Signal focusing through active transport

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Signal focusing through active transport

Aljaž Godec and Ralf Metzler

Phys. Rev. E 92, 010701(R) – Published 2 July 2015

Abstract

The accuracy of molecular signaling in biological cells and novel diagnostic devices is ultimately limited by the counting noise floor imposed by the thermal diffusion. Motivated by the fact that messenger RNA and vesicle-engulfed signaling molecules transiently bind to molecular motors and are actively transported in biological cells, we show here that the random active delivery of signaling particles to within a typical diffusion distance to the receptor generically reduces the correlation time of the counting noise. Considering a variety of signaling particle sizes from mRNA to vesicles and cell sizes from prokaryotic to eukaryotic cells, we show that the conditions for active focusing—faster and more precise signaling—are indeed compatible with observations in living cells. Our results improve the understanding of molecular cellular signaling and novel diagnostic devices.

Received 13 January 2015

DOI:https://doi.org/10.1103/PhysRevE.92.010701

Authors & Affiliations

Aljaž Godec^1,2 and Ralf Metzler^1,3

¹Institute of Physics & Astronomy, University of Potsdam, D-14476 Potsdam-Golm, Germany
²Laboratory for Molecular Modeling, National Institute of Chemistry, Hajdrihova 19, 1000 Ljubljana, Slovenia
³Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland

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Vol. 92, Iss. 1 — July 2015

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Images

Figure 1
Model system. (a) Signaling particles perform passive thermal diffusion (blue phases) interrupted by active ballistic excursions with constant speed and random direction (red phases moving along the black motor tracks). The duration of both phases is distributed exponentially with mean times $τ_{p, a}$ . When the particle reaches the receptor (green sphere) it binds and dissociates with rates $k_{on}$ and $k_{off}$ , respectively. (b) Receptor region.
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Figure 2
(a)–(d) Equilibration times $τ_{0} / τ_{i}$ for various Péclet numbers Pe from numerical inversion of Eq. (5). Yellow lines denote $τ_{0} / τ_{i} = 1$ . When $τ_{0} / τ_{i} > 1$ active trafficking is more efficient. Note the different scales for $τ_{0} / τ_{i}$ . (e) The Pe as a function of cell (domain) size $L$ and diffusivity $D$ for biological cells [19, 35]. Arrows denote increasing particle size from right (small proteins) to left (large vesicles) and cell size from bottom (prokaryotic cells) to top (eukaryotic cells). The border between prokaryotic and eukaryotic cells is around 10 $μ m$ . For $Pe \sim 10^{4}$ (vesicles or large proteins in eukaryotic cells), active transport always improves the speed. For small proteins in eukaryotic cells and large proteins in prokaryotic cells ( $10 ≲ Pe ≲ 10^{2}$ ), in addition to large $x_{a}$ a large $x_{p}$ is necessary for active transport to be more efficient. For $Pe ≲ 10^{- 1}$ active transport is always less efficient.
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Figure 3
(a)–(c) Fractional variance $σ_{a} = {[\bar{δ c_{p}^{2}} / {〈 c_{p} 〉}^{2}]}^{1 / 2}$ of concentration fluctuations at the receptor site for active trafficking compared to thermal diffusion $σ_{0} = {[π D a c_{tot} τ_{m}]}^{- 1 / 2}$ as a function of the average distance traveled in both phases in units of the receptor radius $a$ . In (a)–(c) the total concentration $c_{tot}$ is kept constant, hence decreased sensing accuracy is solely due to exceedingly small equilibrium concentrations in the passive phase. (d) $σ_{a} / σ_{0}$ for equal equilibrium concentrations [sc in (e) and (f)] in the passive phase $〈 c_{p} 〉$ for different total concentrations equal to $c_{tot} (1 + τ_{a} / τ_{p})$ : A strong increase in sensing accuracy is observed for $x_{p} / a \to 0$ . (a)–(d) use the results (13) and (S8) in [20]. (e) Horizontal cross sections of $σ_{a} / σ_{0}$ at large and small $x_{p}$ (symbols) compared to the approximations (15) and (16). (f) Vertical cross sections at large and small $x_{a}$ (symbols) compared to the approximations (15) and (16). Note the different scales for $σ_{a} / σ_{0}$ and the excellent agreement between full and approximate results. The nonmonotonicity at small $x_{p}$ is due to the interplay of signal focusing (smaller $Λ$ ) and decreasing $〈 c_{p} 〉$ .
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