Stochastic models of intracellular transport

The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an overdamped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of adenosine triphosphate hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review a wide range of analytical methods and models of intracellular transport is presented. In the case of diffusive transport, narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion are considered. In the case of active transport, Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi-steady-state reduction methods, and mean-field approximations are considered. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self-organization of subcellular structures.

Figure 1

Example trajectory of a Brownian particle moving in a 2D unit disk with small absorbing windows on an otherwise reflecting circular boundary. Inset: A local coordinate system around the $j$th arc.

Figure 2

(a) Mechanism of facilitated diffusion involving alternating phases of 3D diffusion and 1D diffusion (sliding along the DNA). (b) 1D representation of facilitated diffusion.

Figure 3

(a) Diffusion of a finite size particle (tracer) between obstacles of volume $\nu$ (left) can be modeled as diffusion of a point particle between effective obstacles of volume ${\nu }^{\prime }$ (right). Effective obstacles can partially overlap. (b) Sketch of MSD $⟨R^2⟩$ against time $t$, illustrating three different diffusion regimes: unobstructed diffusion (I), anomalous intermediate diffusion (II), and normal effective diffusion (III). (c) Illustrative plot of the normalized effective diffusion coefficient $D(\varphi )/D_0$ for random spheres. The scale of the curves in (b) and (c) are based on the results of 275.

Figure 4

Schematic illustration of random walk model of sequence-dependent protein diffusion along DNA. Each lattice site corresponds to a base pair with four binding sites that can potentially make hydrogen bonds with the diffusing protein: acceptor sites (black dots), donor sites (gray dots), and missing sites (white dots). In this example, the protein interacts with a base pair sequence of length $r=2$. The energy of protein-DNA interactions determines the transition rates to nearest neighbor lattice sites. From 16, 17.

Figure 5

Basic structure of a neuron. (The inset shows a synaptic connection of strength $w_{ij}$ from an upstream or presynaptic neuron labeled $j$ and a downstream or postsynaptic neuron labeled $i$.) Neurons in the brain communicate with each other by transmitting electrical spikes [action potentials (AP)]. An AP propagates along the axon of a neuron until it reaches a terminal that forms the upstream or presynaptic component of the synaptic connection to a downstream or postsynaptic neuron. The arrival of the action potential induces the release of chemical transmitters into the synapse. These subsequently bind to protein receptors in the postsynaptic membrane resulting in the opening of various ion channels. This generates a synaptic current that flows along the dendritic tree of the postsynaptic neuron and combines with currents from other activated synapses. If the total synaptic current $u(t)$ forces the membrane potential $V(t)$ at a certain location within the cell body to cross some threshold, then the postsynaptic neuron fires an action potential and the process continues. One can thus view the brain as a vast collection of synaptically coupled networks of spiking neurons. Moreover, the strength of synaptic connections within and between networks is modifiable by experience (synaptic plasticity).

Figure 6

An example of a piece of spine studded dendritic tissue (from rat hippocampal region CA1 stratum radiatum). The dendrite on the right-hand side is $\sim 5\mu \mathrm{m}$ in length. From SynapseWeb, Kristen M. Harris, PI, http://synapses.clm.utexas.edu/.

Figure 7

(a) Schematic illustration of the anomalous diffusion model of 319, who carried out detailed 3D simulations of diffusion in a spiny dendrite treated as a system of connected cylinders with the following baseline parameter values: spine neck diameter $0.2\mu \mathrm{m}$, neck length $0.6\mu \mathrm{m}$, head length and diameter $0.6\mu \mathrm{m}$, dendrite diameter $1\mu \mathrm{m}$, and a spine density of $15\mathrm{spines}/\mu \mathrm{m}$. The dendritic spines act as transient traps for a diffusing particle within the dendrite, which leads to anomalous diffusion on intermediate time scales. (b) Schematic illustration of various pathways of AMPA receptor trafficking at a dendritic spine.

Figure 8

Picket-fence model of membrane diffusion. The plasma membrane is parceled up into compartments whereby both transmembrane proteins and lipids undergo short-term confined diffusion within a compartment and long-term hop diffusion between compartments. This corralling is assumed to occur by two mechanisms. (a) The membrane-cytoskeleton (fence) model: transmembrane proteins are confined within the mesh of the actin-based membrane skeleton. (b) The anchored-protein (picket) model: transmembrane proteins, anchored to the actin-based cytoskeleton, effectively act as rows of pickets along the actin fences.

Figure 9

Confined diffusion in a narrow cylindrical channel with a periodically modulated boundary $w(x)$ in the axial direction. (a) Small diffusing particle. (b) Single-file diffusion.

Figure 10

Illustrative sketches of how mobility and diffusivity vary with nondimensionalized applied force $F_{0}L/k_{B}T$ in the case of a 2D channel with a sinusoidally varying half-width (3.56). (a) Effective mobility $\mu$ in units of $\gamma$. In the limit $F_0\to \infty$, $\mu \to {\gamma }^{-1}$. (b) Diffusion coefficient $D$ in units of free diffusivity $D_0$. In the limit $F_0\to \infty$, $D\to D_0$. Sketches are based on numerical results of 301 for $a=L/2\pi$ and $b=1.02$.

Figure 11

Brownian particle moving in a periodic potential $V(x)$. In the absence of tilt ($F_0=0$) the mean velocity in the long time limit is zero. On the other hand, in the presence of a tilt $(F_0\ne 0)$ the net motion of the particle is in the direction of the force.

Figure 12

(a) Single-file diffusion of a tagged particle surrounded by other impenetrable particles. (b) Equivalent noninteracting picture, in which each trajectory is treated as a noninteracting Brownian particle by keeping track of the exchange of particle label.

Figure 13

Schematic illustration of the (a) import and (b) export process underlying the karyopherin-mediated transportation of cargo between the nucleus and cytoplasm via a nuclear pore complex (NPC). See text for details.

Figure 14

Model of 398. Transport of cargo complex through the NPC is modeled as diffusion in an energy landscape. See text for details.

Figure 15

Selective phase model (306; 37), in which the FG repeats within a NPC are treated as a reversible polymer gel. See text for details.

Figure 16

Brownian ratchet model of a molecular motor that can exist in two internal states with associated $l$-periodic potentials $V_1(x)$ and $V_2(x)$. State transition rates are denoted by ${\omega }_1$ and ${\omega }_2$.

Figure 17

State transition diagram for a discrete Brownian ratchet that cycles through $M=3$ internal states and makes a single step of length $\Delta x$.

Figure 18

Schematic diagram of an asymmetric tug-of-war model. Two kinesin and two dynein motors transport a cargo in opposite directions along a single polarized microtubule track. Transitions between two possible motor states are shown.

Figure 19

Transition diagram of the “stop-and-go” model for the slow axonal transport of neurofilaments. See text for definition of different states.

Figure 20

Active transport on a disordered microtubular network. (a) Random orientational arrangement of microtubles. (b) Effective 2D random intermittent search in which a particle switches between diffusion and ballistic motion in a random direction.

Figure 21

Random velocity model of a microtubular network with quenched polarity disorder. Particles move ballistically along parallel tracks in a direction determined by the polarity of the given track. They also hop between tracks according to an unbiased random walk.

Figure 22

Diagram of a 2D radially symmetric cell with radially equidistant microtubles. A virus trajectory is shown that alternates between ballistic motion along a microtubule and diffusion of the cytoplasm. Trajectory starts at the cell membrane and ends at a nuclear pore. From 212.

Figure 23

Schematic diagram of TASEP with Langmuir kinetics, in which particles can spontaneously detach and attach at rates ${\omega }_D$ and ${\omega }_A$, respectively.

Figure 24

Mean-field phase diagram for the TASEP showing the regions of $\alpha$ and $\beta$ parameter space where the low-density (LD), high-density (HD), and maximal-current (MC) phases exist. Schematic illustrations of the density profiles in the various regions are shown as curves.

Figure 25

Formation of a shock for Eq. (4.91). The characteristics are straight lines of speed $1-2\rho$ with $\rho$ constant along a characteristic. The initial density profile evolves into a stationary shock solution.

Figure 26

Diagram showing how the translation of the mRNA and the synthesis of proteins is made by ribosomes. [From LadyofHats (Public domain), via Wikimedia Commons.

Figure 27

A TASEP with extended particles of size $l=3$.

Figure 28

A motor-cargo complex performing a random intermittent search for a hidden target on a one-dimensional track of length $L$. The motor-cargo complex is transported along a microtubule by two populations of molecular motors with opposing directional preferences. When equal numbers of each type of motor are bound to the microtubule, the motor-cargo complex is in a slowly moving search state and can find the target provided that it lies within the target domain of width $2l$ centered at $x=X$. Target detection rate is $k$.

Figure 29

(a) Unbiased $({\beta }_{+}={\beta }_{-}=\beta )$ random intermittent search in the mean-field population model. The MFPT vs (i) the average search time $1/\alpha$ with $\beta =1/ϵ$ and (ii) the average duration $1/\beta$ of the moving (forward and backward) state with $\alpha =2/ϵ$. Each curve has a minimum MFPT for a given value of $\alpha$ and $\beta$. Parameter values used are $ϵ=0.1$, $k=5/N$, $X=50$, and $l=0.25$. (b) First-passage time density for an unbiased search with the same parameters as (a). The solid curve shows the analytical density function in the mean-field limit and the remaining curves are histograms obtained from ${10}^4$ Monte Carlo simulations for different numbers of searchers $N$. (c) Corresponding first-passage time density for a biased search with $\alpha =1/ϵ$, ${\beta }_{+}=1/ϵ$, ${\beta }_{-}=2/ϵ$, $ϵ=0.1$, $k=5/N$, $X=50$, and $l=0.25$. (d) The MFPT vs hitting probability for population model. Each curve is parametrized by $0\le {\beta }_{+}\le {\beta }_{-}$. Analytical results are shown as lines, with the single searcher in gray and the mean-field limit ($N=\infty$) in black. Averaged Monte Carlo simulations (${10}^4$ simulation each) are shown as symbols, sets ranging from gray to black, with a different value of $N$ used in each set. From gray to black there are six sets of simulations with $N=1$, 2, 3, 4, 5, and 25, respectively. Parameter values are the same as (c).

Figure 30

Effects of tau concentration on the tug-of-war model with $N_{+}$ kinesin motors and $N_{-}$ dynein motors. The stall force $F_S$, forward velocity $v_f$, and unbinding rate ${\gamma }_0$ are given by Eqs. (4.31, 4.32, 4.33) with $[\mathrm{ATP}]={10}^3\mu \mathrm{M}$. The other single motor parameters are (267) $F_d=3\mathrm{pN}$, ${\gamma }_0=1{\mathrm{s}}^{-1}$, ${\pi }_0=5{\mathrm{s}}^{-1}$, and $v_b=0.006\mu \mathrm{m}/\mathrm{s}$. The corresponding parameters of the FP equation are obtained using a QSS reduction and plotted as a function of $\tau$. (a) Effective capture rate $\lambda$. (b) Drift velocity $V$. (c) Diffusivity $D$.

Figure 31

Effect of adding tau to the target on the capture probability $\Pi$ and MFPT $\mathcal{T}$ using parameters from Fig. 30. (a) The analytical approximation $P$ (solid line) and results from Monte Carlo simulation. (b) The analytical approximation $T$ along with averaged Monte Carlo simulations. The synaptic trap is located at $X=10\mu \mathrm{m}$, the trapping region has radius $l=2\mu \mathrm{m}$, and the microtubular track (MT) has length $L=20\mu \mathrm{m}$. The capture rate is taken to be $k_0=0.5{\mathrm{s}}^{-1}$.

Figure 32

The effective potential well created by a region of tau coating a MT, and a representative trajectory showing random oscillations.

Figure 33

Stochastic hydrolysis dynamics.

Figure 34

Schematic diagram of intraflagellar transport (IFT), in which IFT particles travel with speed $v_{\pm }$ to the $\pm$ end of a flagellum. When an IFT particle reaches the $+$ end it releases its cargo of protein precursors that contribute to the assembly of the flagellum. Disassembly occurs independently of IFT transport at a speed $V$.

Figure 35

Left: Diagram of secretory pathway including nucleus, ER and Golgi apparatus. 1. Nuclear membrane, 2. nuclear pore, 3. RER, 4. SER, 5. ribosome, 6. protein, 7. transport vesicles, 8. Golgi apparatus, 9. Cis face of Golgi apparatus, Trans face of Golgi apparatus, 11 cisternae of Golgi apparatus, 12. secretory vesicle, 13. plasma membrane, 14. exocytosis, 15 cytoplasm, 16. extracellular domain. From WikiMedia Commons. Right: When two compartments continually exchange products via vesicular transport, a symmetry breaking mechanism is needed to maintain nonidentical compartments ($C_1\ne C_2$).

Figure 36

Signaling molecules can attach and orient actin filaments that deliver vesicles carrying the signaling molecule from the cytoplasm to the plasma membrane. The additional signaling molecules orient more actin filaments that transport more molecules in a positive feedback loop. The local orientation of actin filaments also increases the rate of endocytosis within the cluster. From 245.

Figure 37

(a) A set of receptors transduce an external distribution of chemotactic cues into an internal distribution of activated enzymes $X$ that catalyze the switch of a signaling molecule from an unactivated state ${\Phi }^{-}$ to an activated state ${\Phi }^{+}$. A counteracting enzyme $Y$ transforms ${\Phi }^{+}$ back to ${\Phi }^{-}$. Amplifying feedback loops, in which ${\Phi }^{+}$ activates $X$ and ${\Phi }^{-}$ activates $Y$, result in chemical bistability. The signaling molecules are permanently bound to the plasma membrane, where they exhibit lateral diffusion, while the enzymes are free to move between the membrane and the cytosol. (b) Cell polarization occurs when there is phase separation into two stable chemical states. From 338.