Tunable ultrasensitivity: functional decoupling and biological insights

Sensitivity has become a basic concept in biology, but much less is known about its tuning, probably because allosteric cooperativity, the best known mechanism of sensitivity, is determined by rigid conformations of interacting molecules and is thus difficult to tune. Reversible covalent modification (RCM), owing to its systems-level ingenuity, can generate concentration based, tunable sensitivity. Using a mathematical model of regulated RCM, we find sensitivity tuning can be decomposed into two orthogonal modes, which provide great insights into vital biological processes such as tissue development and cell cycle progression. We find that decoupling of the two modes of sensitivity tuning is critical to fidelity of cell fate decision; the decoupling is thus important in development. The decomposition also allows us to solve the ‘wasteful degradation conundrum’ in budding yeast cell cycle checkpoint, which further leads to discovery of a subtle but essential difference between positive feedback and double negative feedback. The latter guarantees revocability of stress-induced cell cycle arrest; while the former does not. By studying concentration conditions in the system, we extend applicability of ultrasensitivity and explain the ubiquity of reversible covalent modification.


Full model of RCM with positive feedback
The mathematical model has been given by Eqs. (1-10), which describe the time evolution of the system. To obtain the steady states, the left hand side of Eqs. (1-7) are first replaced with 0, which results in ten algebraic equations in total

Response curve
Given an input I, the ten equations are solved numerically to obtain the steady state values, including W * . By sweeping I from small to large and calculating the corresponding W * , the response curve W * (I) is obtained. Note that a given I corresponds to multiple solutions, some of which are meaningless (e.g., E < 0, W * > W tot , etc.). Even with those meaningless solutions discarded, a given I may still correspond to several solutions.
The black response curve in Fig. S.1A gives such an example. If I is in the range (I off , I on ), where I on = 0.0444 and I off = 0.0066, then a single I corresponds to three meaningful W * values.
The middle one of the three is always found to be unstable (it is destroyed by even the slightest 0 00100 = 1, = 0.000100, = 0.000900 = . ( ) random perturbation) according to our stability analysis below. These middle values constitute the middle branch of the response curve. To signify its nonexistence in reality, the middle branch is represented by dashed curve segment. The upper and lower branches are always found to be stable.
The response curve thus manifests bistability, not tristability. The red and green response curves are obtained by reducing K and K * smaller and smaller, while keeping the other parameter values fixed. One sees that as K and K * decrease, the response curve approaches to a limit, which will be determined by an idealized model in the following. Note that the mathematical model does not always produce bistable curves. If positive feedback is not sufficiently strong, and/or the K and K * values are too large, graded response curves may be obtained ( Fig. S.1B ).

Bifurcation analysis
The numerical calculation of thresholds I on and I off is of paramount importance in this paper. For example, the production of Fig. 2 demands numerous calculations of I on and I off . If a pair of I on and I off can be determined only after tracing out the entire response curve (as described above), then the computational costs would be too high. Indeed, if a response curve consists of 1000 points, then the set of 10 equations (Eqs. (S.1-S.9)) have to be solved 1000 times just to plot the response curve. Additional algorithms are then required to determine I on and I off .
Fortunately, singularity and bifurcation theory offers an efficient method, which involves solving a set of 10 equations (Eqs. (S.13-S.22), see below) only once for a single pair of I on and I off . Let W * on and W * off denote the vertical coordinates corresponding to I on and I off , respectively. According to singularity theory [12], the two points (I off , W * off ) and (I on , W * on ) are singularities of the type known as limit point, which satisfy the following normal form equations where G is the combined steady state equation (Eqs. (S.1-S.10) lumped together). The expansion of Eqs. (S.11 and S.12) leads to the following equations: After expansion, the equations become By substituting the values of the parameters k on , k off , k cat , k * on , k * off , k * cat , F max , W 0.5 , n, r, W tot , and E * tot into the above 10 equations, the solutions to the variables (I, If all the solutions are meaningless, then singularities do not exist; and the response curve is a graded curve, not a bistable one. If meaningful solutions exist, then they must present as a pair: one is denoted by (I on , W * on , W on , [W E] on , · · · ) and the other is denoted by . In this way, we have determined the two bifurcation points (I on , W * on ) and (I off ,

Stability analysis
The stability of any point of a response curve can be determined by calculating the eigenvalues of the Jacobian matrix associated with that point. The point is stable if and only if every eigenvalue has negative real part.
There are seven variables in the mathematical model. Because of the three constrains (Eqs. (8-10)), there are only four independent variables, whose time evolutions are described by The associated Jacobian matrix is Notice the expression for Φ 41 . If f ([W * ]) is a nonlinear feedback (Eq. (12)), then Given a point of the response curve, one substitutes the state and parameter values into Eq.
(S.37), resulting in a numerical 4×4 Jacobian matrix. One then obtains its eigenvalues by softwares such as Matlab or Mathematica.
By stability analysis, we found that the graded response curve is always stable. For the bistable response, the upper and lower branches are always stable; while the middle branch is always unstable.

Idealized model
We With this simplification, the steady-state equations (S.1-S.10) reduces to a single equation which is still complex and difficult to analyze.
Equation (S.39) can be reduced by using the limit condition K → 0 and K * → 0, which parallels with the assumption of large W tot . Indeed, from Eq. (11) one sees that the largeness of W tot implies the smallness of K and K * . By this simplification, Eq. (S.39) is further reduced and can be factored into three terms By defining ∆I = I on − I off , we have The above elucidation is also applicable when the feedback is linear (Fig. S.2D,E,F ).
Note that this method of determining I on and ∆I is much simpler than bifurcation analysis of the full model, which embodies the power of idealization. One should also be convinced of the validity of our idealization-the ideal response curve in Fig

RCM with other feedbacks: a summary
We have analyzed RCM with positive feedback W * → E tot in detail. For the other feedbacks, the mathematical model is the same except the part describing regulations to E tot or E * tot (the second column of Fig. S.3). The third column of Fig. S.3 lists results obtained from the idealized model.

Insights into development: examples
In the main text, we mentioned that exclusive differentiation is sometimes necessary. The resultant delay might be sufficient to deprive the cell's differentiation opportunity (Fig. S.4C ).
The above example is about specifying two cell fates by a morphogen gradient. In fact, a single morphogen gradient can even specify three or more cell fates [23,31], by activating many genes. If these genes' I on values are the same, then only two cell fates will result (similar to the example in the main text). To specify more cell fates, these I on values must be well separated so that the on/off combinations are large enough to generate differences. Several mechanisms exist to separate the I on values, e.g., cell-cell communication (Notch/Delta signalling) mentioned in the main text.
Alternatively, I on -tuning can be realized intracellularly through mutual inhibitions among the genes.
Figure S.5 illustrates three genes X, Y, and Z whose on/off combinations determine four cell types.
Through mutual inhibitions among the three genes, their I on values are well separated, which allows for the generation of four cell types by a single morphogen gradient.
Note that decoupling of sensitivity tuning is crucial to the success of the I on -separation strategy. If crosstalks are severe, a nonfeedback inhibition delays the response (desired) but at the same time deforms the response curve (undesired; for the activation quality becomes degraded), leading to inadequate gene expressions and thus abnormal cell types.