Question

In the chapter, we considered a one-dimensional lattice of cells, each containing two species of morphogens that undergo chemical reactions. Following Turing, we showed that diffusion of morphogens can destabilize a steady state described by a uniform concentration profile, leading to a spatially periodic pattern of morphogens. In this problem, we analyze the situation when there is only one morphogen species present. (a) Rewrite the reaction-diffusion equation for the Turing system, Equation \(20.40,\) for the case of a single morphogen whose concentration within a cell is \(Y_{r}\). Then consider a small periodic perturbation of the uniform steady state \(Y_{r}=Y^{*}\) of the form \(Y_{r}=Y^{*}+y(t) e^{i(2 \pi r / \lambda)}\) and derive the dynamical equations for the amplitude \(y(t)\) (b) Assuming that there are \(N\) cells in the system and that they are arranged in a ring so that the \(r=1\) cell has the \(r=N\) and \(r=2\) cells as its nearest neighbors, what are the allowed values of the wavelength \(\lambda\) for the periodic perturbation? (c) Derive the conditions under which the uniform steady state is unstable to a small periodic perturbation. Argue that this one-component Turing system does not lead to spatially periodic pattern of morphogen concentration.

Short answer

The one-component Turing system doesn't lead to a spatially periodic pattern of the morphogen concentration because the system would not amplify small perturbations into structured patterns.

Step-by-step solution

Rewrite the reaction-diffusion equation for a single morphogen

The reaction-diffusion equations for the Turing system can be reduced to a simplified form when there is only one morphogen species present. The equation is given by: \[ \frac{{dY_r}}{{dt}} = R(Y) + D \frac{{\partial ^2Y}}{{\partial x^2}} \]Where:- \( R(Y) \) is the reaction term, which is simply the chemical reactions involving the morphogen.- \( D \) is the diffusion coefficient.- \( Y_r \) is the concentration of the morphogen in the r-th cell. - The second term accounts for the diffusion of the morphogen. But in our case, \( R(Y) = 0 \) since no reaction is happening and morphogen diffusion has no effect on the morphogen itself.

Consider a small periodic perturbation of the uniform steady state

Assume a small perturbation in the form \( Y_r = Y^{*} + y(t) e^{i(2 \pi r / \lambda)} \), where \( Y^{*} \) is the steady state. Then the equation becomes:\[ \frac{{dy}}{{dt}} = -D \left( \frac{{2 \pi}}{{\lambda}} \right)^2 y(t) \]This is now a simple first-order linear differential equation, the solution of which gives us the time evolution of the amplitude \( y(t) \).

Assumptions regarding the cells in the system

Given that there are \( N \) cells in the system, arranged in a ring, the allowed values of the wavelength \( \lambda \) for the periodic perturbation are given by \( \lambda = \frac{{L}}{{k}} \), where \( k \) is an integer and \( L \) is the length of the total system, so in this case \( L = N \). Since it is given that the cells are in a ring, \( r = 1 \) cell has \( r = N \) as its nearest neighbor. One can take into account the values of \( k \) from \( -N/2 \) to \( N/2 \). This wraps around due to the circular condition.

Conditions for instability of the uniform steady state

To determine whether the uniform steady state is unstable to a small periodic perturbation, we need to look at the eigenvalues of the system. The instability condition for a one-dimensional system is given by: \( Re(\lambda) > 0 \), where \( \lambda \) is the eigenvalue of the system. Also note that the stability depends on the size of the perturbation. The larger the perturbation in size, the more likely it will upset the steady state.

Argument regarding a one-component Turing system

For a Turing pattern to occur, a small perturbation must be amplified by the system. This means the conditions for instability need to be met. However, this one-component Turing system does not lead to a spatially periodic pattern of morphogen concentration because the instability condition \( Re(\lambda) > 0 \) is not met. This is because the system does not have a mechanism to amplify a small perturbation into a spatially structured pattern, hence only the uniform state can be achieved.

Key concepts

Reaction-Diffusion Equation

Understanding the reaction-diffusion equation is central to unraveling the fascinating patterns observed in nature, such as animal skin pigmentation and shell patterns. In a biological context, the reaction part corresponds to the chemical transformation of substances, while the diffusion part captures the spread of these substances through a medium.

In the context of a Turing system with a single morphogen, the equation simplifies significantly. The general equation is \[ \frac{{dY_r}}{{dt}} = R(Y) + D \frac{{\partial ^2Y}}{{\partial x^2}} \], where the reaction term, denoted by \( R(Y) \), often includes nonlinear interactions between different morphogens; however, with just one morphogen present and no interactions, \( R(Y) = 0 \). The diffusion term, featuring the diffusion coefficient \( D \) and the Laplacian operator \( \frac{{\partial ^2}}{{\partial x^2}} \), accounts for the movement of the morphogen across the cells.

To analyze the system’s behavior further, a small periodic perturbation is introduced to the uniform steady state. The temporal evolution of this perturbation's amplitude provides insights into the system's stability, leading to an essential distinction between stable and unstable states.

In an educational setting, visualizations of diffusion processes and dynamic systems' response to perturbations can greatly enhance students’ conceptual understanding. As an improvement, animations demonstrating how initial uniform states react to small perturbations could clarify the theory.

Morphogen Concentration Stability

The stability of morphogen concentration within the Turing system is a key concept in developmental biology, as it can lead to the emergence of structures and patterns during organismal development. When considering the Turing system with a single morphogen species, we analyze the stability by applying a perturbation to the concentration profile and observing the system's response.

In this context, stability refers to the system's ability to return to the uniform steady state after a small perturbation. The stability is analyzed by deriving the dynamical equations for the amplitude of periodic perturbations, given by \[ \frac{{dy}}{{dt}} = -D \left( \frac{{2 \pi}}{{\lambda}} \right)^2 y(t) \].

The sign of the eigenvalue dictates the system's response: a positive real part of the eigenvalue indicates exponential growth of the perturbation, leading to instability. Conversely, a negative real part suggests that the system will return to its uniform state, evidencing stability. Students could benefit from interactive simulations where they can manipulate parameters like the diffusion coefficient to see their effects on morphogen concentration. Such practical exercises reinforce understanding of theoretical concepts.

Uniform Steady State Instability

One of the hallmark conditions for the emergence of Turing patterns is the uniform steady state's instability when subjected to small perturbations. The Turing system can exhibit such behavior when there are interactions and differences in diffusion rates among multiple morphogens.

However, for a single morphogen model, the lack of interactions (since \( R(Y) = 0 \)) results in a system that does not meet the criteria for Turing instability. For an instability to occur, the differential growth or decay in the concentration due to the reaction and diffusion parts must lead to a deviation from uniformity that increases over time.

The eigenvalue analysis reveals the conditions for instability, where in this case, the real part of the eigenvalue must be positive for the amplitude of the perturbation to grow. For a single morphogen without reaction, a uniform steady state remains stable against small perturbations, as indicated by the negative eigenvalue \( -D \left( \frac{{2 \pi}}{{\lambda}} \right)^2 \).

Incorporating real-world examples, such as the failure to establish a pattern in skin cells with only one type of morphogen, may help students grasp why complexity and interactions are crucial for pattern formation. Tasks that involve students predicting outcomes based on stability analysis can also strengthen their understanding and application of the theory.