinetic proofreading was first posited separately by Hopfield (1974) and Ninio (1975). It is a method by which biochemical copying systems, such as RNA translation, can improve accuracy by spending extra energy. It is also an excellent example of the motivation [TO1] behind using simple theoretical models to describe systems; while kinetic proofreading was initially posited as a completely theoretical idea, it was widely adopted by the biological community as it gave good agreement with real biological results.

In general, simple copying systems approximate to a system as shown in figure 1. A copy polymer is growing on a template polymer, connected only by its final monomer. A new monomer, of either a matching or non-matching type will bind to the template polymer. It will then polymerise into the chain, and the previous final link between copy and template will break.

Now in a copying system, the most obvious question to ask is about accuracy, how well does the copy match the template, and how does the system discriminate? Discrimination comes in the very first step. Because non-matching monomers are more weakly bound to the template polymer, they fall off more quickly than matching monomers. Thus, if the rates are carefully tuned, the system can polymerise the monomer into the copy polymer chain fast enough that matching monomers are unlikely to fall off before incorporation, but non-matching ones will fall off. Thus the system can generate accuracy.

Kinetic proofreading adds an extra energy driven step. Instead of the system polymerising the monomer into the chain directly, the system first has to spend energy activating the monomer, and only then can the monomer be incorporated into the chain. As long as the activation step is driven energetically towards activation, either through a chemical gradient or more directly, then this effectively gives the incorrect monomer two opportunities to fall off rather than be incorporated; in some limits squaring the discrimination term. Thus you can pay extra energy to increase accuracy.

Now before I move onto the thermodynamic uncertainty relation, I’m going to take a moment to stress that in a kinetic proofreading system, one is usually considering the error, ie. how alike the copy polymer and the template polymer are. This is not the same as the uncertainty in the thermodynamic uncertainty relation.

Figure 1; Left: a simple three step copying reaction in which a monomer binds to the template, is polymerised into the chain and the previous final monomer detaches. Right: the system with an additional proofreading step. The system must be energetically driven to activate the monomer, at which point it has a second opportunity to fall off before incorporation.

So what is the thermodynamic uncertainty relation? The thermodynamic uncertainty relation considers a stochastic process, often visualised by a network of states which outline progression through a process. In the case of biological copying, the process is that of adding a monomer to the end of a growing polymer as shown in figure 2. However, it could equally be the process of a molecular walker taking a step along a track. The thermodynamic uncertainty relation tracks the uncertainty in the net number of times a particular thing  happens . For example, the number of times a system undergoes a specific transition. In a case of a molecular walker  could be the uncertainty in how far the walker had walked with the net number of forward steps being . In the case of a copy process   could be the uncertainty in the net number of correct things the system has added (ie how many times the system has gone round the upper right loop in figure 2) or it could be the uncertainty the net number of incorrect things that have been added (upper left loop). Crucially  would not automatically give you a relationship between the number of right and wrong things added. The form of the uncertainty relationship  tells us that we can again spend energy to reduce the uncertainty; here  is the energy cost of the process per unit time multiplied by the time.

So while both relationships have a form of uncertainty, and this uncertainty can be reduced in both cases by paying energy, the two uncertainties; the error  and the uncertainty , are not immediately related to each other and should not be conflated.

Now the aspect of kinetic proofreading which is most straightforward to link to the thermodynamic uncertainty is not the error but the speed. Banerjee et al discusses how for many simple copying processes such as those found in T7 DNAP enzymes acting on DNA and TRNA selection in E. coli ribosomes, in general systems are willing to tolerate a certain amount of error in order to maximise speed. While the thermodynamic uncertainty relationship doesn’t directly measure speed, it does characterise the uncertainty in progress, ie how reliable said speed is. Someone with a stronger molecular biology background than I might be able to convince me that predictability in copying speed is important, sadly the paper fails to do so.

Figure 2 The left hand side represents the network for adding a monomer in DNA related actions, the right hand side represents that for RNA related actions. In both the green “reduced system” at the bottom shows the path for adding a new monomer and extending the chain whereas the blue cycle adds and removes a monomer through kinetic proofreading. Reactions a and b are gradually turned off later in the paper.

The paper focusses on the number of times the system above goes round the green cycles in figure 2, with this being   .  The thermodynamic uncertainty variable  is defined relative to this. They define a lower bound on  by considering the reduced cycle corresponding to that system which contains only the green cycle. For this cycle which is unicyclic the uncertainty is well defined and here labelled; . They thus define a quantity  which defines the thermodynamic uncertainty relative to the minimum uncertainty and compare this quantity for a number of different systems. However it should be clear that it is not true that  means lower error, it merely states that it the net number of times the system goes round that particular cycle becomes more predictable. They compare this variable for a series of biological systems, with two subtly different networks shown in figure 2. These are Err ribosome (right) WT ribosome (right), Acc ribosome (right) and T7 polymerase (left). Recall that  . They present these results and suggest that a lower  means that the system is capable of better trading off error and speed, although this isn’t necessarily persuasively explained.

In the next section the authors ask how turning off two of the reactions, which reduce accessibility to parts of the network, changes . They relate this to error  and , a time constant which quantifies the time taken to go round the cycle.

Removing reaction a) as labelled in figure 2 prevents the system from attempting to add incorrect monomers, and removes access to the whole left hand side of the cycle. Removing reaction b) prevents the system from removing correct monomers via kinetic proofreading, but does not change the overall topology of the network in the way removing a) does.

The authors state that in both cases  is reduced when the reactions are removed. This would seem self-evident. They also state that energy cost for faster speeds is minimized when reactions are removed, because you don’t spend extra energy being pushed around futile cycles.

The authors point out that in case a) Q decouples from  as the reaction is removed, but the network retains it’s dependence on  in case b). They then explore how  depends on Q. In both cases they find that as the reactions are removed,  decouples from Q. This tells us that in the accurate limit where only the reduced cycle happens, Q is not related to . So the measurement of Q seems like a curious choice when it is decoupled from both  and  in the accurate limit.

The thermodynamic uncertainty relation seems like a tool unsuited to answer the important questions about accurate copying. While there may be good reasons to perform this analysis, the authors have failed to provide them.

A full house of papers combining DNA with other biomolecules to stabilise, direct and monitor assembly

Coating and Stabilization of Liposomes by Clathrin-Inspired DNA Self-Assembly 
doi:10.1021/acsnano.9b09453

Assembling a DNA triskelion array layer on liposomes stabilized the membrane while keeping the fluidic nature of the lipid molecules. The vesicle did not rupture on the mica surface an nor was it dissolved by adding triton-X 100 detergent.


Peptide Assembly Directed and Quantified Using Megadalton DNA Nanostructures 
https://doi.org/10.1021/acsnano.9b04251

Sequence-structure relationships are sufficiently well understood for alpha-helical polypeptides to enable bottom-up design of simple alpha helix complexes. Two halves of a heterodimeric peptide were each tethered to large DNA nanostructures. The DNA nanostructures could be clearly differentiated with TEM. The peptide sequences were intentionally designed with a hydrophobic seam along which two alpha-helices could form a bond, bringing together two DNA nanostructures. To prevent DNA sticking to the peptide complexes, the sequences of peptides were chosen to be charge neutral at pH 7. Multiple peptide halves can be attached to each DNA nanostructure, altering the valency of the alpha helix bonding interaction. The authors measure the Kd disassociation constant for the peptide interactions using CD melting and by counting samples using TEM. This semi-quantitative technique is a first step toward the creation of complex rationally-designed peptide-oligonucleotide nanostructures but is not the most promising route toward measuring peptide interactions.


Directed Energy Transfer through DNA-Templated J-Aggregates
https://pubs.acs.org/doi/10.1021/acs.bioconjchem.9b00043

DNA photonic wires are DNA duplexes labelled with fluorophores that are able to transfer optical excitation through long distances by FRET interactions. In Nature, optical excitation transference is usually achieved by dye clusters templated over polymeric chains, such as proteins. Cyanine dyes are able to produce such clusters by stacking their aromatic groups together.

In the present paper, the authors template the formation of dye clusters, out of the fluorophore pseudoisocyanine, over a DNA duplex of poly(A)-poly(T).The DNA scaffold is able to produce a continuous cyanine aggregate across 48 base pairs. The optical transfer of the continuous cyanine cluster is compared with a DNA scaffold that produces a gap in the cluster. A single base pair gap in the cluster results in a sensible decrease of the optical transference efficiency, remarking the importance of the continuous templating.