Formation of a Morphogen Gradient: Acceleration by Degradation

I is known that several classes of signaling molecules stimulate complex concentration-dependent responses that are critical for growth, development, and tissue formation in multicellular organisms. 4,11 In recent years there have been many quantitative studies that revealed important features and details of morphogen gradients in different biological systems. 10 However, fundamental mechanisms of morphogen gradient formation are still not well understood. The current view of the development of morphogen gradients suggests that this process is a result of a combination of physical and chemical processes. The simplest and the most popular proposedmechanism, known as a synthesis diffusion degradation (SDD) model, assumes that locally produced signaling molecules diffuse freely along the tissue while being uniformly degraded. This model has been used successfully to describe temporal evolution of signaling molecule profiles in different biological systems.However, the application of thismechanism to the formation of bicoid morphogen gradient led to controversial observations. Measured mobility of bicoid molecules was too low to explain fast establishment of the stationarystate profile by a simple unbiased diffusion. Several ideas how to resolve this paradox, such as biased diffusion due to active processes and the effect of the advective transport, have been proposed. However, experimentally all these suggestions have not yet been supported. Furthermore, a recent theoretical study of the formation of morphogen gradients has provided a systematic approach to explicitly evaluate time to reach steady-state concentration profiles. The most surprising observation of this theoretical analysis is a linear scaling as a function of the distance from the source for the time to establish a morphogen gradient in the SDDmodel. It led to a conclusion that stationary concentration profiles could be formed faster than was thought previously, but the mechanism of such acceleration remains unclear. In the system with the unbiased diffusion, a quadratic scaling with the distance and slow reaching of steady-state conditions are expected. In this work, we propose a microscopic model that allows one to explain this paradoxical behavior. Our theoretical picture provides a physical chemical mechanism for the fast formation of morphogen gradients and linear scalings in the relaxation times. To analyze the formation of the morphogen gradient, we utilize a discrete-state version of the SDD model, as shown in Figure 1. The signaling molecule can be found in one of the discrete sites n (ng 0) that might be associated with an underlying line of cells. It is analogous to a compartmental model developed recently for the bicoid gradient. In addition, a single-molecule view of the process is adopted here, i.e., the concentration of molecules is equivalent to the probability of finding a single particle at a given site.We assume that particles are produced at the origin, n = 0, with a rate Q, then they spread to the right (ng 0) via a free diffusion along the lattice of discrete sites with a diffusion rate u (see Figure 1). At any position, the particle might also be degraded with a rate k. The continuum description of the system is obtained when the diffusion rate is much faster that the degradation processes, i.e., u. k. In the discrete case, the probability Pn(t) of finding the particle at the position n at time t is governed by a set of master equations: