Powering a burnt bridges Brownian ratchet: A model for an extracellular motor driven by proteolysis of collagen

Biased diffusion of collagenase on collagen fibrils may represent the first observed adenosine triphosphate-independent extracellular molecular motor. The magnitude of force generated by the enzyme remains unclear. We propose a propulsion mechanism based on a burnt bridges Brownian ratchet model with a varying degree of coupling of the free energy from collagen proteolysis to the enzyme motion. When constrained by experimental observations, our model predicts $0.1\mathrm{pN}$ stall force for individual collagenase molecules. A dimer, surprisingly, can generate a force in the range of $5\mathrm{pN}$, suggesting that the motor can be of biological significance.

Figure 1

(Color online) The powered burnt bridges model: The motor molecule performs a random walk on its substrate as shown in (a). The solid circle represents a potential cleavage site while the open circle represents a site already cleaved. Cleavage sites are positioned at a distance $\Delta$ along the track. Once cleavage occurs, the molecule is subjected to a constant force up to a distance $\delta$ from the digested site as shown in (b). A representation of collagen fibrils is shown in (c). A collagen fibril is composed of triple-helical monomers that line up to form asymmetric tracks. Each arrow on the fibril shows one triple helical monomer with the arrow pointing from the C to the N terminus of the monomer. Each fibril consists of many such parallel tracks that are offset by $60\mathrm{nm}$ from one another. Solid circles show cleavage sites which are positioned approximately three-quarter of the length from the N terminus of the monomers. In this context, the “spring” in the caption title and in Fig. 1a should not be regarded as a conventional Hookean spring, but rather as a device that produces constant force independent of compression.

Figure 2

The force-velocity response under varying RFFL: The force-velocity relationship is calculated from eq. (5). The force is normalized to the Brownian force which equals $1∕\beta \Delta$ and the velocity is normalized to $D∕\Delta$. The $F_{\mathrm{int}}=100$ in all, $\delta ∕\Delta =0$ (open squares), $\delta ∕\Delta =0.3$ (open triangles), $\delta ∕\Delta =0.6$ (solid diamonds), $\delta ∕\Delta =0.9$ (solid triangle), and $\delta ∕\Delta =1$ (solid squares). The inset shows the force activation of the motor when $P_c<1$. The force-velocity relationship is calculated using the Monte-Carlo simulation outlined in the text $\delta ∕\Delta =1$ in all and the curves show $P_c=1$, 0.8, 0.6, 0.4, 0.2, and 0.1 in descending order.

Figure 3

The experimental and modeled correlation functions: Open circles represent an experimental correlation function measured previously [7] for the active MMP–1 molecule on a collagen fibril. The gray solid line is a fit to the correlation function using $P_c=10\%$ with $F_{\mathrm{int}}=0$. The top and bottom dashed lines represent the correlation function with $P_c=5\%$ with no force coupling for the upper curve and with $F_{\mathrm{int}}=\frac{100}{\beta \Delta }$ and $\delta ∕\Delta =1$ at the lower curve. The best fit is achieved when $\delta ∕\Delta =0.5$ and $F_{\mathrm{int}}>\frac{10}{\beta \Delta }$ as shown by the black line. The correlation functions for any internal force magnitude greater than 10 overlap (data not shown). The inset however shows the force velocity prediction with an $\mathrm{RFFL}=1$, $P_c=0.05$, $F_{\mathrm{int}}=0.13\mathrm{pN}$ (solid circles), $0.65\mathrm{pN}$ (open triangles), $1.3\mathrm{pN}$ (open circles) and $1.95\mathrm{pN}$ (closed squares) demonstrating that under the condition of external load the different internal forces would yield dramatically different behaviors.

Figure 4

The predicted force velocity behavior of MMP-1: Single MMP–1 molecule with $P_c=0.05$ and $F_{\mathrm{int}}=0$ (solid squares), with $P_c=1$ and $F_{\mathrm{int}}=0$ (open triangles), $P_c=1$, $F_{\mathrm{int}}=100$, and $\delta ∕\Delta =0.5$ (open circles). A group of two MMP–1 molecules with $P_c=1$, $\delta ∕\Delta =0.5$ and $F_{\mathrm{int}}=100$ (solid circles), the two molecules move on tracks that are offset by $\delta$ thus creating an effective $\delta ∕\Delta =1$.