List of publications in chronological order:

• SF Longfield, RS Gormal, M Feller, P Parutto, J Reingruber, TP Wallis, M Joensuu, GJ Augustine, R Martínez-Mármol, D Holcman, FA Meunier: Synapsin 2a tetramerisation selectively controls the presynaptic nanoscale organisation of reserve synaptic vesicles, Journal of Mathematical Biology, Nat Commun 15, 2217 (2024)
  Abstract, pdf
  Neurotransmitter release relies on the regulated fusion of synaptic vesicles (SVs) that are tightly packed within the presynaptic bouton of neurons. The mechanism by which SVs are clustered at the presynapse, while preserving their ability to dynamically recycle to support neuronal communication, remains unknown. Synapsin 2a (Syn2a) tetramerization has been suggested as a potential clustering mechanism. Here, we used Dual-pulse sub-diffractional Tracking of Internalised Molecules (DsdTIM) to simultaneously track single SVs from the recycling and the reserve pools, in live hippocampal neurons. The reserve pool displays a lower presynaptic mobility compared to the recycling pool and is also present in the axons. Triple knockout of Synapsin 1-3 genes (SynTKO) increased the mobility of reserve pool SVs. Re-expression of wild- type Syn2a (Syn2aWT), but not the tetramerization-deficient mutant K337Q (Syn2aK337Q), fully rescued these effects. Single-particle tracking revealed that Syn2aK337QmEos3.1 exhibited altered activity-dependent presynaptic translo- cation and nanoclustering. Therefore, Syn2a tetramerization controls its own presynaptic nanoclustering and thereby contributes to the dynamic immobi- lisation of the SV reserve pool.
• A Abtout, J Reingruber: Analysis of dim-light responses in rod and cone photoreceptors with altered calcium kinetics, Journal of Mathematical Biology, 87:69 (2023)
  Abstract, pdf
  Rod and cone photoreceptors in the retina of vertebrates are the primary sensory neurons underlying vision. They convert light into an electrical current using a signal transduction path- way that depends on Ca2+ feedback. It is known that manipulating the Ca2+ kinetics affects the response shape and the photoreceptor sensitivity, but a precise quantification of these effects remains unclear. We have approached this task in mouse retina by combining numerical simu- lations with mathematical analysis. We consider a parsimonious phototransduction model that incorporates negative Ca2+ feedback onto the synthesis of cyclic GMP, and fast buffering reac- tions to alter the Ca2+ kinetics. We derive analytic results for the photoreceptor functioning in sufficiently dim light conditions depending on the photoreceptor type. We exploit these results to obtain conceptual and quantitative insight into how response waveform and amplitude depend on the underlying biophysical processes and the Ca2+ feedback. With a low amount of buffering, the Ca2+ concentration changes in proportion to the current, and responses to flashes of light are monophasic. With more buffering, the change in the Ca2+ concentration becomes delayed with respect to the current, which gives rise to a damped oscillation and a biphasic waveform. This shows that biphasic responses are not necessarily a manifestation of slow buffering reactions. We obtain analytic approximations for the peak flash amplitude as a function of the light intensity, which shows how the photoreceptor sensitivity depends on the biophysical parameters. Finally, we study how changing the extracellular Ca2+ concentration affects the response.
• J Reingruber, A Papale, S Ruckly, J-F Timsit, D Holcman: Data-driven multiscale dynamical framework to control a pandemic evolution with non-pharmaceutical interventions, PLoS One 18.1 (2023): e0278882, (2023)
  Abstract, pdf
  Before the availability of vaccines, many countries have resorted multiple times to drastic social restrictions to prevent saturation of their health care system, and to regain control over an otherwise exponentially increasing COVID-19 pandemic. With the advent of data-sharing, computational approaches are key to efficiently control a pandemic with non-pharmaceutical interventions (NPIs). Here we develop a data-driven computational framework based on a time discrete and age-stratified compartmental model to control a pandemic evolution inside and outside hospitals in a constantly changing environment with NPIs. Besides the calendrical time, we introduce a second time-scale for the infection history, which allows for non-exponential transition probabilities. We develop inference methods and feedback procedures to successively recalibrate model parameters as new data becomes available. As a showcase, we calibrate the framework to study the pandemic evolution inside and outside hospitals in France until February 2021. We combine national hospitalization statistics from governmental websites with clinical data from a single hospital to calibrate hospitalization parameters. We infer changes in social contact matrices as a function of NPIs from positive testing and new hospitalization data. We use simulations to infer hidden pandemic properties such as the fraction of infected population, the hospitalisation probability, or the infection fatality ratio. We show how reproduction numbers and herd immunity levels depend on the underlying social dynamics.
• A Abtout, GL Fain, J Reingruber: Analysis of waveform and amplitude of mouse rod and cone flash responses, Journal Physiology 599(13), (2021)
  Abstract, pdf
  Vertebrate eyes have rod and cone photoreceptors, which use a complex transduction pathway comprising many biological processes to transform the absorption of light into an electrical response. A fundamental question in sensory transduction is how these processes contribute to the response. To study this question, we use a well-accepted phototransduction model, which we analyze with a novel method based on the log-transform of the current. We derive an analytical solution that describes the entire time course of the photoreceptor response to dim flashes of light. We use this solution to dissect and quantify the contribution of each process to the response. We find that the entire dim-flash response is proportional to the flash intensity. By normalizing responses to unit amplitude, we define a waveform that is independent of the light intensity and characterizes the invariant shape of dim-flash responses. We show that this waveform is exclusively determined by deactivation rates; activation rates only scale the waveform and affect the amplitude. This analysis corrects a previous assumption that the rising phase is determined entirely by activation rates. We further show that the rising phase depends on Ca2+ feedback to the cyclase, contrary to current belief. We identify the deactivation rates that control the recovery phase of the response, and we devise new methods to extract activation and deactivation rates from an analysis of response shape and response amplitude. In summary, we provide a comprehensive understanding of how the various transduction processes produce the cellular response.
• J Reingruber, NT Ingram, KG Griffis , GL Fain: A kinetic analysis of mouse rod and cone photoreceptor responses, Journal Physiology 598(17), (2020)
  Abstract, pdf
  Most vertebrates have rod and cone photoreceptors, which differ in their sensitivity and response kinetics. We know that rods evolved from cone-like precursors through the expression of different transduction genes or the same genes at different levels, but we do not know which molecular differences were most important. We have approached this problem in mouse retina by analysing the kinetic differences between rod flash responses and recent voltage-clamp recordings of cone flash responses, using a model incorporating the principal features of photoreceptor transduction. We apply a novel method of analysis using the log-transform of the current, and we ask which of the model's dynamic parameters need be changed to transform the flash response of a rod into that of a cone. The most important changes are a decrease in the gain of the response, reflecting a reduction in amplification of the transduction cascade; an increase in the rate of turnover of cGMP in darkness; and an increase in the rate of decay of activated phosphodiesterase, with perhaps also an increase in the rate of decay of light-activated visual pigment. Although we cannot exclude other differences, and in particular alterations in the Ca2+ economy of the photoreceptors, we believe that we have identified the kinetic parameters principally responsible for the differences in the flash responses of the two kinds of photoreceptors, which were likely during evolution to have resulted in the duplex retina.
• J Reingruber, A Papale, D Holcman: Monitoring and predicting SARS-CoV-2 epidemic in France after deconfinement using a multiscale and age-dependent model, Preprint, MedRxiv, (2020)
  Abstract, pdf
  The world and France were strongly impacted by the SARS-COV-2 epidemic. Finding appropriate measures that effectively contain the epidemic without putting severe pressure on social and economic life is a major challenge for modern predictive approaches. We developed an analytical framework to precisely monitor and predict the spread of the epidemic together with its impact on the health care system. The current implementation accounts for interactions between five age-stratified population groups, and predicts disease progression and hospitalization status using eight different categories such as infected, hospitalized, occupancy of intensive care units, deceased, recovered from hospitalization and more. We use a variety of public health care data for the five most infected regions of France during lockdown (March 18th till May 11th) to validate and calibrate the model. At day of deconfinement (May 11th), we find that around $14\%$ (around 4.8M) of the population is infected in the five most affected regions of France (extrapolating to 5.8M for France). We then apply the calibrated model to explore different deconfinement scenarios. We find that wearing of masks and social distancing can prevent a significant second peak. In the context of school openings with limited testing capacities, we argue that testing should focus on children, but without tracing it will have only a limited impact. Finally, we explore a complementary deconfinement scenario where the fragile elderly population initially remains confined, while the rest of the population becomes gradually deconfined to achieve herd immunity within few month before deconfining also the elderly.
• Y Zhang, TK Tsang, E A Bushong, L-A Chu, A-S Chiang, MH Ellisman, J Reingruber, CY Su : Asymmetric ephaptic inhibition between compartmentalized olfactory receptor neurons, Nature Communications 10, 1560 (2019).
  Abstract, pdf
  In the Drosophila antenna, different subtypes of olfactory receptor neurons (ORNs) housed in the same sensory hair (sensillum) can inhibit each other non-synaptically. However, the mechanisms underlying this underexplored form of lateral inhibition remain unclear. Here we use recordings from pairs of sensilla impaled by the same tungsten electrode to demonstrate that direct electrical (“ephaptic”) interactions mediate lateral inhibition between ORNs. Intriguingly, within individual sensilla, we find that ephaptic lateral inhibition is asymmetric such that one ORN exerts greater influence onto its neighbor. Serial block-face scanning electron microscopy of genetically identified ORNs and circuit modeling indicate that asymmetric lateral inhibition reflects a surprisingly simple mechanism: the physically larger ORN in a pair corresponds to the dominant neuron in ephaptic interactions. Thus, morpho- metric differences between compartmentalized ORNs account for highly specialized inhibi- tory interactions that govern information processing at the earliest stages of olfactory coding.
• J. Reisert, J. Reingruber : Ca2+-activated Cl- current ensures robust and reliable signal amplification in vertebrate olfactory receptor neurons, PNAS 116(3), 1053-1058 (2019).
  Abstract, pdf
  Activation of most primary sensory neurons results in transduc- tion currents that are carried by cations. One notable exception is the vertebrate olfactory receptor neuron (ORN), where the trans- duction current is carried largely by the anion Cl . However, it remains unclear why ORNs use an anionic current for signal amplification. We have sought to provide clarification on this topic by studying the so far neglected dynamics of Na+, Ca2+, K+, and Cl in the small space of olfactory cilia during an odor- ant response. Using computational modeling and simulations we compared the outcomes of signal amplification based on either Cl or Na+ currents. We found that amplification produced by Na+ influx instead of a Cl efflux is problematic for several rea- sons: First, the Na+ current amplitude varies greatly, depending on mucosal ion concentration changes. Second, a Na+ current leads to a large increase in the ciliary Na+ concentration dur- ing an odorant response. This increase inhibits and even reverses Ca2+ clearance by Na+ /Ca2+ /K+ exchange, which is essential for response termination. Finally, a Na+ current increases the ciliary osmotic pressure, which could cause swelling to damage the cilia. By contrast, a transduction pathway based on Cl efflux circum- vents these problems and renders the odorant response robust and reliable.
• Wang T, Reingruber J, Woodruff ML, Majumder A, Camarena A, Artemyev NO, Fain GL, Chen J : The PDE6 mutation in the rd10 retinal degeneration mouse model causes protein mislocalization and instability and promotes cell death through increased ion influx, J Biol Chem 293(40), 15332-15346 (2018).
  Abstract, pdf
  The retinal degeneration model rd10 contains a missense mutation of the catalytic PDE6 beta subunit, which hydrolyzes cGMP in response to light. This model produces cell death more slowly than others caused by PDE6 loss of function, making it of particular interest for studying potential therapeutics. We used morphology, biochemistry, and single-cell physiology to examine the mechanism of rd10 degeneration. Our results show that the mutation produces no alteration of Pde6b RNA but does dramatically decrease maximal and basal PDE6 activity, apparently caused by a decrease in protein stability and transport. The enzymatic properties of the remaining mutant PDE6 appear to be nearly normal. We demonstrate that an increase in free cGMP, which would result from decreased PDE6 activity and serve to increase opening of the cGMP-gated channels and calcium influx, is an underlying cause of cell death: degeneration of rd10/Cngb1-/- double mutants is slower than the parent rd10 line. Paradoxically, degeneration in rd10/Cngb1-/- is also slower than in Cngb1-/-. This rescue is correlated with a lowering of cGMP content in Cngb1-/- retinas and suggests that it may be caused by mislocalization of active PDE6. Single-cell recordings from rd10 rods show that the rates of rise and decay of the response are significantly slower; simulations indicate that these changes are primarily the result of the decrease in PDE6 concentration and rod collecting area. Together, these results provide insights into the complex mechanisms that underlie rd10-mediated retinal degeneration and a cautionary note for analysis of therapeutic interventions.
• J. Reingruber, D. Holcman, GL. Fain: How rods respond to single photons: Key adaptations of a G-protein cascade that enable vision at the physical limit of perception, Bioessays 37(11), 1243 (2015).
  Abstract, pdf
  Rod photoreceptors are among the most sensitive light detectors in nature. They achieve their remarkable sensitivity across awide variety of species through a number of essential adaptations: a specialized cellular geometry, a G-protein cascade with an unusually stable receptor molecule, and a low-noise transduction mechanism, a nearly perfect effector enzyme, and highly evolved mechanisms of feedback control and receptor deactivation. Practically any change in protein expression, enzyme activity, or feedback control can be shown to impair photon detection, either by decreasing sensitivity or signal-to-noise ratio, or by reducing temporal resolution. Comparison of mammals to amphibians suggests that rod outer-segment morphology and the molecules and mechanism of transduction may have evolved together to optimize light sensitivity in darkness, which culminates in the extraordinary ability of these cells to respond to single photons at the ultimate limit of visual perception.
• J. Cartailler, J. Reingruber: Facilitated diffusion framework for transcription factor search with conformational changes, Phys Biol. 12(4), 046012 (2015).
  Abstract, pdf
  Cellular responses often require the fast activation or repression of specific genes, which depends on transcription factors (TFs) that have to quickly find the promoters of these genes within a large genome. TFs search for their DNA promoter target by alternating between bulk diffusion and sliding along the DNA, a mechanism known as facilitated diffusion. We study a facilitated diffusion framework with switching between three search modes: a bulk mode and two sliding modes triggered by conformational changes between two protein conformations. In one conformation (search mode) the TF interacts unspecifically with the DNA backbone resulting in fast sliding. In the other conformation (recognition mode) it interacts specifically and strongly with DNA base pairs leading to slow displacement. From the bulk, a TF associates with the DNA at a random position that is correlated with the previous dissociation point, which implicitly is a function of the DNA structure. The target affinity depends on the conformation. We derive exact expressions for the mean first passage time (MFPT) to bind to the promoter and the conditional probability to bind before detaching when arriving at the promoter site. We systematically explore the parameter space and compare various search scenarios. We compare our results with experimental data for the dimeric Lac repressor search in E. coli bacteria. We find that a coiled DNA conformation is absolutely necessary for a fast MFPT. With frequent spontaneous conformational changes, a fast search time is achieved even when a TF becomes immobilized in the recognition state due to the specific bindings. We find a MFPT compatible with experimental data in presence of a specific TF-DNA interaction energy that has a Gaussian distribution with a large variance.
• J. Reingruber, D. Holcman: Computational and mathematical methods for morphogenetic gradient analysis, boundary formation and axonal targeting, Sem Cell Dev Biol 35, 189 (2014).
  Abstract, pdf
  Morphogenesis and axonal targeting are key processes during development that depend on complex interactions at molecular, cellular and tissue level. Mathematical modeling is essential to bridge this multi-scale gap in order to understand how the emergence of large structures is controlled at molecular level by interactions between various signaling pathways. We summarize mathematical modeling and computational methods for time evolution and precision of morphogenetic gradient formation. We discuss tissue patterning and the formation of borders between regions labeled by different morphogens. Finally, we review models and algorithms that reveal the interplay between morphogenetic gradients and patterned activity for axonal pathfinding and the generation of the retinotopic map in the visual system.
• J. Reingruber, J. Pahlberg, ML. Woodruff, AP. Sampath, GL. Fain, D. Holcman: Detection of single photons by toad and mouse rods, PNAS 110, 19378 (2013).
  Abstract, pdf
  Amphibian and mammalian rods can both detect single photons of light even though they differ greatly in physical dimensions, mammalian rods being much smaller in diameter than amphibian rods. To understand the changes in physiology and biochemistry required by such large differences in outer segment geometry, we developed a computational approach, taking into account the spatial organization of the outer segment divided into compartments, together with molecular dynamics simulations of the signaling cascade. We generated simulations of the single-photon response together with intrinsic background fluctuations in toad and mouse rods. Combining this computational approach with electrophysiological data from mouse rods, we determined key biochemical parameters. On average around one phosphodiesterase (PDE) molecule is spontaneously active per mouse compartment, similar to the value for toad, which is unexpected due to the much smaller diameter in mouse. A larger number of spontaneously active PDEs decreases dark noise, thereby improving detection of single photons; it also increases cGMP turnover, which accelerates the decay of the light response. These constraints explain the higher PDE density in mammalian compared with amphibian rods that compensates for the much smaller diameter of mammalian disks. We further find that the rate of cGMP hydrolysis by light-activated PDE is diffusion limited, which is not the case for spontaneously activated PDE. As a consequence, in the small outer segment of a mouse rod only a few activated PDEs are sufficient to generate a signal that overcomes noise, which permits a shorter lifetime of activated rhodopsin and greater temporal resolution.
• YX. Bouchoucha*, J. Reingruber*, C. Labalette, MA Wassef, E. Thierion, C. Desmarquet-Trin Dinh, D Holcman, P Gilardi-Hebenstreit, P Charnay: Dissection of a Krox20 positive feedback loop driving cell fate choices in hindbrain patterning, Mol Syst Biol. 9, 690 (2013).
  Abstract, pdf
  Although feedback loops are essential in development, their molecular implementation and precise functions remain elusive. Using enhancer knockout in mice, we demonstrate that a direct, positive autoregulatory loop amplifies and maintains the expression of Krox20, a transcription factor governing vertebrate hindbrain segmentation. By combining quantitative data collected in the zebrafish with biophysical modelling that accounts for the intrinsic stochastic molecular dynamics, we dissect the loop at the molecular level. We find that it underpins a bistable switch that turns a transient input signal into cell fate commitment, as we observe in single cell analyses. The stochasticity of the activation process leads to a graded input–output response until saturation is reached. Consequently, the duration and strength of the input signal controls the size of the hindbrain segments by modulating the distribution between the two cell fates. Moreover, segment formation is buffered from severe variations in input level. Finally, the progressive extinction of Krox20 expression involves a destabilization of the loop by repressor molecules. These mechanisms are of general significance for cell type specification and tissue patterning.
• O. Stettler, R. Joshi, A. Wizenmann, J. Reingruber, D. Holcman, C. Bouillot, A. Prochiantz, K.L. Moya: Engrailed homeoprotein recruits the adenosine A1 receptor to potentiate ephrin A5 function in retinal growth cones, Development 139, 215-24 (2012).
  Abstract, pdf
  Engrailed 1 and engrailed 2 homeoprotein transcription factors (collectively Engrailed) display graded expression in the chick optic tectum where they participate in retino-tectal patterning. In vitro, extracellular Engrailed guides retinal ganglion cell (RGC) axons and synergises with ephrin A5 to provoke the collapse of temporal growth cones. In vivo disruption of endogenous extracellular Engrailed leads to misrouting of RGC axons. Here we characterise the signalling pathway of extracellular Engrailed. Our results show that Engrailed/ephrin A5 synergy in growth cone collapse involves adenosine A1 receptor activation after Engrailed-dependent ATP synthesis, followed by ATP secretion and hydrolysis to adenosine. This is, to our knowledge, the first evidence for a role of the adenosine A1 receptor in axon guidance. Based on these results, together with higher expression of the adenosine A1 receptor in temporal than nasal growth cones, we propose a computational model that illustrates how the interaction between Engrailed, ephrin A5 and adenosine could increase the precision of the retinal projection map.
• J. Reingruber, D. Holcman: Transcription factor search for a DNA promoter in a three-state model, Phys Rev E 84, 020901 (2011).
  Abstract, pdf
  To ensure fast gene activation, transcription factors (TFs) use a mechanism known as facilitated diffusion to find their DNA promoter site. Here we analyze such a process where a TF alternates between three- and onedimensional diffusion. In the latter (TF bound to the DNA), the TF further switches between a fast translocation state dominated by interaction with the DNA backbone, and a slow examination state where interaction with DNA base pairs (bp) is predominant. We derive a formula for the mean search time, and show that it is faster and less sensitive to the binding-energy fluctuations as compared to the case with a single sliding state. We find that for an optimal search, the time spent bound to the DNA is larger compared to the three-dimensional time, in agreement with recent experimental data.
• C. Ribrault, J. Reingruber, M. Petkovic, N.E. Ziv, D. Holcman, A Triller: Syntaxin1A lateral diffusion reveals transient and local SNARE interactions, J. Neuroscience 31(48), 17590 (2011).
  Abstract, pdf
  At the synapse, vesicles stably dock at the active zone. However, in cellular membranes, proteins undergo a diffusive motion. It is not known how the motion of membrane proteins involved in vesicle exocytosis is compatible with both vesicle docking and the dynamic remodeling ofthe plasma membrane imposed by cycles of exocytosis and endocytosis. To addressthis question, we studiedthe motion of the presynaptic membrane protein syntaxin1A at both the population and single-molecule levels in primary cultures of rat spinal cord neurons. Syntaxin1A was rapidly exchanged between synaptic and extrasynaptic regions. Changes in syntaxin1A mobility were associated with interactions related to the formation of the exocytotic complex. Finally, we propose a reaction-diffusion model reconciling the observed diffusive properties of syntaxin at the population level and at the molecular level. This work allows us to describe the diffusive behavior and kinetics of interactions between syntaxin1A and its partners that lead to its transient stabilization at the synapse.
• U. Pannasch, L. Vargova, J. Reingruber, P. Ezan, D. Holcman, C. Giaume, E. Sykova, N. Rouach: Astroglial networks scale synaptic activity and plasticity, PNAS 108, 8467 (2011).
  Abstract, pdf
  Astrocytes dynamically interact with neurons to regulate synaptic transmission. Although the gap junction proteins connexin 30 (Cx30) and connexin 43 (Cx43) mediate the extensive network organization of astrocytes, their role in synaptic physiology is unknown. Here we show, by inactivating Cx30 and Cx43 genes, that astroglial networks tone down hippocampal synaptic transmission in CA1 pyramidal neurons. Gap junctional networking facilitates extracellular glutamate and potassium removal during synaptic activity through modulation of astroglial clearance rate and extracellular space volume. This regulation limits neuronal excitability, release probability, and insertion of postsynaptic AMPA receptors, silencing synapses. By controlling synaptic strength, connexins play an important role in synaptic plasticity. Altogether, these results establish connexins as critical proteins for extracellular homeostasis, important for the formation of functional synapses.
• J. Reingruber, D. Holcman: The Narrow Escape problem in a flat cylindrical microdomain with application to diffusion in the synaptic cleft, SIAM Multiscale Modeling and Simulation 9(2), 793 (2011).
  Abstract, pdf
  The mean first passage time (MFPT) for a Brownian particle to reach a small target in cellular microdomains is a key parameter for chemical activation. Although asymptotic estimations of the MFPT are available for various geometries, these formulas cannot be applied to degenerated structures where one dimension is much smaller compared to the others. Here we study the narrow escape time problem for a Brownian particle to reach a small target located on the surface of a flat cylinder, where the cylinder height is comparable to the target size, and much smaller than the cylinder radius. When the cylinder is sealed, we estimate the MFPT for a Brownian particle to hit a small disk located centrally on the lower surface. For a laterally open cylinder, we estimate the conditional probability and the conditional MFPT to reach the small disk before exiting through the lateral opening. We apply our results to diffusion in the narrow synaptic cleft, and we compute the fraction and the mean time for neurotransmitters to find their specific receptors located on the postsynaptic terminal. Finally, we confirm our formulas with Brownian simulations.
• J. Reingruber, D. Holcman: Narrow escape for a stochastically gated Brownian ligand, J. Phys. Cond. Matter 22, 065103 (2010).
  Abstract, pdf
  Molecular activation in cellular microdomains is usually characterized by a forward binding rate, which is the reciprocal of the arrival time of a ligand to a key target. Upon chemical interactions or conformational changes, a Brownian ligand may randomly switch between different states, and when target activation is possible in a specific state only, switching can significantly alter the activation process. The main goal of this paper is to study the mean time for a switching ligand to activate a small substrate, modeled as the time to exit a microdomain through a small absorbing window on the surface. We present the equations for the mean sojourn times the ligand spends in each state, and study the escape process with switching between two states in dimension one and three. When the ligand can exit in only one of the two states, we find that switching always decreases its sojourn time in the state where it can exit. Moreover, the fastest exit is obtained when the ligand diffuses most of the time in the state with the maximal diffusion coefficient, although this may imply that it spends most of the time ‘hidden’ in the state where it cannot exit. We discuss the physical mechanisms responsible for this apparent paradox. In dimension three we confirm our results with Brownian simulations. Finally, we suggest possible applications in cellular biology.
• J. Reingruber, D. Holcman: Gated Narrow Escape Time for Molecular Signaling, Phys. Rev. Lett. 103, 148102 (2009).
  Abstract, pdf
  The mean time for a diffusing ligand to activate a target protein located on the surface of a microdomain can regulate cellular signaling. When the ligand switches between various states induced by chemical interactions or conformational changes, while target activation occurs in only one state, this activation time is affected. We investigate this dynamics using new equations for the sojourn times spent in each state. For two states, we obtain exact solutions in dimension one, and asymptotic ones confirmed by Brownian simulations in dimension 3. We find that the activation time is quite sensitive to changes of the switching rates, which can be used to modulate signaling. Interestingly, our analysis reveals that activation can be fast although the ligand spends most of the time ‘‘hidden’’ in the nonactivating state. Finally, we obtain a new formula for the narrow escape time in the presence of switching.
• J. Reingruber, D. Holcman: Diffusion in narrow domains and application to phototransduction, Phys. Rev. E 79, 030904 (2009).
  Abstract, pdf
  The mean time for a Brownian particle to find a small target inside a narrow domain is a key parameter for many chemical reactions occurring in cellular microstructures. Although current estimations are given for a large class of domains, they cannot be used for narrow domains often encountered in cellular biology, such as the synaptic cleft, narrow compartments in the outer segment of vertebrate photoreceptors, or neuron-glia contact. We compute here the mean time for a Brownian particle to hit a small target placed on the surface of a narrow cylinder. We then use this result to estimate the rate constant of cyclic-GMP cGMP hydrolysis by the activated enzyme phosphodiesterase PDE in the narrow microdomains that build up the outer segment of a rod photoreceptor. By controlling the cGMP concentration, PDE activity is at the basis of the early photoresponse chemical reaction cascade. Our approach allows us to compute the cGMP rate constant as a function of biophysical parameters.
• J. Reingruber, E. Abad, D. Holcman: Narrow escape time to a structured target located at the boundary of a microdomian, J. Chem. Phys 130, 094909 (2009).
  Abstract, pdf
  The forward binding rate of chemical reactions is the reciprocal of the mean time for a Brownian molecule to hit its molecular target. When the target is embedded in the surface of a microdomain, this time is known as the narrow escape time, and it has been computed for various geometries. However, for large targets that extend from the surface far into the cytosol the classical computations do not apply and new ones are needed. In this work we generalize the narrow escape time formula to a three-dimensional spine attached to the boundary. We treat in detail the case of an ellipsoidal spine and validate our analysis by Brownian simulations. Finally, we compute the narrow escape time when the spine is uniformly covered by small partially absorbing disks and estimate the homogenized trapping rate of such a patchy surface.
• E. Abad, J. Reingruber, M.S.P. Sansom: On a novel rate theory for transport in narrow ion channels and its application to the study of flux optimization via geometric effects, . J. Chem. Phys 130, 085101 (2009).
  Abstract, pdf
  We present a novel rate theory based on the notions of splitting probability and mean first passage time to describe single-ion conduction in narrow, effectively one-dimensional membrane channels. In contrast to traditional approaches such as transition state theory or Kramers theory, transitions between different conduction states in our model are governed by rates which depend on the full geometry of the potential of mean force PMF resulting from the superposition of an equilibrium free energy profile and a transmembrane potential induced by a nonequilibrium constraint. If a detailed theoretical PMF is available e.g., from atomistic molecular dynamics simulations, it can be used to compute characteristic conductance curves in the framework of our model, thereby bridging the gap between the atomistic and the mesoscopic level of description. Explicit analytic solutions for the rates, the ion flux, and the associated electric current can be obtained by approximating the actual PMF by a piecewise linear potential. As illustrative examples, we consider both a theoretical and an experimental application of the model. The theoretical example is based on a hypothetical channel with a fully symmetric sawtooth equilibrium PMF. For this system, we explore how changes in the spatial extent of the binding sites affect the rate of transport when a linear voltage ramp is applied. Already for the case of a single binding site, we find that there is an optimum size of the site which maximizes the current through the channel provided that the applied voltage exceeds a threshold value given by the binding energy of the site. The above optimization effect is shown to arise from the complex interplay between the channel structure and the applied electric field, expressed by a nonlinear dependence of the rates with respect to the linear size of the binding site. In studying the properties of current-voltage curves, we find a double crossover between sublinear and superlinear behaviors as the size of the binding site is varied. The ratio of unidirectional fluxes clearly deviates from the Ussing limit and can be characterized by a flux ratio exponent which decreases below unity as the binding site becomes wider. We also explore effects arising from changes in the ion bulk concentration under symmetric ionic conditions and the presence of additional binding sites in the hypothetical channel. As for the experimental application, we show that our rate theory is able to provide good fits to conductance data for sodium permeation through the gramicidin A channel. Possible extensions of the theory to treat the case of an asymmetric equilibrium PMF, fluctuations in the mean number of translocating ions, the case of fluctuating energy barriers, and multi-ion conductance are briefly discussed.
• J. Reingruber, D. Holcman: Estimating the rate constant of cyclic GMP hydrolysis by activated phosphodiesterase in photoreceptors, J. Chem. Phys. 129, 145102 (2008).
  Abstract, pdf
  The early steps of light response occur in the outer segment of rod and cone photoreceptor. They involve the hydrolysis of cGMP, a soluble cyclic nucleotide, that gates ionic channels located in the outer segment membrane. We shall study here the rate by which cGMP is hydrolyzed by activated phosphodiesterase PDE. This process has been characterized experimentally by two different rate constants beta_d and beta_sub: beta_d accounts for the effect of all spontaneously active PDE in the outer segment, and beta_sub characterizes cGMP hydrolysis induced by a single light-activated PDE. So far, no attempt has been made to derive the experimental values of beta_d and beta_sub from a theoretical model, which is the goal of this work. Using a model of diffusion in the confined rod geometry, we derive analytical expressions for beta_d and beta_sub by calculating the flux of cGMP molecules to an activated PDE site. We obtain the dependency of these rate constants as a function of the outer segment geometry, the PDE activation and deactivation rates and the aqueous cGMP diffusion constant. Our formulas show good agreement with experimental measurements. Finally, we use our derivation to model the time course of the cGMP concentration in a transversally well-stirred outer segment.
• J. Reingruber, D. Holcman: The Dynamics of Phosphodiesterase Activation in Rods and Cones, Biophys. J. 94, 1954 (2008).
  Abstract, pdf
  Phototransduction starts with the activation of a rhodopsin (respectively, coneopsin) molecule, located in the outer segment of rod (respectively, cone) photoreceptors. The subsequent amplification pathway proceeds via the G-protein transducin to the activation of phosphodiesterase (PDE), a G-protein coupled effector enzyme. In this article, we study the dynamics of PDE activation by constructing a Markov model that is based on the underlying chemical reactions including multiple rhodopsin phosphorylations. We derive explicit equations for the mean and the variance of activated PDE. Our analysis reveals that a low rhodopsin lifetime variance is neither necessary nor sufficient to achieve reliable PDE activation. The numerical simulations show that during the rising phase the variability of PDE activation is much lower compared to the recovery phase, and this property depends crucially on the transducin activation rates. Furthermore, we find that the dynamics of the activation process greatly differs depending on whether rhodopsin or PDE deactivation limits the recovery of the photoresponse. Finally, our simulations for cones show that only very few PDEs are activated by an excited photopigment, which might explain why in S-cones no single photon response can be observed.
• B. Bergerhoff, J. Reingruber: A consistent nonperturbative approach to thermal damping-rates, Phys. Lett. B 488, 435 (2000).
  Abstract, pdf
  We propose a nonperturbative scheme for the calculation of thermal damping-rates using exact renormalization group (RG) -equations. Special emphasis is put on the thermal RG where first results for the rate were given by Pietroni [Phys. Rev. Lett. 81 1998. 2424]. We point out that in order to obtain a complete result that also reproduces the known perturbative behaviour one has to take into account effects that were neglected by Pietroni. We propose a well-defined way of doing the calculations that reproduces perturbation theory in lowest order but goes considerably beyond perturbative results and should be applicable also at second order phase-transitions.
• B. Bergerhoff, J. Manus, J. Reingruber: Thermal renormalization group for fermions, universality, and the chiral phase transition, Phys. Rev. D  61, 125005 (2000).
  Abstract, pdf
  We formulate the thermal renormalization group, an implementation of the Wilsonian RG in the real-time (CTP) formulation of finite temperature field theory, for fermionic fields. Using a model with scalar and fermionic degrees of freedom which should describe the two-flavor chiral phase transition, we discuss the mechanism behind fermion decoupling and universality at second order transitions. It turns out that an effective mass-like term in the fermion propagator which is due to thermal fluctuations and does not break chiral symmetry is necessary for fermion decoupling to work. This situation is in contrast with the high-temperature limit, where the dominance of scalar over fermionic degrees of freedom is due to the different behavior of the distribution functions. The mass-like contribution is the leading thermal effect in the fermionic sector and is missed if a derivative expansion of the fermionic propagator is performed. We also discuss results on the phase transition of the model considered where we find good agreement with results from other methods.
• B. Bergerhoff, J. Reingruber: Thermal renormalization group equations and the phase transition of scalar O(N) theories, Phys. Rev. D 60, 105036 (1999).
  Abstract, pdf
  We discuss the formulation of ‘‘thermal renormalization group equations’’ and their application to the finite temperature phase transition of scalar O(N) theories. Thermal renormalization group equations allow for a computation of both the universal and the nonuniversal aspects of the critical behavior directly in terms of the zero-temperature physical couplings. They provide a nonperturbative method for a computation of quantities such as real-time correlation functions in a thermal environment, where in many situations straightforward perturbation theory fails due to the bad infrared behavior of the thermal fluctuations. We present results for the critical temperature, critical exponents and amplitudes as well as the scaling equation of state for selfinteracting scalar theories.
• J. Reingruber, A. Actor, I. Bender: Casimir effect on a finite lattice, Fortschritte der Physik 48, 303 (2000).
  Abstract, pdf
  Lattice quantum field theory is a well established branch of modern quantum field theory (QFT). However, it has only peripherally been used for the investigation of Casimir systems i.e. for systems in which quantum fields are distorted by their interaction with classical background objects. This article presents a Hamiltonian lattice formulation of static Casimir systems at a level of generality appropriate for an introductory investigation. Background structure represented by a lattice potential is introduced along one spatial direction with translation invariance in all other spatial directions. It is simple to extend this formulation to include arbitrary background structure in more than one spatial direction.