Calibration Procedure
Calibration procedure
To obtain concise asymptotic formulas for the probability and the mean first passage time of a virus to a nuclear pore, we first coarse grain the viral intermittent trajectories into a stochastic continuous equation that contains both a drift term and a diffusion term. Using this Langevin description of the viral velocity and modeling the cell degradation activity with a steady state killing field (see Figure 1), we can derive the Kolmogorov equation for the survival probability density function of the viral particle inside the cytoplasm and compute asymptotically the probability and the MFPT of the virus to a nuclear pore.
The problem is then to calibrate the drift term as a function of the microtubules network organization, the diffusion constant of the virus and the dynamical properties of the active transport along the microtubules. To calibrate the drift we considered a fundamental step: The virus starts at a given position x0, diffuses during a mean time τ(x0) until it reaches a microtubule at a mean binding site; It is then transported during a mean time tm, over a mean distance dm and unbinds at a mean final position xf. The drift b(x0) is then calibrated such that the mean time from x0 to xf with the continuous Langevin description is equal to τ(x0)+tm. That is, in the limit of a small diffusion coefficient D<<1:
Calibration in a radial geometry
To explicitly compute the drift amplitude, we first consider a two-dimensional radial cell with N uniformly distributed microtubules, and we assume, in a first approximation, that the microtubules converge to the nucleus instead of the centrosome (see Figure 2). In particular, this two-dimensional approximation stands for culture cells that are quite flat due to their adhesion to the substrate.
Due to the uniform distribution of the microtubules, we reduce our analysis to the fundamental wedge domain (angle Θ=2π/N) between two neighboring microtubules (see Figure 3).
In that domain, a viral particle starting at an initial radial distance r0 from the cell center will diffuse during a mean time τ(r0) until it reaches a microtubule at a mean position r(r0). It is then actively transported over a mean distance dm during a mean time tm before detaching at a mean final radius rf. In the small diffusion limit, the radial dependent drift amplitude b(r0) is then approximated by:
Solving explicitly the heat and the dynkin equations in the wedge domain, with lateral absorbing boundaries and an external reflecting one, we asymptotically found for N>>1:
which leads to
Interestingly, we tested the steady state density into a closed reflecting disk for a viral particle whose trajectories are solution of the continuous Langevin equation with the drift computed above against the steady state distribution of many intermittent Brownian trajectories, and both distributions matched very nicely:
Calibration in a cylindrical geometry
Transport of vesicles, proteins or virus (such as the Herpes virus) is ubiquitous in neurons’ dendrites or axons. In a first approximation, we approximate the dendrite geometry with a long thin cylinder (radius R) that contains N microtubules (radius ε) uniformly distributed along the dendrite principal axis (see Figure 5).
Due to the cylindrical symmetry, for any position x, the steady state drift b(x) is equal to Bz where B is a constant and z the principal axis unit vector along the dendrite. In a small diffusion approximation, the leading order term of B is given by B=dm/(tm+τN), where τN is the mean first passage time of a diffusing virus to one of the N microtubules. In a first approximation, we computed:
which leads to the drift amplitude:
Perspectives
In a future analysis, it should be very interesting to:
-define a drift calibration procedure in spherical geometries, which are close to most cells geometries
-compute the second order term of the MFPT τN to a microtubule in a dendrite to obtain a more accurate formula for the drift amplitude.
Calibration procedure
To obtain concise asymptotic formulas for the probability and the mean first passage time of a virus to a nuclear pore, we first coarse grain the viral intermittent trajectories into a stochastic continuous equation that contains both a drift term and a diffusion term. Using this Langevin description of the viral velocity and modeling the cell degradation activity with a steady state killing field (see Figure 1), we can derive the Kolmogorov equation for the survival probability density function of the viral particle inside the cytoplasm and compute asymptotically the probability and the MFPT of the virus to a nuclear pore.
Figure
1: The intermittent trajecories of the virus (left-hand side) are
coarse-grained in a Langevin equation that contains both a diffusion
term and a drift term. The degradation activity is modeled by a
steady-state killing rate.
The problem is then to calibrate the drift term as a function of the microtubules network organization, the diffusion constant of the virus and the dynamical properties of the active transport along the microtubules. To calibrate the drift we considered a fundamental step: The virus starts at a given position x0, diffuses during a mean time τ(x0) until it reaches a microtubule at a mean binding site; It is then transported during a mean time tm, over a mean distance dm and unbinds at a mean final position xf. The drift b(x0) is then calibrated such that the mean time from x0 to xf with the continuous Langevin description is equal to τ(x0)+tm. That is, in the limit of a small diffusion coefficient D<<1:
Calibration in a radial geometry
To explicitly compute the drift amplitude, we first consider a two-dimensional radial cell with N uniformly distributed microtubules, and we assume, in a first approximation, that the microtubules converge to the nucleus instead of the centrosome (see Figure 2). In particular, this two-dimensional approximation stands for culture cells that are quite flat due to their adhesion to the substrate.
Figure
2: The cell geometry is approximated by a two-dimensional disk with N
uniformly distributed microtubules converging to the nucleus.
Due to the uniform distribution of the microtubules, we reduce our analysis to the fundamental wedge domain (angle Θ=2π/N) between two neighboring microtubules (see Figure 3).
Figure 3: Fundamental
wedge domain between two neighbouring microtubules. A fundamental
intermittent step of the virus is represented.
In that domain, a viral particle starting at an initial radial distance r0 from the cell center will diffuse during a mean time τ(r0) until it reaches a microtubule at a mean position r(r0). It is then actively transported over a mean distance dm during a mean time tm before detaching at a mean final radius rf. In the small diffusion limit, the radial dependent drift amplitude b(r0) is then approximated by:
Solving explicitly the heat and the dynkin equations in the wedge domain, with lateral absorbing boundaries and an external reflecting one, we asymptotically found for N>>1:
which leads to
Interestingly, we tested the steady state density into a closed reflecting disk for a viral particle whose trajectories are solution of the continuous Langevin equation with the drift computed above against the steady state distribution of many intermittent Brownian trajectories, and both distributions matched very nicely:
Figure 4: Viral
distribution with a continous Langevin description of the velocity
(dashed line) against empirical steady state distribution obtained by
running 10,000 intermittent Brownian trajectories (solid line). The
cell radius is R=20μm and Θ=π/6.
Calibration in a cylindrical geometry
Transport of vesicles, proteins or virus (such as the Herpes virus) is ubiquitous in neurons’ dendrites or axons. In a first approximation, we approximate the dendrite geometry with a long thin cylinder (radius R) that contains N microtubules (radius ε) uniformly distributed along the dendrite principal axis (see Figure 5).
Figure 5: Dendrite cross-section. N microtubules (radius ε) are uniformly distributed along the dendrite principal axis.
Due to the cylindrical symmetry, for any position x, the steady state drift b(x) is equal to Bz where B is a constant and z the principal axis unit vector along the dendrite. In a small diffusion approximation, the leading order term of B is given by B=dm/(tm+τN), where τN is the mean first passage time of a diffusing virus to one of the N microtubules. In a first approximation, we computed:
Perspectives
In a future analysis, it should be very interesting to:
-define a drift calibration procedure in spherical geometries, which are close to most cells geometries
-compute the second order term of the MFPT τN to a microtubule in a dendrite to obtain a more accurate formula for the drift amplitude.