Appendix B: general solution of an N-cycle kinetic scheme
We wish to find the null space of matrices of the form:
We start from the equations given detailed balance. In this case, For all i (modulus n). This system of n equations generally has no trivial solution for an arbitrary set of coefficients, unless
Again, note the indexes are modulus n; this indexing convention will be used throughout this solution. We proceed to dump the nth equation and solve for v1, resulting in the non-trivial solution space:
To obtain simpler expressions, we define by
.The solution space becomes
With this solution, there is a residual current from v
n to
v1:
We note the current zeroes out exactly when detailed balance holds. We could have obtained the same solution by dumping any other equation. This can be seen by applying a cyclic permutation on the indices. The expression for
however
will remain invariant. By rotating the indices, we need to express products that might wrap through the nth index to the 1st. We adopt the following notation to express this kind of product
Thus the solution obtained by omitting the pth equation is:
And the sum over p produces the required solution:
Appendix C - full expression for the ratio of active state population
The solution for the ratio of the population of active states between the system with the flexible linker and the corresponding system without, is given in general by the ratio of the steady states (see Kinetic
scheme):
With the rates ki,j assigned according to Figure 5 and the following text. Without making any further assumptions, the resulting expression is very long and is given here in its full length: