**My general research interests lie in the fields of Statistical Physics, Physical Chemistry, Biological Physics, and Stochastic Processes. In short, this means that I am broadly interested in complex systems that are governed by statistical laws and random events and in the scientific interfaces between Physics, Chemistry, Biology, Probability and Statistics. A brief review of my work is given below. For an updated list of publications please see my page on Google Scholar.**

**Ribosomes are Optimized for Autocatalytic Production**

**Many fine-scale features of ribosomes have been explained in terms of function, revealing an elegant molecular machine optimized for information transfer, error correction, speed and control. However, several large-scale features are not well understood, e.g. why a few rRNA molecules of varying size can dominate the ribosome mass or why r-protein is divided into 55-80 small and similarly sized r-proteins. We mathematically demonstrate that these features, and many differences between ribosomes, could be explained by optimization for efficient autocatalytic biogenesis.**

**S. Reuveni,**

**M. Ehrenberg and Johan Paulsson,**

*Ribosomes are optimized for autocatalytic production,*Nature, 547, 7663, (2017).**Michaelis-Menten Kinetics: a**

**Unified**

**Approach to First Passage Under Restart**

**In 1913 Michaelis & Menten published a seminal paper in which they presented a mathematical model of an enzymatic reaction and demonstrated how it can be utilized for the analysis and interpretation of kinetic data. More than a century later, the work of Michaelis & Menten is considered classic textbook material, and their reaction scheme is widely applied both in (see our work on s**

**ingle-molecule enzymology**

**below) and out of its original context. At its very core, the scheme can be seen as one which describes a generic first-passage process that has further become subject to stochastic restart with (or without) an additional time overhead. This context free standpoint is not the standard one, but it has recently allowed us to treat a wide array of seemingly unrelated processes on equal footing, thus deepening our understanding on the fundamental problem of first passage under restart and on its various applications.**

**Q: "Optimal restart?"**

**Stopping a process in its midst—only to start it all over again—may prolong, leave unchanged, or even shorten the time taken for its completion. Among these three possibilities the latter is particularly interesting as it suggests that restart can be used to expedite the completion of complex processes involving strong elements of chance. This observation has long been made in the field of computer science where the use of restart is now routine as it drastically improves performance of randomized algorithms, but is also relevant to many physical, chemical, and biological processes since restart is an integral part of the renowned Michaelis-Menten reaction scheme and because**

**Michaelian**

**processes are prevalent in nature.**

We looked into the effect of restart, and retry, on the completion time of a generic process, asking: what is the optimal restart rate that brings the mean completion time of a process to a minimum? First, the governing equation for this problem was derived and solved exactly for cases of particular interest. And it was then demonstrated that there are regimes at which solutions to the general problem take on universal, details independent forms, which further give rise to optimal scaling laws. T

We looked into the effect of restart, and retry, on the completion time of a generic process, asking: what is the optimal restart rate that brings the mean completion time of a process to a minimum? First, the governing equation for this problem was derived and solved exactly for cases of particular interest. And it was then demonstrated that there are regimes at which solutions to the general problem take on universal, details independent forms, which further give rise to optimal scaling laws. T

**he results obtained are widely applicable as they allow one to take into account the effect of restart on an existing, generic, first passage time problem in an almost plug & play manner.**

**T. Rotbart, S. Reuveni and M. Urbakh,**

*Michaelis-Menten Reaction Scheme as a Unified Approach Towards the Optimal Restart Problem*, Phys. Rev. E 92, 060101(R) (2015).**Q: "Are there any**

**commonalities**

**shared among processes that are restarted at an optimal rate?"**

**A moderate restart rate will bring the mean completion time to a minimum for just about any process in which start and finish are separated by an unknown, and highly variable, time spread. The underlying stochastic nature of the process will still cause completion times to fluctuate above and below their expected value but we show that when the restart rate is optimal the relative magnitude of these fluctuations is universal. Strikingly, this relative level of uncertainty is (literally) one and the same**

**irrespective of the process at hand.**

**S. Reuveni,**

*Optimal Stochastic Restart Renders Fluctuations in First Passage Times Universal*, Phys. Rev. Lett. 116, 170601 (2016).**Q: "Diffusion with resetting to the origin is a quintessential example of first passage under restart. What general lessons can be learned from it?**

**"**

The problem of first passage under restart emerges in multiple contexts and in various shapes and forms, but the endless variety of ways in which restart mechanisms and first passage processes mix and match hindered the identification of unifying principles and the discovery of general truths. And so, while many particular instances of first passage under restart have been analyzed and studied in great detail, very little was said about the problem in general. |

**We develop a theoretical framework for first passage under restart. With this at hand, we explain why certain features of diffusion under restart are not at all unique to this particular process, but rather universal characteristics of the phenomena at large. In particular, sharp (deterministic) restart is then revealed as a superior restart strategy whose first-passage time fluctuations at optimality are governed by a simple universal inequality.**

**A. Pal & S. Reuveni,**

*First Passage Under Restart*, Physical Review Letters 118 (3), 030603, (2017).**Also see : Coverage at phys.org.**

**Single-Molecule**

**Enzymology**

**Enzymes are biological catalysts vital to all life processes, and the quest to determine their inner workings continues to attract and fascinate scientists over a broad range of disciplines. With the advent of cutting edge single-molecule spectroscopy methods, it is now possible to directly observe and manipulate the behavior of individual enzymes in the course of a chemical reaction.**

**Chemistry at the single-molecule level is, however, inherently stochastic and, at times, extremely unintuitive.**

**Q: "What is the role of substrate unbinding in enzymatic reactions?**

**"**

**The Michaelis–Menten equation provides a hundred-year-old prediction by which any increase in the rate at which a substrate unbinds its enzyme will**

**inevitably**

**decrease the rate of enzymatic turnover (the conversion of substrate into product). Surprisingly, this prediction was never tested experimentally nor was it scrutinized using modern theoretical tools. We show that unbinding may also speed up enzymatic turnover --- t**

**urning a spotlight to the fact that its actual role in enzymatic catalysis remains to be determined experimentally.**

**Analytically constructing the unbinding phase-space, we identify four distinct categories of unbinding:**

*inhibitory, excitatory, super-excitatory*and*restorative*. A transition, in which the effect of unbinding changes from*inhibitory*to*excitatory*as substrate concentrations increase, and an overlooked trade-off between the speed and efficiency of enzymatic reactions, are naturally unveiled as a result. Our theory is independent of the enzymological context within which it was developed and is relevant to any system described by a stochastic Michaelis–Menten reaction scheme (see our work on first passage under restart above)**.**

**S. Reuveni,**

**M. Urbakh and J. Klafter,**

*"*

*Role of substrate unbinding in Michaelis–Menten enzymatic reactions*"*,*PNAS 111 (12), 4391, (2014).

**Also see : "In this Issue" PNAS highlight.**

**Q:**

**"How would the rate at which an enzyme converts substrate into product change in the presence of a molecule whose binding**

__completely__shuts down its ability to catalyze?"**The classical theory of enzymatic inhibition aims to quantitatively describe the effect of certain molecules—called inhibitors—on the progression of enzymatic reactions, but growing signs indicate that it must be revised to keep pace with the single-molecule revolution that is sweeping through the sciences. Here, we take the single enzyme perspective and rebuild the theory of enzymatic inhibition from the bottom up. We find that accounting for multi-conformational enzyme structure and intrinsic randomness cannot undermine the validity of classical results in the case of competitive inhibition; but that it should strongly change our view on the uncompetitive and mixed modes of inhibition. There, stochastic fluctuations on the single-enzyme level could give rise to inhibitor-activator duality—a phenomenon in which, under some conditions, the introduction of a molecule whose binding shuts down enzymatic catalysis will counter intuitively work to facilitate product formation. We state—in terms of experimentally measurable quantities—a mathematical condition for the emergence of inhibitor-activator duality, and propose that it could explain why certain molecules that act as inhibitors when substrate concentrations are high elicit a non-monotonic dose response when substrate concentrations are low. The fundamental and practical implications of our findings are thoroughly discussed.**

**T. Rotbart, S. Reuveni and M. Urbakh,**

*Single-molecule theory of enzymatic inhibition predicts the emergence of inhibitor-activator duality*, bioRxiv 095562.**Vibrational Dynamics of Fractals and the Fractal-Like Nature of Proteins**

**Proteins**

**(and enzymes as a subclass) are large organic**

**molecules that play a vital role in all biological organisms.**

**Fractals**

**are geometrical objects that possess self-similarity. Recent studies have shown that proteins resemble fractals, and this similarity manifests itself in the way proteins thermally vibrate and in**

**the manner in which they fill space. This observation allowed us to harness the vast mathematical and physical machinery, originally devised to describe the physical properties of fractals, in order to quantitatively analyze protein structure and dynamics**.

**Q: "Universality in a realm of specificity?"**

**We introduce an "equation of state" for protein native topology based on recent analysis of data from the Protein Data Bank and on a generalization of the Landau-Peierls instability criterion for fractals. The equation relates the protein fractal dimension, the spectral dimension, and the number of amino acids. The fractal nature of proteins is shown to bridge their seemingly conflicting properties of stability and flexibility. Thousands of proteins are analyzed and shown to obey this equation of state.**

**S. Reuveni, R. Granek and J. Klafter,**

*"**Proteins: coexistence of stability and flexibility*"*,*Phys. Rev. Lett. 100, 208101, (2008).**M. de Leeuw, S. Reuveni, J. Klafter and R. Granek,**

*"**Coexistence of flexibility and stability of proteins: an equation of state*"*,*PLoS ONE 4(10), (2009).**Q: "Can a simple harmonic energy landscape lead to the anomalous vibrational dynamics that proteins display**

**?"**

**Proteins have been shown to exhibit strange/anomalous dynamics displaying non-Debye density of vibrational states, anomalous spread of vibrational energy, large conformational changes, non-exponential decay of correlations, and non-exponential unfolding times. The anomalous behavior may, in principle, stem from various factors affecting the energy landscape under which a protein vibrates. Investigating the origins of such unconventional dynamics, we focus on the structure-dynamics interplay and introduce a stochastic approach to the vibrational dynamics of proteins. We use diffusion, a method sensitive to the structural features of the protein fold and them alone, in order to probe protein structure. Conducting a large-scale study of diffusion on over 500 Protein Data Bank structures we find it to be anomalous, an indication of a fractal-like structure. Taking advantage of known and newly derived relations between vibrational dynamics and diffusion, we demonstrate the equivalence of our findings to the existence of structurally originated anomalies in the vibrational dynamics of proteins. We conclude that these anomalies are a direct result of the fractal-like structure of proteins.**

**S. Reuveni, R. Granek and J. Klafter,**

*"**Anomalies in the vibrational dynamics of proteins are a consequence of fr**actal like structure"**,*PNAS*107 (31), 13696, (2010).***Q: "The static structure factor of fractals is well understood, what about the dynamic structure factor?"**

**Naturally occurring fractals are ubiquitous and scattering experiments provide a valuable way to characterize fractal structure and dynamics. A key player in these experiments is the structure factor (SF). The static SF of fractals is well understood but a theory that**

**explains dynamic SF measurements from**

**low dimensional fractals in solution (e.g.**

**low temperature glasses and porous materials, proteins, sol-gel branched polymer clusters, colloidal aggregates and more...) was lacking for up until recently. We develop this theory and apply it to the study of the dynamic**

**structure factor of proteins in solution.**

**S. Reuveni, J. Klafter and R. Granek,**

*"*

*Dynamic structure factor of vibrating fractals*"*,*Phys. Rev. Lett. 108*,***068101, (2012).**

**S. Reuveni, J. Klafter and R. Granek,**

*"**Dynamic Structure Factor of Vibrating Fractals: Proteins as a Case Study*"*,*Phys. Rev. E. 85, 011906 (2012).**Mapping Random Walks onto Thermal Vibrations**

Mapping two different physical problems onto one another has proven very useful in physics. One such well-known mapping is the mapping between random walks and electrical networks, another is the mapping between random walks and thermal vibrations in the scalar elasticity model. While the latter has long been recognized and some of its consequences have already been exploited, vibrational analogs for several basic quantities arising in the theory of random walks remained unknown for up until recently.

Mapping two different physical problems onto one another has proven very useful in physics. One such well-known mapping is the mapping between random walks and electrical networks, another is the mapping between random walks and thermal vibrations in the scalar elasticity model. While the latter has long been recognized and some of its consequences have already been exploited, vibrational analogs for several basic quantities arising in the theory of random walks remained unknown for up until recently.

**Q: "Solving the mean first passage time problem with elementary bead-spring physics?"**

**What is the average time a random walker takes to get from A to B on a fractal structure and how does this mean time scale with the size of the system and the distance between source and target? We take a non-probabilistic approach towards this problem and show how the solution is readily obtained using an analysis of thermal vibrations on fractals.**

**S. Reuveni, R. Granek and J. Klafter,**

*"**Vibrational Shortcut to the mean-first-passage-time problem*"*,*Phys. Rev. E 81, 040103(R), (2010).

**Q:**

**"More tricks from the same old dog?**

**"**

**We present an approach to the mapping between random walks and vibrational dynamics on general networks. Random walk occupation probabilities, first passage time distributions and passage probabilities between nodes are expressed in terms of thermal vibrational correlation functions. Recurrence is demonstrated equivalent to the Landau-Peierls instability. Fractal networks are analyzed as a case study. In particular, we show that the spectral dimension governs whether or not the first passage time distribution is well represented by its mean.**

**S. Reuveni, R. Granek and J. Klafter,**

*"**General Mapping between Random Walks and Thermal Vibrations in Elastic Networks: Fractal Networks as a Case Study*"*,*Phys. Rev. E 82, 041132, (2010).**Tandem Stochastic Systems**

**Many complex processes contain strong elements of randomness which are intrinsic to their description. Some of these complex processes can be decomposed into simpler processes which occur sequentially, i.e., one after the other. Such processes are usually referred to as tandem stochastic systems. More specifically, tandem stochastic systems are systems in which a stochastic input flow (of jobs, molecules, particles, etc...) progresses through a serial array of compartments. The act of motion from one compartment to the consecutive compartment is governed by a combination of system specific rules and is subject to the occurrence of random events.**

**Q: "What happens when particles on a 1D lattice hop unidirectionally while being**

**subject to inclusion interactions?**

**" or**

**"What happens in**

**a tandem system of Markovian queues with unbounded capacity and unlimited batch service?**

**"**

**In order to answer this question we have recently introduced the Asymmetric Simple Inclusion Process (ASIP),**

**a lattice-gas model for unidirectional transport with aggregation. The ASIP is an**

**`inclusion' counterpart of the Asymmetric Simple Exclusion Process (ASEP)**

**and can also be looked at as a**

**Tandem Jackson Network (TJN) with the additional property of unlimited batch service.**

**S. Reuveni, I. Eliazar and U. Yechiali,**

*"**Asymmetric Inclusion Process*"*,*Phys. Rev. E. 84, 041101, (2011).**S. Reuveni, I. Eliazar and U. Yechiali,**

*"*

*Asymmetric Inclusion Process as a Showcase of Complexity*", Phys. Rev. Lett. 109, 020603, (2012).**S. Reuveni, I. Eliazar and U. Yechiali**

*, "Limit Laws for the Asymmetric Inclusion Process*"*,*Phys. Rev. E. 86, 061133, (2012).

**Q: "Catalan's Numbers, Catalan's Triangle, Catalan's Trapezoids?**

**"**

**Named after the French-Belgian mathematician Eugène Charles Catalan,**

**Catalan's numbers**

**arise in various combinatorial problems.**

**Catalan's triangle**

**, a triangular array of numbers somewhat similar to**

**Pascal's triangle**

**, extends the combinatorial meaning of Catalan's numbers and facilitates the solution to Bertrand's famous**

**ballot problem**

**. The entries of Catalan's triangle, a.k.a ballot numbers, appear in the exact solution for the steady state probability distribution of the ASEP.**

**Catalan's trapezoids, a countable set of number trapezoids whose first element is Catalan's triangle,**

**naturally arise as part of the exact solution for the**

**steady state probability distribution of the ASIP**

**.**

**Catalan's trapezoids facilitate the solution to a generalized ballot problem.**

**S. Reuveni,**

*"Catalan's Trapezoids*"*,*Probability in the Engineering and Informational Sciences, 28 (03), 353, (2014).

**S. Reuveni, O. Hirschberg, I. Eliazar and U. Yechiali**

*, "Occupation Probabilities and Fluctuations in the Asymmetric Simple Inclusion Process*"*,*Phys. Rev. E., 89, 042109, (2014).**Q:**

**"Gene translation, can we think of it as unidirectional**

**ribosomal motion on a 1D lattice with excluded volume interactions?**

**"**

**Gene translation is a central process in all living organisms. This process is, however, still enigmatic with**

**different studies stating**

**contradicting conclusions regarding essential parameters that determine translation rates. We describe the first large scale analysis of gene translation that is based on a model that takes into account the physical and dynamical properties of this process. The Ribosomal Flow Model (**

*RFM*) predicts fundamental features of the translation process, including translation rates, protein abundance levels, ribosomal densities and the relation between all these variables, better than alternative ('non-physical') approaches.**Popular Science**

**S. Reuveni,**

*"? הטבלה המחזורי**ת - עד 120*" (Hebrew), Galileo 139, (2010).

**S. Reuveni,**

*"**חוקי התנועה של מולקולות בתוך התא*" (Hebrew), Galileo 341, (2010).**Dissertations**

**S. Reuveni, Tandem Stochastic Systems: The Asymmetric Simple Inclusion Process, Te-Aviv University, (2014).**

**S. Reuveni, The Fractal-Like Nature of Proteins: Foundations, Applications and Ramifications, Tel-Aviv University, (2012).**

**S. Reuveni, Proteins: Unraveling Universality in a Realm of Specificity, Tel-Aviv University, (2007).**