Kinetic proofreading and the limits of thermodynamic uncertainty

William D. Piñeros and Tsvi Tlusty

Phys. Rev. E 101, 022415 – Published 24 February 2020

Abstract

To mitigate errors induced by the cell's heterogeneous noisy environment, its main information channels and production networks utilize the kinetic proofreading (KPR) mechanism. Here, we examine two extensively studied KPR circuits, DNA replication by the T7 DNA polymerase and translation by the E. coli ribosome. Using experimental data, we analyze the performance of these two vital systems in light of the fundamental bounds set by the recently discovered thermodynamic uncertainty relation (TUR), which places an inherent trade-off between the precision of a desirable output and the amount of energy dissipation required. We show that the DNA polymerase operates close to the TUR lower bound, while the ribosome operates $\sim 5$ times farther from this bound. This difference originates from the enhanced binding discrimination of the polymerase which allows it to operate effectively as a reduced reaction cycle prioritizing correct product formation. We show that approaching this limit also decouples the thermodynamic uncertainty factor from speed and error, thereby relaxing the accuracy-speed trade-off of the system. Altogether, our results show that operating near this reduced cycle limit not only minimizes thermodynamic uncertainty, but also results in global performance enhancement of KPR circuits.

Received 11 November 2019
Accepted 24 January 2020

DOI:https://doi.org/10.1103/PhysRevE.101.022415

Physics Subject Headings (PhySH)

Biomolecular dynamics Nonequilibrium fluctuations

Biological networks Biomolecules Nonequilibrium systems Stochastic networks

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

William D. Piñeros¹ and Tsvi Tlusty ^1,2,*

¹Center for Soft and Living Matter, Institute for Basic Science (IBS), Ulsan 44919, Korea
²Department of Physics, Ulsan National Institute of Science and Technology (UNIST), Ulsan 44919, Korea

^*tsvitlusty@gmail.com

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Vol. 101, Iss. 2 — February 2020

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Images

Figure 1
Chemical reaction networks for (a) T7 DNA polymerase and (b) E. coli ribosome. Half-arrows indicate reversible forward and backward paths for both the right (R) and wrong (W) cycles. Kinetic rates are labeled by $k_{i, W / R}^{(-)}$ , where $i = 1, 2, 3, p$ for each relevant path. Note that transitions through $p$ in green (curved gray) half-arrows are implied to reset the enzyme to its initial state after product formation. Blue half-arrows (straight gray) indicate the proofreading transitions. Shown to the right of the black arrows are the reduced kinetic cycles (RC) where only relevant paths and rates leading to correct product formation are included.
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Figure 2
Product cycle current $J_{p}^{R}$ as a function of generalized rate constant $k$ , defined as $k_{1 R} = a_{1} k$ , $k_{2 R} = a_{2} k$ , $k_{p R} = a_{p} k$ with $a_{1} = 1$ , $a_{2} = k_{2 R}^{phys} / k_{1 R}^{phys}$ and $a_{p} = k_{p}^{phys} / k_{1 R}^{phys}$ , where $k_{i}^{phys}$ are the physiological values. The physiological points are shown as dots. (a) The current of correct substrate production $J_{p}^{R}$ for Wild-type ribosome (WT, blue top), more erroneous (Err, red bottom) and more accurate (Acc, yellow middle) mutants and the ideal RC current (dashed line). (b) Same as (a) but for T7 polymerase (solid line). The difference between the actual T7 current and RC is in the order of $1 %$ and not noticeable at this scale.
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Figure 3
Parametric plots of the normalized TUR measure $q \equiv Q / Δ μ_{p}$ vs. mean production time $τ$ ( $s$ ) as a function of $k$ for decreasing values of discrimination factor $f_{1}$ (a,c) and increasing values of the proofreading discrimination factor $f_{3}$ (b,d) for the wild-type ribosome and DNA polymerase, respectively. Each corresponding factor $f_{i}$ is scaled from its physiological value $f_{i}^{phys}$ as shown in the gradient scale on top. Dashed black lines indicate ideal RC limit. Dotted gray lines indicate curves of constant $k$ scaled from its physiological value in powers of 2.
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Figure 4
(a) Parametric plot of $q \equiv Q / Δ μ_{p}$ vs. error $η$ as a function of $f_{1}$ . Lines indicate T7 DNA polymerase (purple leftmost), WT ribosome (blue bottom), erroneous (red top), and more accurate (yellow middle) ribosome mutants. Thick points indicate physiological values. Dashed line is a guide to the eye showing the value of $Q$ achieved at the ideal RC limit of WT ribosome which is approximately the same for all systems shown. (b) Same as (a) but for $f_{3} \to \infty$ . Inset: WT ribosome scaled from $k = k_{1 R}^{phys}$ to $k = 10 k_{1 R}^{phys}$ illustrating that only $f_{1} \to 0$ guarantees $Q$ goes to the RC limit.
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Figure 5
Parametric plots of the error $η$ versus mean production time $τ$ for wild-type ribosome as a function of $k$ for (a) decreasing values of binding discrimination $f_{1}$ , and (b) increasing values of proofreading discrimination $f_{3}$ . Each respective factor $f_{i}$ is scaled from its physiological value $f_{i}^{phys}$ as shown in the gradient scale on top. Dotted gray lines indicate curves of constant $k$ value scaled from its physiological value in powers of 2.
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Figure 6
Parametric plots of normalized dissipation differences between the actual $\dot{q} = \dot{Q} / Δ μ_{p}$ and the ideal RC limit ${\dot{q}}_{RC} = {\dot{Q}}_{RC} / Δ μ_{p}$ versus the mean production time $τ$ as a function of $k$ . Lines indicate T7 DNA polymerase (purple left), WT ribosome (blue middle), erroneous (red top), and more accurate (yellow bottom) ribosome mutants. Err mutant plot has been scaled down by a factor of 2 to fit the figure. Points indicate physiological values for each line respectively. Dashed line marks the RC difference which is zero by definition.
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