Understanding non-linear effects from Hill-type dynamics with application to decoding of p53 signaling

Analytical equations are derived depicting four possible scenarios resulting from pulsed signaling of a system subject to Hill-type dynamics. Pulsed Hill-type dynamics involves the binding of multiple signal molecules to a receptor and occurs e.g., when transcription factor p53 orchestrates cancer prevention, during calcium signaling, and during circadian rhythms. The scenarios involve: (i) enhancement of high-affinity binders compared to low-affinity ones, (ii) slowing reactions involving high-affinity binders, (iii) transfer of the clocking of low-affinity binders from the signal molecule to the products, and (iv) a unique clocking process that produces incremental increases in the activity of high-affinity binders with each signal pulse. In principle, these mostly non-linear effects could control cellular outcomes. An applications to p53 signaling is developed, with binding to most gene promoters identified as category (iii) responses. However, currently unexplained enhancement of high-affinity promoters such as CDKN1a (p21) by pulsed signaling could be an example of (i). In general, provision for all possible scenarios is required in the design of mathematical models incorporating pulsed Hill-type signaling as some aspect.


S1. Pulsed Hill-model: variable and parameter definitions
where   S denotes the (time dependent) signal-molecule concentration, 1 k , 2 k are the binding and dissociation rate constant, respectively, and the Hill coefficient indicates the number of signal molecules that must bind to the receptor. The signal-molecule concentration is controlled externally by the signal encoder and is not affected by reactions with the receptor as its concentration is assumed to be very low (as would be the case for binding to DNA). For pulsed signaling, we assume that the signal-molecule concentration takes on the square-wave oscillation ( 1) S 1,2,..... 0 ( 1 ) as illustrated in Figure 2, where A is the pulse amplitude, T is the pulse period,  is the pulse duration, and / T    is the duty cycle. For sustained signaling, 1   and the signal-molecule concentration is taken to jump from zero at t = 0 to a constant value of   S A  . Equation (5) is then easily solved to yield the time evolution of the binding probability for t > 0: is the dissociation constant of RS n . For pulsed signaling, analytical solutions are known for square-wave signaling 2 and for sinusoidal signaling 3 ) that include also adaptions for downstream product production and delays 4,5 . For square-wave signaling, the results are best represented by considering changes in binding that occurs during each individual period of the signaling. Introducing ( 1) i t i T     as the time elapsed since the beginning of the i-th pulse, the binding change during this pulse can be expressed as a rising component appropriate for the time interval 0 i     during which the signal-molecule concentration is high,   (7), the binding probability i P averaged over the i-th period can be determined to be 2 1 2 1 2 At long times, the average value of the binding for pulsed signaling therefore becomes For fast pulsing (Eqn. (9), the oscillation period is much less than the dissociation time constant so    and the probability of association at long times becomes simply These equations are the same form as those deduced for calcium signaling 2 .
Supplementary Figure S1 | Promoter binding probabilities P(t) (t in h) shown for n = 4 over the range of 4 -64 nM in dissociation constants K A , for various values of A (in nM) and k 1 (in 10 -6 nM -4 h -1 ), rows, and for various duty cycles  ( = 1 indicates sustained signaling), columns. The pulsing period is T = 6 h. The dashed lines indicate, when feasible, the maximum values of K A satisfying inequalities Eqn. 9 (brown, fast pulsing limit), Eqn. 22 (grey, pulsing slows rise time), and Eqn. 24 (magenta, clocking such that binding increases with each pulse). Marked regimes are: a-K-independent initial binding, b-changeover, c-asymptotic regime, d-slow pulsing, e-competitive pulsing, f-fast pulsing, g-graded activation per pulse.
Supplementary Figure S2a. Values for n = 2 of k 2 over parameter space of Figure 3, where the scale shows log 10 (k 2 / h) (i.e., there is a seven-order change in magnitude in k 2 from 10 -4 h -1 (2.810 -8 Hz) to 1000 h -1 (0.28 Hz). A range of 1 -64 nM in dissociation constants K A is shown for various values of A (in nM) and k 1 (in 10 -3 nM -2 h -1 ), with the pulsing period taken to be T = 6 h.

S4. The relationships between the steady mean binding probability, the average signalmolecule concentration, and the duty cycle
From. Eqn.
(1), the average signal-molecule concentration is Using this relationship and Eqn. (S5), Eqn. (S6) can be rewritten to express the average binding probability as is the apparent dissociation constant for pulsed signaling. Equation (14) shows that the average binding for pulsed signaling is reduced from that for signaling sustained at signal-molecule concentrations equal to the maximum amplitude A of the pulsed signal. However, this binding is much more than that obtained using signaling sustained at the average signal molecule concentration in the pulsed experiments. This occurs because n signal molecules must bind in order to activate gene expression, making periods of relatively high concentration very much more effective if n > 1. When n = 4, this non-linear effect becomes highly pronounced. An illustrative example of the principle can be seen by taking in arbitrary units A = 1, K = 0.01, and  = 0.1. First, we consider pulsed signaling which, from Eqn. (14), leads to pulsed P = 0.9. Then we consider sustained signaling at either the peak signal-molecule concentration A = 1, or else the average signal-molecule concentration A = 0.1: the peak concentration yields sus P = 1.0 whereas the average concentration yields sus P = 0.01. Hence pulsed signaling increases the average binding hundredfold compared to sustained signaling at the low average signal-molecule concentration. However, if the same calculations are repeated using n = 1, then Eqn. (14) indicates that for pulsed signaling P = 0.01, the same as for sustained signaling at the average signal-molecule concentration. These results are captured in Eqn. (16) which tells that pulsed binding reduces the apparent dissociation constant by the factor  (n-1)/n .
Another way of considering these results is through consideration of pulsed P for changes in S : which shows signal pulses enhance binding with the receptor at low signal-molecule levels. A similar mechanism was found that calcium oscillations increase calcium sensitivity of gene transcription at low levels of stimulation 6 .

S5. Times to reach steady-state binding driven by pulsed and sustained signaling
For sustained signaling, the relaxation time to reach the steady binding probability is given simply from Eqn. (S1) as which can be expanded in a Taylor expansion in the fast-pulsing limit 2 0 k T  for fixed duty cycle  to give 2 2 1 1 ( Comparing Eqns. (6) and (7), it is clear that pulsed sus    as 1   , but ignoring the higher order terms in Eqn. (S12), steady state is achieved much slower ( pulsed sus    ) for pulsed signaling when the pulse frequency is high, i.e., when This period may be less than or greater than that required for the fast-pulsing approximation used in the above derivation, so always the most stringent condition needs to be considered.

S6. The relationship between binding probability and number of pulses during the initial stages of relaxation
From Eqn. (6), the ratio of the binding probability at the end of the i-th pulse to that at the end of the previous pulse is given by where, from Eqn. (7), analogous to the equations describing Ca 2+ signaling 7,8 . Let 1    , then from Eqn. (S15) 0 whenever allowing a Taylor expression of Eqn. (S16) yielding Hence the binding probability under the i-th pulse steadily increases whenever Eqn. (24) holds. Using Eqn. (3), this condition can also be written as 2 1 1 n k T k A T    (S19) and so is stricter than the limit for fast pulsing, Eqn. (9).

S7. Cumulative signal molecule exposure and cumulative binding
There is much interest in the "cumulative signal" c S and associated "cumulative binding probability" c P associated with pulsed signaling. Cumulative signal is defined as the total time exposure of the system to signal molecules after reaction time t : gives the cumulative binding. Of concern is the ratio of cumulative binding from sustained signaling to that from pulsed signaling given the same amount of cumulative signal exposure in each case. This involves comparing the cumulative binding at different times t for sustained signaling and t for pulsed signaling that produce the same cumulative signal-molecule level, which from Eqns. (5) and (S20) is at the end of each pulse. The solution of this equation is simply t t     , (S23) which states nothing more than the relative rates at which pulsed and sustained signaling delivers the signal molecule. The ratio of the cumulative bindings therefore becomes This result for fast pulsing approaches  for high-affinity reactions and 1 for low-affinity reactions. Pulsing therefore always increases somewhat the cumulative binding at the same level of cumulative signal molecule. In Supplementary Figure S4, the ratio of the cumulative binding is shown as a function of the ratio of cumulative signal molecule from pulsed signaling to that manifested in the simulations at t = 48 h, its characteristic features defined by Eqn. (18). Figure S4a. The ratio of the cumulative binding for n = 2 from sustained signaling to that from pulsed signaling is shown for the parameter space of Figure 3: a range of 1 -64 nM in dissociation constants K A , for various values of A (in nM) and k 1 (in 10 -3 nM -2 h -1 ), with the pulsing period taken to be T = 6 h. The brown dashed lines indicate the crossover between fast and slow pulsing, Eqn. (9). Figure S4b. The ratio of the cumulative binding for n = 4 from sustained signaling to that from pulsed signaling is shown for the parameter space of Supplementary Figure S1: a range of 4 -64 nM in dissociation constants K A , for various values of A (in nM) and k 1 (in 10 -6 nM -4 h -1 ), with the pulsing period taken to be T = 6 h. The brown dashed lines indicate the crossover between fast and slow pulsing, Eqn. (9).

S8. Some other ways in which Hill-type signaling can manifest when 4 signal molecules must bind to the receptor a la p53 binding.
In the main text, the equations are presented showing how binding of p53 dimers to DNA can be interpreted using a simple Hill equation, focusing on the limiting situations of weak and strong dimerization. Alternatively, if all four p53 molecules pre-associate into a tetramer that binds as a single unit to DNA, then the reactions can be described most simply as     respectively. In the strong binding limit, the first p53 binds immediately but inhibits subsequent binding, giving a Hill coefficient of 3, whereas in the weak binding limit the first p53 is difficult to bind but then the remaining three bind cooperatively, generating a Hill coefficient of 4.
Expanding on this solution, the sequential binding process can be represented generally as