Thermodynamics of Error Correction

Information processing at the molecular scale is limited by thermal fluctuations. This can cause undesired consequences in copying information since thermal noise can lead to errors that can compromise the functionality of the copy. For example, a high error rate during DNA duplication can lead to cell death. Given the importance of accurate copying at the molecular scale, it is fundamental to understand its thermodynamic features. In this paper, we derive a universal expression for the copy error as a function of entropy production and work dissipated by the system during wrong incorporations. Its derivation is based on the second law of thermodynamics; hence, its validity is independent of the details of the molecular machinery, be it any polymerase or artificial copying device. Using this expression, we find that information can be copied in three different regimes. In two of them, work is dissipated to either increase or decrease the error. In the third regime, the protocol extracts work while correcting errors, reminiscent of a Maxwell demon. As a case study, we apply our framework to study a copy protocol assisted by kinetic proofreading, and show that it can operate in any of these three regimes. We finally show that, for any effective proofreading scheme, error reduction is limited by the chemical driving of the proofreading reaction.

Popular Summary

From the operation of digital computers to the duplication of genetic material, accurate copying of information is crucial for the functioning of complex organisms and devices. A common feature of all copying machines is that they consume energy to operate at a finite speed, making copying a thermodynamically irreversible process. Our work reveals the existence of a universal relation between irreversibility and accuracy, which is independent of the specific system and the copying protocol. We apply this relation to study kinetic proofreading, an important error-correction mechanism employed by biological systems. We find that, in addition to increasing precision, proofreading can also extract work while correcting errors.

Thermal fluctuations affect copying on the molecular level, and significant deleterious effects can result from, for example, errors associated with DNA copying. Here, we focus on the second law of thermodynamics and how thermodynamics affects the accuracy of copying. We examine how a polymer strand with two different kinds of monomers is copied when free monomers are selected for the copying process. We recover an equilibrium error rate and three copying regimes, all of which are possible in nature. We note that the microscopic details of the copying process are important; they determine both the error rate and thermodynamic quantities such as dissipated work. Although each of these quantities is detail dependent, there are relations among these quantities that are independent of the details.

We expect that our findings will pave the way for future studies of other cellular information-processing tasks (e.g., chemosensing).

Figure 1

Template-assisted polymerization. The template strand is a preexisting polymer made up of two different kinds of monomers (gray and white circles). A molecular copying machine (red circle) assists the growth of a copy strand by incorporating freely diffusing monomers of two different types, trying to match them with those of the template strand. Right and wrong matches are noted $r$ and $w$.

Figure 2

Transition network of template-assisted polymerization and examples. (a) State space of the template-assisted polymerization model. Monomer incorporation occurs via a network of intermediate states represented inside the dashed circles. The two colors distinguish networks leading to incorporation of right and wrong monomers. The structure is repeated in a tree-shaped structure as the polymer grows by addition of more and more monomers. (b) Examples of networks of intermediate states. First example: template-assisted polymerization without intermediate states (see, e.g., Refs. [8, 11, 12, 13, 15]). Second example: kinetic proofreading, where after an intermediate state a backwards-driven pathway removes errors to improve the overall accuracy of the copy [4, 13]. Third example: messenger RNA translation, where the three copying steps represent initial binding, GTP hydrolysis, and final accommodation; a proofreading reaction is also present [18].

Figure 3

Energy landscape and kinetic rates. (a) Energetic diagram of a single transition in the reaction network. (b) Corresponding kinetic rates. The transition $j\to i$ can be driven by energy differences and the chemical driving ${\mu }_{ij}$. Transitions involving a right and a wrong monomer can be characterized by different kinetic barriers ${\delta }_{ij}$, as well as different energetic landscapes $\mathrm{\Delta }{E}_j^w\ne \mathrm{\Delta }{E}_j^r$. The bare rate ${\omega }_{ij}$ is the inverse characteristic time scale of each reaction.

Figure 4

Template-assisted polymerization without intermediate states. (a) Excess work $\mathrm{\Delta }W-\mathrm{\Delta }{F}_{\mathrm{eq}}$, entropy production, and Kullback-Leibler term of Eq. (2) as a function of the error. Notice that the excess work dominates over the information term. (b) Same terms as in (a) but for wrong monomers only. In this case, the information term dominates the entropy production. (c) Relation between error and entropy production of wrong monomers, together with thermodynamic (red dashed line) and hardware-specific (black dashed line) bounds. In all panels, the driving ${\mu }_{10}$ is varied to vary the error. Parameters are ${\delta }_{10}=10T$, $\mathrm{\Delta }{E}_1^r=0$, $\mathrm{\Delta }{E}_1^w=3T$.

Figure 5

Regimes and bounds of kinetic proofreading. (a) Scheme of a generic proofreading scheme. Copying occurs at a net speed $v_c>0$ through an arbitrary reaction network of intermediate states. After the copy is finalized, a proofreading reaction removes errors at a speed $v_p<0$. The net average speed is $v=v_c+v_p\ge 0$. (b) Thermodynamic regimes of kinetic proofreading. The model combines a copying scheme with one intermediate state with kinetic proofreading, as represented in Fig. 2. The shaded regions denote the three thermodynamic regimes discussed in the Sec. 2b. Parameters are ${\delta }_{10}=5T$, ${\delta }_{21}=0$, ${\delta }_{02}=5T$, $\mathrm{\Delta }{E}_2^w=\mathrm{\Delta }{E}_1^w=2T$, $\mathrm{\Delta }{E}_2^r=\mathrm{\Delta }{E}_1^r=0$. We remind the reader that states 0, 1, and 2 represent the state before monomer incorporation, the intermediate state, and the final state where monomer has been incorporated, respectively [see also Fig. 3]. For each value of the error $\eta$, the other free parameters (${\mu }_{10}$, ${\mu }_{21}$, ${\omega }_{21}$, ${\omega }_{02}$) are determined by minimizing the entropy production per copied wrong monomer $\mathrm{\Delta }{S}_{\mathrm{tot}}^w$. (c) Minimum error as a function of the proofreading work $\mathrm{\Delta }{W}_p={\mu }_{02}$. For each curve, energies and activation barriers are fixed parameters as in (b) (except for ${\delta }_{02}$, which varies, as in the captions). For each value of ${\mu }_{02}$, the other free parameters (${\mu }_{10}$, ${\mu }_{21}$, ${\omega }_{21}$, ${\omega }_{02}$) are determined by numerically minimizing the error $\eta$. Red dashed and black dashed lines represent thermodynamic and hardware-specific bounds, respectively.