Proofreading through spatial gradients

  1. Vahe Galstyan
  2. Kabir Husain
  3. Fangzhou Xiao
  4. Arvind Murugan
  5. Rob Phillips

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Abstract

Key enzymatic processes use the nonequilibrium error correction mechanism called kinetic proofreading to enhance their specificity. The applicability of traditional proofreading schemes, however, is limited because they typically require dedicated structural features in the enzyme, such as a nucleotide hydrolysis site or multiple intermediate conformations. Here, we explore an alternative conceptual mechanism that achieves error correction by having substrate binding and subsequent product formation occur at distinct physical locations. The time taken by the enzyme–substrate complex to diffuse from one location to another is leveraged to discard wrong substrates. This mechanism does not have the typical structural requirements, making it easier to overlook in experiments. We discuss how the length scales of molecular gradients dictate proofreading performance, and quantify the limitations imposed by realistic diffusion and reaction rates. Our work broadens the applicability of kinetic proofreading and sets the stage for studying spatial gradients as a possible route to specificity.

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  1. Version published to 10.7554/elife.60415 on eLife
  2. eLife

    ###Reviewer #3:

    I like this paper. It clearly and succinctly presents an interesting and (to my knowledge) novel mechanism for proofreading that is distinct from typical formulations, that decouples the enzyme itself from the proofreading functionality (essentially modularizing the proofreading mechanism). The derivations and figures explore its possibilities and physical limits in a fairly convincing fashion (subject to several minor quibbles I detail below), supporting the conclusions. This mechanism significantly broadens the scope of systems that could enact proofreading, and allows tuning of the proofreading by regulating activity or concentration of gradient maintainers or enzyme, thus promising significant implications.

    My two main suggestions are to give more context about (1) the effect of enzymatic catalysis on the resulting spatial distributions and (2) the relative costs of the two most prominent energy-consuming processes needed for this scheme. Specifically:

    1. The entire manuscript assumes that catalysis is negligible and thus need not be explicitly modeled in solving for the steady-state distributions. How would incorporating a boundary condition at the right that involves non-negligible catalysis change (even qualitatively) your findings?

    2. When quantifying the energetic costs, the main text solely focuses on the cost of counteracting the enzyme binding substrate, diffusing, and releasing. The SI explores some theory for the other cost of maintaining the substrate gradients, but without reporting any absolute numbers. For the biologically plausible kinase/phosphatase substrate-maintenance mechanism explored in the main text, how does its cost compare to the cost that you study quantitatively in the main text?

  3. eLife

    ###Reviewer #2:

    In the manuscript by Galstyn et al on "Proofreading through spatial gradients", the authors proposed and studied a new kinetic proofreading (KP) model/scheme based on having a spatial gradient of the substrate (both "correct" and "wrong" ones) and the diffusive transport of the substrate-bound enzyme molecules to a spatially localized production site. The authors did an excellent job in explaining their new model and its connection and difference w.r.t. the classical Hopfield-Ninos KP mechanism. The key insight is that with spatial inhomogeneity, e.g., in the presence of a persistent spatial gradient for the enzyme or the substrate, one can consider spatial location as a state-variable. By having the substrate and product (or production site) at different spatial locations, these spatial degrees of freedom of the enzyme, i.e., enzymes at different physical location, can be considered as the intermediate states that are necessary for kinetic proofreading - each intermediate state contributes a certain probability for error-correction. In the original Hopfield-Ninos KP scheme, the intermediate state is provided by additional enzyme(s), whereas in this new KP scheme, it depends on having a spatial gradient, which the authors argue is more tunable. I like the theory for its simplicity and elegance. I have only a few mostly technical questions/comments.

    My main concern for this study, however, is about how relevant this mechanism is for realistic biological systems. The original Hopfield-Ninos KP mechanism was motivated by specific and important biological problems (puzzles), namely the unusually high fidelity in biochemical synthesis process (in comparison with its equilibrium value). In this MS, the theory is developed without a specific biological system or specific biological question in mind. It is true that spatial gradient exists across biological systems and the authors also showed that typical kinetic rates may fall in the functional range of this new gradient-dependent KP mechanism. But, what is the function of the original system that such a kinetic proofreading process can help improve? Is it biochemical synthesis? Do the authors envision "correct" and "wrong" biomolecules being produced at the production site (x=L) like in the original setting of Hopfield-Ninos? Or is it signaling like in the T-cell signaling case? If so, do the authors envision that both the correct signaling molecule and the incorrect signaling molecule have a spatial gradient and they can both be carried by the same enzyme to their functional sites? I am not asking for a detailed comparison with a specific system, but I think a known but unsolved biological phenomenon that may be explained by this new mechanism would really help motivate a biologist audience. Furthermore, a connection to a specific biological system could also lead to testable predictions that would ultimately verify (or falsify) the existence of this mechanism.

    Questions related to the model/theory:

    1. In this study, there is a production r for the enzymatic reaction at x=L where the enzyme is active. However, the effect of this reaction, which change ES-->E+P, is not considered in the model equations (1-3). Is it because r is considered to be small? If so, smaller than what? Since speed is directly related to r, how does the value of r affect the speed and the speed-accuracy trade-off?

    2. The nonmonotonic dependence of fidelity on the diffusion time for finite gradient as shown in Fig. 3c is intriguing. What determines the optimal diffusion constant (or diffusion time) when the fidelity is maximum for a given gradient length scale?

    3. The study of trade-off among energy dissipation, speed, and fidelity is quite nice and adds to a growing list of study on performance trade-off's in nonequilibrium systems. For example, a similar energy-speed-accuracy (ESA) trade-off was studied systematically in the context of adaptation in bacterial chemotaxis (Lan et al, Nature Physics 8, 422-428, 2012) and chemosensory adaptation in eukaryotic cells (Lan and Tu, J R Soc Interface 10 (87), 2013). In particular, the exponential dependence of the fidelity on power consumption (energy dissipation) shown in Fig. 4 in this MS agrees well with results in these earlier studies (see Fig. 3c and Eq. 5 in Lan et al, 2012; Fig. 4 in Lan&Tu, 2103). It would be informative to discuss the trade-off found here for the gradient-dependent KP scheme in comparison with similar trade-off relations in other systems.

    4. The power dissipation P is computed by Eq.8 in this MS. Where does Eq. 8 come from? What's the physical meaning of P? The standard way to compute energy dissipation is by computing the entropy production rate S', which is well defined. Then by assuming the internal energy does not change with time in steady state, we equate energy dissipation with kT*S'. The form of entropy production rate is known and can be found in text book (such as those from T. Hill) and papers (e.g., those from H. Qian and collaborators; and from U. Seifert and collaborators), and the formula given in Eq. 8 does not seem to be consistent with the known form of entropy production. In particular, for a given reaction with forward flux J+ and backward flux J-, the entropy production rate is: (J+-J-)ln(J+/J-), which can be easily shown to be positive definite and only =0 when detailed balance J+=J- is satisfied.

    Overall, the MS provided a new gradient-dependent scheme for error correction in chemical systems. The study of trade-off among energy dissipation, speed, and fidelity (accuracy) in this new mechanism is also valuable for the general study of cost-performance relation in non-equilibrium systems. My main concern is the lack of examples of specific biological systems where this gradient-dependent error correction mechanism could be at work to enhance the specific biological functions of these systems.

  4. eLife

    ###Reviewer #1:

    The authors proposed a new theoretical mechanism of kinetic proofreading based on spatially distributed biochemical systems. This concept is novel and distinctive from existing models of proofreading, although it is not yet proved experimentally. The writing is clear, concise and elegant. There are no logical flaws, and I really enjoyed reading this manuscript. Yet, I have a number of comments to be addressed, which will substantially increase the quality of this manuscript.

    1. P. 1. The same concentration profiles are assumed for the right substrate R and the wrong substrate W. This is a strong assumption, could the authors consider the case where the concentration gradient length of the wrong substrate profile is larger than this length for the right substrate but still smaller that the distance L? They may calculate a series of the fidelity curves with increasing Lambda_W and the same Lambda_R. How will proofreading change?

    2. P. 2. "The scheme proposed here does not rely on any proofreading-specific structural features in the enzyme; indeed, any 'equilibrium' enzyme with a localized effector can proofread using our scheme if appropriate concentration gradients of the substrates or enzymes can be set up. As a result, spatial proofreading is easy to overlook in experiments and suggests another explanation for why reconstitution of reactions in vitro can be of lower fidelity than in vivo." The key is the difference in the off rates for the right substrate R and the wrong substrate W, k^W_off >k ^R_off because W & R compete for E. This has to be mentioned in the above statement.

    3. P. 2. "To demonstrate the proofreading capacity of the model, we first analyze the limiting case where substrates are highly localized to the left end of the compartment, lambda S << L." However, Eq. 5 is derived assuming that not only lambda s << L, but also lambda S << lambda ES (see Appendix).

    4. P. 3. "... a red curve on the plot, is reached in the limit of ideal sequestration, ... " The word sequestration has a different meaning in biochemistry, e.g., it is used to describe 'sequestration' of an enzyme by the substrate/product or an inhibitor, which is not what the authors have in mind. They use 'sequestration' to describe the ideal substrate localization, Lambda_S -> 0. Put aside that this use of 'sequestration' is not the best choice, the authors need, at least, to explicitly define what they mean under 'sequestration'.

    5. Fig. 3. Please explicitly define Veq speed (when k^W_off = k^R_off). In addition, how a black dotted curve is obtained is not explained, and the corresponding parameters are not given.

    6. P. 5. "an enzyme E that acts on active forms of cognate (R) and non-cognate (W) substrates which have off rates 0.1 s−1 and 1 s−1, respectively (hence, theta eq = 10)." This implies a large difference in the free energy of binding of more than 1kcal/mol. In the absence of ATP/GTP hydrolysis, the difference in the binding energies is usually small. Can the authors give a specific example for an enzyme system where the difference in the free energy of binding is more than 1kcal/mol with no ATP/GTP hydrolysis?

    7. Pp 5- 6. "As expected, proofreading by these gradients is most effective when the enzyme-substrate binding is very slow, in which case the exponential substrate profile is maintained and the system attains the fidelity predicted by our earlier explanatory model (Fig. 5b). .... If the binding rate constant (kon) or the enzyme's expression level (r_E) is any higher, then enzymatic reactions overwhelm the ability of the kinase/phosphatase system to keep the active forms of substrates sufficiently localized (Fig. 5c) and proofreading is lost." This is not entirely clear because the gradients depend on the phosphatase activity, whereas the authors did not mention that they likely assumed that when the substrate is bound to the enzyme, it is protected against the phosphatase.

    8. Appendix D. The authors have to also consider or at least discuss the different diffusivities for phosphorylated and unphosphorylated substrates, a feature of many spatially distributed system and cite [FEBS Letters 583 (2009) 4006-4012] where this case was considered for dynamically stable spatial gradients.

  5. eLife

    ##Preprint Review

    This preprint was reviewed using eLife’s Preprint Review service, which provides public peer reviews of manuscripts posted on bioRxiv for the benefit of the authors, readers, potential readers, and others interested in our assessment of the work. This review applies only to version 1 of the manuscript. Ahmet Yildiz (University of California) served as the Reviewing Editor.

    ###Summary:

    In the manuscript by Galstyn et al on "Proofreading through spatial gradients", the authors proposed and studied a new kinetic proofreading (KP) model/scheme based on having a spatial gradient of the substrate (both "correct" and "wrong" ones) and the diffusive transport of the substrate-bound enzyme molecules to a spatially localized production site. The authors did an excellent job in explaining their new model and its connection and difference w.r.t. the classical Hopfield-Ninos KP mechanism. The key insight is that with spatial inhomogeneity, e.g., in the presence of a persistent spatial gradient for the enzyme or the substrate, one can consider spatial location as a state-variable. By having the substrate and product (or production site) at different spatial locations, these spatial degrees of freedom of the enzyme, i.e., enzymes at different physical location, can be considered as the intermediate states that are necessary for kinetic proofreading - each intermediate state contributes a certain probability for error-correction. In the original Hopfield-Ninos KP scheme, the intermediate state is provided by additional enzyme(s), whereas in this new KP scheme, it depends on having a spatial gradient, which the authors argue is more tunable. The reviewers were enthusiastic about the theoretical model presented in this study because of its simplicity and elegance. However, the reviewers have also raised serious concerns that need to be addressed. In summary, the panel feels that discussion of possible biological example(s) where this novel type of proofreading may be occurring would significantly improve the manuscript's appeal to a broad audience. In addition, the reviewers ask for more explicit explanation of the effect of enzymatic catalysis rates, and discussion of the full dissipation cost.

  6. Version published to 10.1101/2020.05.23.112664v1 on bioRxiv