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## Instructions:

While working exactly is often a good starting point, it is often not enough to solve the problem completely. In many cases, we encounter integrals that cannot be evaluated analytically, or equations that are too complex to solve by hand. This is where numerical methods come in, as they allow us to approximate solutions to these problems using iterative processes that are well-suited to computers.

One of the advantages of numerical methods is that they allow us to handle real-world data that do not fit neatly into simple mathematical models. For example, if we have data from an experiment that we want to analyze, we may need to use numerical methods to fit a curve to the data or to approximate an integral that describes some aspect of the system being studied.

When using numerical methods, it is important to be aware of how errors propagate through computations. Even small errors in initial data or in the numerical algorithms can lead to significant errors in the final result. Therefore, it is important to understand the sources of errors and to develop techniques for controlling and minimizing them.

Some of the most important numerical methods that are commonly used in applied mathematics include methods for finding roots of equations, approximating integrals, and interpolating data. In each case, there are many different methods to choose from, each with its own advantages and disadvantages in terms of accuracy and efficiency. It is important to choose the right method for the problem at hand, taking into account factors such as the size of the problem, the desired level of accuracy, and the available computational resources.

Biological signaling and regulation networks often involve cycles in which a protein backbone is transformed through a collection of modified states with different numbers of phosphate groups attached. A basic cycle might be described by the reaction network:

$$

\mathrm{A} \stackrel{k_1}{\rightarrow} \mathrm{B} \stackrel{k_2}{\rightarrow} \mathrm{C} \stackrel{k_3}{\rightarrow} \mathrm{A}

$$

where A, B and $\mathrm{C}$ have the same protein backbone with different numbers of phosphate groups. Of course, some kind of energy input is required to maintain a cycle, which is not represented above.

Write down the stoichiometry matrix $\mathbf{S}$ for this reaction network.

$\mathbf{S}=\left(\begin{array}{ccc}-1 & 0 & 1 \ 1 & -1 & 0 \ 0 & 1 & -1\end{array}\right)$

Characterize the null space of $\mathbf{S}$ in terms of a dimension and a basis. What does this tell you about the fluxes (reaction rates) in the network at steady state? What physical interpretation can you provide for this?

The null space has dimension 1 and a basis: $(1,1,1)$. This indicates that the fluxes are equal at steady state.

Characterize the left null space of $\mathbf{S}$ in terms of a dimension and a basis. What does this tell you about the time evolution of the protein concentrations? What physical interpretation can you provide for this?

The left null space has dimension 1 and a basis: $(1,1,1)$. This indicates that the sum of the protein concentrations is constant in time.