Intra-cellular active transport by native cargos is ubiquitous. We investigate the motion of sphe... more Intra-cellular active transport by native cargos is ubiquitous. We investigate the motion of spherical nano-particles (NPs) grafted with flexible polymers that end with a nuclear localization signal peptide. This peptide allows the recruitment of several mammalian dynein motors from cytoplasmic extracts. To determine how motor–motor interactions influenced motility on the single microtubule level, we conducted bead-motility assays incorporating surface adsorbed microtubules and combined them with model simulations that were based on the properties of a single dynein. The experimental and simulation results revealed long time trajectories: when the number of NP-ligated motors Nm increased, run-times and run-lengths were enhanced and mean velocities were somewhat decreased. Moreover, the dependence of the velocity on run-time followed a universal curve, regardless of the system composition. Model simulations also demonstrated left- and right-handed helical motion and revealed self-reg...
The motion of small cargo particles on microtubules by means of motor proteins in disordered micr... more The motion of small cargo particles on microtubules by means of motor proteins in disordered microtubule networks is investigated theoretically using both analytical tools and computer simulations. Different network topologies in two and three dimensions are considered, one of which has been recently studied experimentally by Salman et al. ͓Biophys. J. 89, 2134 ͑2005͔͒. A generalization of the random velocity model is used to derive the mean-square displacement of the cargo particle. We find that all cases belong to the class of anomalous superdiffusion, which is sensitive mainly to the dimensionality of the network and only marginally to its topology. Yet in three dimensions the motion is very close to simple diffusion, with sublogarithmic corrections that depend on the network topology. When details of the thermal diffusion in the bulk solution are included, no significant change to the asymptotic time behavior is found. However, a small asymmetry in the mean microtubule polarity affects the corresponding long-time behavior. We also study a three-dimensional model of the microtubule network in living animal cells. Three first-passage-time problems of intracellular transport are simulated and analyzed for different motor processivities: ͑i͒ cargo that originates near the nucleus and has to reach the membrane, ͑ii͒ cargo that originates from the membrane and has to reach the nucleus, and ͑iii͒ cargo that leaves the nucleus and has to reach a specific target in the cytoplasm. We conclude that while a higher motor processivity increases the transport efficiency in cases ͑i͒ and ͑ii͒, in case ͑iii͒ it has the opposite effect. We conjecture that the balance between the different network tasks, as manifested in cases ͑i͒ and ͑ii͒ versus case ͑iii͒, may be the reason for the evolutionary choice of a finite motor processivity.
We introduce an equation for protein native topology based on recent analysis of data from the Pr... more We introduce an equation 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 d f , the spectral dimension d s , and the number of amino acids N. Deviations from the equation may render a protein unfolded. The fractal nature of proteins is shown to bridge their seemingly conflicting properties of stability and flexibility. Over 500 proteins have been analyzed (d f , d s , and N) and found to obey this equation of state.
... The Helmholtz free-energy density (per unit area) for a membrane confined between two paralle... more ... The Helmholtz free-energy density (per unit area) for a membrane confined between two parallelwalls (in the absence of flow) has been evaluated in reference [11] for small deviations of ... AG Zilman and R. Granek: Lamellar undulation instability and onions induced by shear ...
What is the average time a random walker takes to get from A to B on a fractal structure and how ... more 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 nonprobabilistic approach toward this problem and show how the ...
Compartmentalization of the cytoplasm by membranes should have a strong influence on the diffusio... more Compartmentalization of the cytoplasm by membranes should have a strong influence on the diffusion of macromolecules inside a cell, and we have studied how this could be reflected in fluorescence correlation spectroscopy (FCS) experiments. We derived the ...
Many essential processes in eukaryotic cells depend on regulated molecular exchange between its t... more Many essential processes in eukaryotic cells depend on regulated molecular exchange between its two major compartments, the cytoplasm and the nucleus. In general, nuclear import of macromolecular complexes is dependent on specific peptide signals and their recognition by receptors ...
The effect of diffusion on energy transfer from excited donor to acceptor molecules in liquid sol... more The effect of diffusion on energy transfer from excited donor to acceptor molecules in liquid solutions is studied with particular attention focused on vibrational energy transfer between solute molecules in dilute solutions. Such processes are often discussed in the independent binary collision (IBC) framework and diffusion effects are assumed to be negligible. We introduce the concept of diffusive collisions (encounters between acceptor and donor molecule within an effective energy transfer range) and investigate the conditions under which the cross section for the energy transfer process may be affected by the cross section for the diffusive collision as opposed to the more common fast diffusion limit where the energy transfer is dominated by direct binary collisions. We conclude that while in most common situations vibrational energy transfer is indeed dominated by binary collision events, pronounced diffusion effects should exist at moderately high pressures. Explicit estimates are provided for the H...
tncer particle in a System of oth~ pa~ie[~ characterized by a different jump ~te [ 14] am do~ly ~... more tncer particle in a System of oth~ pa~ie[~ characterized by a different jump ~te [ 14] am do~ly ~lated to the ~rk described below. Refs. [ [0-14] focus on s~llng relations n~r the undedying latti~ pe~olation thr~ho]d. 2. Diff~ion and conductivity in dynamlcat;l~ disordered s}~tems The following dlscussion focuses on the diffuslon ~ffiei~t D associated with a single moving particle in a dynamically disordered nelwork. The ~esponding conductivity ~ is obtained from D using the N~st-Einslein ~lation ~=(nq21kBT)D, (I) where n is the density, q is the charge ~f the (assumed non-int ~acting) carder~ and T is the lemp~ture. We begin with the DNR model, Consider f'~t de diffusion. For lhe de diffusion ~ fficient DNR derive the ~markably simple ~sult [ 5] D~ n~<r=>~.l:..,
... in solid ionic conductors2 and in polymer conductors3 and elec-tronic conductivity in disorde... more ... in solid ionic conductors2 and in polymer conductors3 and elec-tronic conductivity in disordered materials.4 Although an extensive literature exists on the diffusion of particles in such systems,5 comparatively few studies directly address the is-sue of particle ... J. Chern. ...
Dynamic percolation theory is used to obtain the tracer diffusion coefficient in binary mixtures ... more Dynamic percolation theory is used to obtain the tracer diffusion coefficient in binary mixtures of ‘‘noninteracting’’ lattice gas (with only the blocking interactions, i.e., double occupancy of a lattice site is forbidden) within the effective medium approximation (EMA). Our approach is based on regarding the background particles as a changing random environment. The result is expressed in terms of two
We suggest a mathematical model for the spread of an infectious disease in human population, with... more We suggest a mathematical model for the spread of an infectious disease in human population, with particular attention to the COVID-19. Common epidemiological models, e.g., the well-known susceptible-exposed-infectious-recovered (SEIR) model, implicitly assume fast mixing of the population relative to the local infection rate, similar to the regime applicable to many chemical reactions. However, in human populations, especially under different levels of quarantine conditions, this assumption is likely to fail. We develop a continuous spatial model that includes five different populations, in which the infectious population is split into latent (or pre-symptomatic) and symptomatic. Based on nearest-neighbor infection kinetics, we arrive into a “reaction-diffusion” model. Our model accounts for front propagation of the infectious population domains under partial quarantine conditions, which is present on top of the common local infection process. Importantly, we also account for the v...
We deduce the directional step distribution of yeast dynein motor protein on the microtubule surf... more We deduce the directional step distribution of yeast dynein motor protein on the microtubule surface by combing intrinsic features of the dynein and microtubule. These include the probability distribution of the separation vector between the two microtubule binding domains (MTBDs), the angular probability distribution of a single MTBD translation, the existence of a microtubule seam defect, microtubule binding sites, and theoretical extension that accounts for a load force on the motor. Our predictions are in excellent accord with the measured longitudinal step size distributions at various load forces. Moreover, we predict the side-step distribution and its dependence on longitudinal load forces, which shows a few surprising features. First, the distribution is broad. Second, in the absence of load, we find a small right-hand bias. Third, the side-step bias is susceptible to the longitudinal load force; it vanishes at a load equal to the motor stalling force and changes to a left-h...
The thermally driven dynamics of a polymer network are studied by direct view observation of the ... more The thermally driven dynamics of a polymer network are studied by direct view observation of the motion of a single point within a single polymer. Taking advantage of rather rigid biological microtubules as a case study, we expand the space and time scales of the system to those accessible by optical microscopy and standard video tools. Tracking is achieved by chemically attaching an optically resolved micro-sphere to a single point on the filament. We measure a sequence of instantaneous positions, from which we compute the mean squared displacement vs time. Typical behavior in a network shows two distinct regimes. At short times, we observe anomalous diffusion with power law 3/4. At long times (seconds) the undulation amplitude saturates as a consequence of the finite length of the polymer. In the network, entanglements effectively divide the filament into segments shorter than the physical length. In a pre-stressed network we find a short time sub-diffusive regime with power law 0...
We report an experimental study of the dynamics of particles driven along biopolymer tracks by ac... more We report an experimental study of the dynamics of particles driven along biopolymer tracks by active protein motors. We use extracts of Xenopus laevis eggs, which support the formation of cytoskeletal networks (microtubules and actin filaments) in vitro and provide all the ...
We study the diffusion (and conductivity) associated with the random walk of noninteracting parti... more We study the diffusion (and conductivity) associated with the random walk of noninteracting particles on a disordered lattice characterized by bond disorder, temporal rearrangement, and spatial correlations. This paper extends previous works on dynamic bond percolation processes to situations where spatial correlations in the rearrangement process are important. Many bond effective-medium theory is used to obtain the effective diffusion coefficient
Intra-cellular active transport by native cargos is ubiquitous. We investigate the motion of sphe... more Intra-cellular active transport by native cargos is ubiquitous. We investigate the motion of spherical nano-particles (NPs) grafted with flexible polymers that end with a nuclear localization signal peptide. This peptide allows the recruitment of several mammalian dynein motors from cytoplasmic extracts. To determine how motor–motor interactions influenced motility on the single microtubule level, we conducted bead-motility assays incorporating surface adsorbed microtubules and combined them with model simulations that were based on the properties of a single dynein. The experimental and simulation results revealed long time trajectories: when the number of NP-ligated motors Nm increased, run-times and run-lengths were enhanced and mean velocities were somewhat decreased. Moreover, the dependence of the velocity on run-time followed a universal curve, regardless of the system composition. Model simulations also demonstrated left- and right-handed helical motion and revealed self-reg...
The motion of small cargo particles on microtubules by means of motor proteins in disordered micr... more The motion of small cargo particles on microtubules by means of motor proteins in disordered microtubule networks is investigated theoretically using both analytical tools and computer simulations. Different network topologies in two and three dimensions are considered, one of which has been recently studied experimentally by Salman et al. ͓Biophys. J. 89, 2134 ͑2005͔͒. A generalization of the random velocity model is used to derive the mean-square displacement of the cargo particle. We find that all cases belong to the class of anomalous superdiffusion, which is sensitive mainly to the dimensionality of the network and only marginally to its topology. Yet in three dimensions the motion is very close to simple diffusion, with sublogarithmic corrections that depend on the network topology. When details of the thermal diffusion in the bulk solution are included, no significant change to the asymptotic time behavior is found. However, a small asymmetry in the mean microtubule polarity affects the corresponding long-time behavior. We also study a three-dimensional model of the microtubule network in living animal cells. Three first-passage-time problems of intracellular transport are simulated and analyzed for different motor processivities: ͑i͒ cargo that originates near the nucleus and has to reach the membrane, ͑ii͒ cargo that originates from the membrane and has to reach the nucleus, and ͑iii͒ cargo that leaves the nucleus and has to reach a specific target in the cytoplasm. We conclude that while a higher motor processivity increases the transport efficiency in cases ͑i͒ and ͑ii͒, in case ͑iii͒ it has the opposite effect. We conjecture that the balance between the different network tasks, as manifested in cases ͑i͒ and ͑ii͒ versus case ͑iii͒, may be the reason for the evolutionary choice of a finite motor processivity.
We introduce an equation for protein native topology based on recent analysis of data from the Pr... more We introduce an equation 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 d f , the spectral dimension d s , and the number of amino acids N. Deviations from the equation may render a protein unfolded. The fractal nature of proteins is shown to bridge their seemingly conflicting properties of stability and flexibility. Over 500 proteins have been analyzed (d f , d s , and N) and found to obey this equation of state.
... The Helmholtz free-energy density (per unit area) for a membrane confined between two paralle... more ... The Helmholtz free-energy density (per unit area) for a membrane confined between two parallelwalls (in the absence of flow) has been evaluated in reference [11] for small deviations of ... AG Zilman and R. Granek: Lamellar undulation instability and onions induced by shear ...
What is the average time a random walker takes to get from A to B on a fractal structure and how ... more 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 nonprobabilistic approach toward this problem and show how the ...
Compartmentalization of the cytoplasm by membranes should have a strong influence on the diffusio... more Compartmentalization of the cytoplasm by membranes should have a strong influence on the diffusion of macromolecules inside a cell, and we have studied how this could be reflected in fluorescence correlation spectroscopy (FCS) experiments. We derived the ...
Many essential processes in eukaryotic cells depend on regulated molecular exchange between its t... more Many essential processes in eukaryotic cells depend on regulated molecular exchange between its two major compartments, the cytoplasm and the nucleus. In general, nuclear import of macromolecular complexes is dependent on specific peptide signals and their recognition by receptors ...
The effect of diffusion on energy transfer from excited donor to acceptor molecules in liquid sol... more The effect of diffusion on energy transfer from excited donor to acceptor molecules in liquid solutions is studied with particular attention focused on vibrational energy transfer between solute molecules in dilute solutions. Such processes are often discussed in the independent binary collision (IBC) framework and diffusion effects are assumed to be negligible. We introduce the concept of diffusive collisions (encounters between acceptor and donor molecule within an effective energy transfer range) and investigate the conditions under which the cross section for the energy transfer process may be affected by the cross section for the diffusive collision as opposed to the more common fast diffusion limit where the energy transfer is dominated by direct binary collisions. We conclude that while in most common situations vibrational energy transfer is indeed dominated by binary collision events, pronounced diffusion effects should exist at moderately high pressures. Explicit estimates are provided for the H...
tncer particle in a System of oth~ pa~ie[~ characterized by a different jump ~te [ 14] am do~ly ~... more tncer particle in a System of oth~ pa~ie[~ characterized by a different jump ~te [ 14] am do~ly ~lated to the ~rk described below. Refs. [ [0-14] focus on s~llng relations n~r the undedying latti~ pe~olation thr~ho]d. 2. Diff~ion and conductivity in dynamlcat;l~ disordered s}~tems The following dlscussion focuses on the diffuslon ~ffiei~t D associated with a single moving particle in a dynamically disordered nelwork. The ~esponding conductivity ~ is obtained from D using the N~st-Einslein ~lation ~=(nq21kBT)D, (I) where n is the density, q is the charge ~f the (assumed non-int ~acting) carder~ and T is the lemp~ture. We begin with the DNR model, Consider f'~t de diffusion. For lhe de diffusion ~ fficient DNR derive the ~markably simple ~sult [ 5] D~ n~<r=>~.l:..,
... in solid ionic conductors2 and in polymer conductors3 and elec-tronic conductivity in disorde... more ... in solid ionic conductors2 and in polymer conductors3 and elec-tronic conductivity in disordered materials.4 Although an extensive literature exists on the diffusion of particles in such systems,5 comparatively few studies directly address the is-sue of particle ... J. Chern. ...
Dynamic percolation theory is used to obtain the tracer diffusion coefficient in binary mixtures ... more Dynamic percolation theory is used to obtain the tracer diffusion coefficient in binary mixtures of ‘‘noninteracting’’ lattice gas (with only the blocking interactions, i.e., double occupancy of a lattice site is forbidden) within the effective medium approximation (EMA). Our approach is based on regarding the background particles as a changing random environment. The result is expressed in terms of two
We suggest a mathematical model for the spread of an infectious disease in human population, with... more We suggest a mathematical model for the spread of an infectious disease in human population, with particular attention to the COVID-19. Common epidemiological models, e.g., the well-known susceptible-exposed-infectious-recovered (SEIR) model, implicitly assume fast mixing of the population relative to the local infection rate, similar to the regime applicable to many chemical reactions. However, in human populations, especially under different levels of quarantine conditions, this assumption is likely to fail. We develop a continuous spatial model that includes five different populations, in which the infectious population is split into latent (or pre-symptomatic) and symptomatic. Based on nearest-neighbor infection kinetics, we arrive into a “reaction-diffusion” model. Our model accounts for front propagation of the infectious population domains under partial quarantine conditions, which is present on top of the common local infection process. Importantly, we also account for the v...
We deduce the directional step distribution of yeast dynein motor protein on the microtubule surf... more We deduce the directional step distribution of yeast dynein motor protein on the microtubule surface by combing intrinsic features of the dynein and microtubule. These include the probability distribution of the separation vector between the two microtubule binding domains (MTBDs), the angular probability distribution of a single MTBD translation, the existence of a microtubule seam defect, microtubule binding sites, and theoretical extension that accounts for a load force on the motor. Our predictions are in excellent accord with the measured longitudinal step size distributions at various load forces. Moreover, we predict the side-step distribution and its dependence on longitudinal load forces, which shows a few surprising features. First, the distribution is broad. Second, in the absence of load, we find a small right-hand bias. Third, the side-step bias is susceptible to the longitudinal load force; it vanishes at a load equal to the motor stalling force and changes to a left-h...
The thermally driven dynamics of a polymer network are studied by direct view observation of the ... more The thermally driven dynamics of a polymer network are studied by direct view observation of the motion of a single point within a single polymer. Taking advantage of rather rigid biological microtubules as a case study, we expand the space and time scales of the system to those accessible by optical microscopy and standard video tools. Tracking is achieved by chemically attaching an optically resolved micro-sphere to a single point on the filament. We measure a sequence of instantaneous positions, from which we compute the mean squared displacement vs time. Typical behavior in a network shows two distinct regimes. At short times, we observe anomalous diffusion with power law 3/4. At long times (seconds) the undulation amplitude saturates as a consequence of the finite length of the polymer. In the network, entanglements effectively divide the filament into segments shorter than the physical length. In a pre-stressed network we find a short time sub-diffusive regime with power law 0...
We report an experimental study of the dynamics of particles driven along biopolymer tracks by ac... more We report an experimental study of the dynamics of particles driven along biopolymer tracks by active protein motors. We use extracts of Xenopus laevis eggs, which support the formation of cytoskeletal networks (microtubules and actin filaments) in vitro and provide all the ...
We study the diffusion (and conductivity) associated with the random walk of noninteracting parti... more We study the diffusion (and conductivity) associated with the random walk of noninteracting particles on a disordered lattice characterized by bond disorder, temporal rearrangement, and spatial correlations. This paper extends previous works on dynamic bond percolation processes to situations where spatial correlations in the rearrangement process are important. Many bond effective-medium theory is used to obtain the effective diffusion coefficient
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Papers by Rony Granek