Unraveling Adaptation in Eukaryotic Pathways: Lessons from Protocells

Eukaryotic adaptation pathways operate within wide-ranging environmental conditions without stimulus saturation. Despite numerous differences in the adaptation mechanisms employed by bacteria and eukaryotes, all require energy consumption. Here, we present two minimal models showing that expenditure of energy by the cell is not essential for adaptation. Both models share important features with large eukaryotic cells: they employ small diffusible molecules and involve receptor subunits resembling highly conserved G-protein cascades. Analyzing the drawbacks of these models helps us understand the benefits of energy consumption, in terms of adjustability of response and adaptation times as well as separation of cell-external sensing and cell-internal signaling. Our work thus sheds new light on the evolution of adaptation mechanisms in complex systems.


1
Single-component model The main hypothesis of the work is that adaptation can be obtained even in the absence of energy consumption by the cell, i.e. through equilibration processes.
The idea of the main text is to build a model for a protocell in which the receptors distributed on the cell membrane are activated (change conformation) by the binding of an extracellular ligand and deactivated by the binding on its intracellular domain of the same type of ligand, which has diused through the membrane into the cell.
Figs. S1-S4 show additional adaptation time courses of the one-component model with A = 1/(1 + e N ∆f ) from Eq. (3), extended by the receptor-complex size N . In particular, the internal concentration, the single-receptor free-energy dierence and the receptor-complex activity are presented for dierent complex sizes (N =1, 4, 10) and dierent length of the receptors (r=0.5, 0.7, 0.9).

Adaptation time: simulations
We can dene the adaptation time as the time required for the activity to return to half of the displacement from steady-state level. From the time course of the simulations we can estimate the adaptation time from the activity prole, both for the response to a positive and a negative step in external concentration ∆c. (S1) with ∆c the dierence between the maximal external concentration c e and the minimal c 0 , and K 1 and K 2 the ligand-dissociation constants of the receptor. In particular, for the external domain of the receptor, K 1 corresponds to the dissociation constant for the o state and K 2 for the on state, while for the internal binding K 1 correspond to the on state and K 2 to the o state. In and notice that all the concentration time courses obtained for dierent c e and c 0 in response to positive and negative steps in external concentration collapse onto a single curve. This allows us to generically analyze the adaptation time (see Fig. S6). As a consequence, the time courses of the internal concentration c i are symmetrical in response to positive and negative steps in c e , and are proportional to the maximal external concentration c e (see Figs. S1-S4b).
Special case. The steady-state of the activity A ss is equal to 1/2 as can be seen in Figs. S1-S4. For the case in which the activity response saturates we can calculate the adaptation time as the time required for the activity to reach 1/4 in response to a positive step and 3/4 in response to a negative step.
Based on Eq. (3), the corresponding values for the free-energy dierence are ln 3 and − ln 3, respectively. From Eq.
(2) we estimate the corresponding internal Note for the rst equation we set c 0 = 0, and for the second c e = 0 (see Fig. S7).
Next, we would like to see the locations of these two points on the collapsed curve ∆c (t) /∆c 0 , in order to compare them and to nd out whether their position depends on particular parameters: We notice that the value of the rst point, ∆c (A = 1/4) /∆c 0 , increases with increasing external concentration c e . This means that under our assumptions the adaptation time of the response of the system to increasing positive steps in external concentration decreases (as seen in Fig. 5a). When, instead, we consider the response to increasing negative steps we notice that ∆c (A = 3/4) /∆c 0 decreases with increasing c e , so the adaptation time consequently increases. General case. We now take a more general perspective by considering a generic c 0 , as well as adding an external contribution ∆f 0 to the free-energy dierence. The free-energy dierence becomes ∆f = ln 1 + c e (t)/K 1 1 + c e (t)/K 2 + ln 1 + c i (t)/K 2 1 + c i (t)/K 1 + ∆f 0 , and the new steady-state value of the activity is given by    which is now asymmetric, i.e. A ss ≷ 1/2 for ∆f 0 ≶ 0. As before, assuming a saturating response, we can calculate the activity values A 1 and A 2 corresponding to the half-displacement from the adapted steady-state level in response to a positive and negative step in external concentration. The corresponding free-energy values are given by: where for the second equation c e has been substituted with c 0 . Calculating the concentration-dierence ratios on the collapsed curve it is possible to proceed with further considerations. In particular, for Eq. (S10): if c e c 0 , increasing ∆f 0 causes ∆c (A = A 1 ) /∆c 0 to decrease, thus leading to longer adaptation times. The dependence of the system on the external concentration is not aected (still decreases with increasing c e , as for the special case); if c e c 0 the assumption of a saturating response does not hold, so we cannot study this case with the above procedure.
For Eq. (S11): if c e c 0 , ∆c (A = A 2 ) /∆c 0 increases with increasing ∆f 0 , which means that the adaptation time decreases.
As for the special case, ∆c (A = A 2 ) /∆c 0 decreases with increasing c e , leading to longer adaptation times with increasing external concentrations.
if c e c 0 as before the saturation assumption is not satised.
In summary, in response to a positive step in external concentration, the adapta- Figs. S8-S11 show results of the simulations obtained in response to dierent steps in the outer concentration, for dierent complex sizes (N 2 =1, 4, 10) and lengths for the second signaling protein (r=0.5, 0.7, 0.9). Specically, outer concentrations range from 0 to 0.1mM (Fig. S8), from 0 to 0.5mM (Fig. S9), from 0 to 1mM (Fig. S10), and from 0 to 5mM (Fig. S11). Generally, panel a shows the concentrations of a and b secreted by the rst receptor, and panel b shows the ratio in concentrations of a and b perceived by the signaling protein after diusion. In panel c, the free-energy dierence ∆f 2 for the signaling protein is presented. On the right, nal signaling activities proles for the second transmembrane protein and dierent complex sizes are shown.

Adaptation time: simulations and analytical study
As done for the one-component model, it is possible to study the adaptation time and its dependence on the amplitude of the concentration step. Specically, Figs. S8-S11 show the activity of the signaling protein A 2 = 1/(1 + e N2∆f2 ) in response to stimulus c e (t), with N 2 the size of the signaling complex (cf. Eq. (7)). Concentration c r is the released concentration of a (and hence b) at the receptor.
For simplicity, we assume this released concentration to be constant during the stimulation, i.e. unaected by diusion. The results are summarized in Fig. S12, in which the adaptation time is shown as functions of the concentration step in c r (t) and the step in the free-energy dierence, given by where c r,min and c r,max represent, respectively, the released concentrations of a and b before and after the step in the activity of the receptor (Figs. S8-S11a).
Notice from Fig. S12 that the adaptation time is almost independent of the change in the external concentration in response to a positive step, while it increases with increasing concentration for a negative step. Also for this model, it is possible to obtain analytical understanding for this dependence. From Eq. (7) we can calculate the free-energy dierences corresponding to A 2 = 1/4 and A 2 = 3/4 to be ∆f 2 = ln 3 and ∆f 2 = − ln 3, respectively. Assuming that diusion of a is fast enough, relative to b, to be approximated with c r (t), we can calculate from Eq. (S14) Also in this case, we can consider the collapsed curves where c r (t) represents the current value of the released concentration and c r,0 the value before the step (thus corresponding to c r,min in the case of a positive step and to c r,max for a negative step). These curves are qualitatively similar to the curve presented in Fig Regarding the positive step response, the point corresponding to A = 1/4 on the collapsed curve does not depend on c r (under our assumptions), and consequently on the external concentration c e . This explains the (nearly) constant adaptation time in response to dierent positive steps of external concentration.
Note instead that c r,min corresponds to c e = 0 independent of the amplitude of the step. Thus the value of the point described by Eq. (S18) for the negative step decreases with increasing c r,max (and thus with increasing maximal external concentration c e ), and consequently the adaptation time increases. These results can be observed in Fig. S12.

Fold-change detection
Similar to the one-component model, the two-component model also satises FCD. Indeed, we can consider the change in free-energy dierence δf 2 of Eq. (S12) and, assuming the diusion of a much faster than that of b, approximate c a with the maximal value of the released concentration and c b with the minimal. The released concentration of a and b corresponds to the activity of the receptor A 1 rescaled by their proportionality factor (10 −2 ). The equation we obtain from this substitution is in which, for special case L D = K on , P = K on K off + 100K 2 on K off + 100K 2 on K off e σ 100K on K off + 100K 2 on e σ + K off , Q = K on K off + K on K off e σ K off + K on e σ .
As for the one-component model, we can nd values for c 0 with P c 0 Q, such that the conditions for FCD are satised (see Fig. S13 for an example).

Gradient sensing
As