Rectified directional sensing in long-range cell migration

How spatial and temporal information are integrated to determine the direction of cell migration remains poorly understood. Here, by precise microfluidics emulation of dynamic chemoattractant waves, we demonstrate that, in Dictyostelium, directional movement as well as activation of small guanosine triphosphatase Ras at the leading edge is suppressed when the chemoattractant concentration is decreasing over time. This ‘rectification’ of directional sensing occurs only at an intermediate range of wave speed and does not require phosphoinositide-3-kinase or F-actin. From modelling analysis, we show that rectification arises naturally in a single-layered incoherent feedforward circuit with zero-order ultrasensitivity. The required stimulus time-window predicts ~5 s transient for directional sensing response close to Ras activation and inhibitor diffusion typical for protein in the cytosol. We suggest that the ability of Dictyostelium cells to move only in the wavefront is closely associated with rectification of adaptive response combined with local activation and global inhibition.


The basis of rectification in the ultrasensitive LEGI model
In order for the response R to adapt to spatially-uniform stimuli (equation (1)), the level of R at the steady state should not depend on the value of S. Here, adaptation refers to the recovery of the response to the pre-stimulus level under persistent stimulation ( Supplementary Fig. 5a). Under constant and spatially-uniform stimuli S(x,t) = S 0 , one obtains By substituting dR/dt = 0 in the third equation of equation (1), we see that the fixed point R = R 0 is determined . When the adaptation response to the sustained stimulus is nearly perfect, θ a and θ i should be negligibly small. In this case, we see that  Fig. 5g and j). As described in the main text, the extended model (equation (1)- (3)) is ultrasensitive to the activator-inhibitor ratio (A/I) ( Fig. 6c and Supplementary Fig. 5k). This is a natural outcome of the near-zero-order kinetics of the inhibitory reaction G(R) that outcompetes the activation reaction F(R) when K I is small relative to R tot 3 . As a consequence, the stationary value of R (= R 0 ) becomes insensitive to changes in the stationary value of Q (= Q 0 ) for small Q 0 ( Supplementary   Fig. 5k). This is in sharp contrast to the basic LEGI model where changes in Q 0 are proportionately transferred to the response in R 0 (Fig. 6c).
The asymmetric R(t) response to changes in Q(t) can be understood from the trajectories in the Q-R plane ( Supplementary Fig. 5b, e, and h). The response R(t) moves along the Q 0 -R 0 curve after an abrupt change in Q(t) induced by the change in S. In the basic LEGI model, the transients in R(t) from the resting state are almost symmetric in magnitude for increasing and decreasing Q(t) (Supplementary Fig. 5b; orange for increasing and cyan for decreasing stimuli) due to linearity in its response function (Q 0 -R 0 curve) (broken black line in Supplementary Fig. 5b). The symmetry is evident in the transient changes of Q×F(R) in response to S (Supplementary Fig. 5c).  Fig. 5h and i). To summarize, the 'rectified' response is based on near zero change in R to decrease in ratio Q = A/I and that is due to the 'hemi-' zero-order sensitivity (i.e. only the reverse reaction of the push-pull network operates near the zero-order kinetics).

Other possible routes for rectification
The observed Ras response in cells exposed to traveling wave stimulus of cAMP predicts a rectification mechanism that filters out temporally decreasing signal stimuli. Earlier works have experimentally tested and demonstrated various aspects of the LEGI-framework such adaptive temporal and spatial sensing 4 and its response to complex stimuli 1,5 , local production of the inhibitor 6 and incoherent feed-forward type network topology 7 . Our analysis suggests that rectification is separable from downstream amplification 1 and/or excitable 8 circuit thus arises at or very close to the level of LEGI -like circuitry. We introduced ultra-sensitive LEGI to study the effect of strong nonlinearity independent of amplification. Nevertheless, chemotactic signaling pathways are highly redundant, and many details await future experimental analysis. Here, to survey other possible implementation of rectification, we analyzed earlier models to study how the required filtering characteristic as clarified by the ultra-sensitive LEGI model can be embedded. Levchenko and Iglesias (2002) 9 proposed a circuit that integrates adaptation and signal amplification in a single layer of signal transduction (Supplementary Fig. 8a). The activator 'A' and the inhibitor 'I' are governed by equation (1), whereas the output 'R' (R) and its precursor 'R inactive ' (R in ) obey the following rate equations.
Here, the output R is further amplified because 'R inactive ' is replenished by 'A' (the first term in the RHS of the second equation), hence the output 'R' is no longer mass conserved. As discussed in Ref 9 , for a given Q = A/I, stationary value of R follows R = (σk A /ρk I )×Q 2 ( Supplementary Fig. 8b), meaning that amplification is in the second order thus nonlinearity is weak to support rectified response. As expected from the R-Q curve, both the positive and negative response of R occurs to spatially uniform change of the stimulus S ( Supplementary Fig. 8c). In line with this, the directional response appears not only in the wavefront but in the waveback ( Supplementary Fig. 8d). For comparison, the parameter values associated with A and I are the same in the basic and ultrasensitive LEGI (Supplementary Table 1). Other parameters are k A = 3.0, k I = 1.6, σ = 3.0, and ρ = 4.5.
The model has an architecture similar to the ultra-sensitive LEGI; i.e. it introduces nonlinearity to the basic LEGI scheme in the same layer of signaling cascade, however due to its weak nonlinearity it does not support rectification. The other nonlinear scheme proposed by Levchenko and Iglesias (2002) 9 integrates adaptation and amplification by connecting in sequence the basic LEGI module and a positive feedback signal-amplification module (Supplementary Fig. 8e). The kinetics of 'A', 'I', and 'R' obey the basic LEGI model with F(R) = k A (R tot -R) and G(R) = k I R (equation (1)). Downstream of the basic LEGI circuit lies another module governed by Here, the final output 'T' is activated by 'R' from the inactive state T in . Furthermore, T positively regulates production of T in , thereby providing a positive feedback loop that amplifies T (Supplementary Fig. 8e). The stationary response curve of T for a given When σ is small and k σ is large, the relationship between Q and T realizes the characteristic rectification curve as shown in Supplementary Figure 8f (k T = 1.1, K T = 7.5, k ρ = 1.5, σ = 0.03, k σ = 50, γ σ = 9.5) similar to that for the ultrasensitive LEGI (see Fig. 6c and Supplementary Fig. 5k for comparison).
Accordingly, the model simulation shows the rectified response to spatially uniform stimulation ( Supplementary Fig. 8g). To the wave stimulation, the response T was elevated at the side facing the higher concentration of S (T + ) only when S is rising in time ( Supplementary Fig. 8h). Although the model can implement rectification, due to additional layers of regulation, the timescale of final output T (which should correspond to Ras activity) may deviate from the timescale of A and I (dictated by i !! − a !! ) . As discussed in the main text, the timescale of temporal sensing estimated from directionality of cell migration suggests a close match with that of transient Ras activation. Based on the observations that localized PIP3 synthesis and the resulting PH-domain protein localization are strongly amplified with respect to the imposed gradient steepness, the amplified LEGI model 1 extends the basic LEGI circuit with an additional downstream amplification reaction (Supplementary Fig. 8i). The model scheme is similar to that of Levchenko-Iglesias Model B 9 (Supplementary Fig. 8e). The variables 'A', 'I' and 'R' follow the basic LEGI model with F(R) = k A (R tot -R) and G(R) = k I R (equation (1)).
Downstream of the basic LEGI module lies a switching module that amplifies R as follows For small K ρ , the stationary relationship between Amp and Q = A/I obeys the characteristic rectifying curve (Supplementary Fig. 8j; R tot = 2.0, k A = 3.0, k I = 1.6, Amp tot = 2.0, k Amp = 2.2, K Amp = 1.6, k ρ = 2.4, and K ρ = 0.02) similar to that shown for the ultrasensitive LEGI (see Fig. 6c and Supplementary Fig. 5k for comparison). Rectified response appeared to the uniform stimulation ( Supplementary Fig. 8k). The variable Amp behaves in the rectifying manner in the wave stimulation as shown in Supplementary Figure 8l. Note that relationship between R and Amp at the stationary state is similar to that between Q and R in the ultra-sensitive LEGI model. In the amplified LEGI model, Supplementary Fig. 8j, dotted line), indicating that the rectification curve at the level of the final output 'Amp' (Supplementary Fig. 8j, solid line) is stretched out much towards higher Q. Thus, compared to the ultra-sensitive LEGI model, the response is less sensitive, and the dynamic range is more restricted. As a result, the amplified LEGI model requires careful parameter tuning to achieve rectification.  Fig. 8m). The governing equations are 'B' is assumed to diffuse throughout the cell, thus after its membrane translocation, 'B m ' effectively acts as the global inhibitor. The model follows the basic scheme of adaptation supported by an incoherent feedforward circuit as in LEGI; 'A' and 'B' are both activated by signal 'S'. A is subsequently suppressed by B with some delay incurred by its membrane translocation. The original analysis 2 studied a case k A = k B and A AB A B ≫ 1. In the following analysis, to satisfy the latter condition, we assume that k AB is large without loss of generality. The stationary value of A for fixed S and B is given by In the balanced inactivation model, plotting A as a function of q ≡ k A S − k M B, serves the equivalent of the R-Q curve in the LEGI models (Supplementary Fig. 8n; k A = k B = 1.5, γ A = 0.1, γ B = 0.02, k AB = 1000, and k M = 0.75). Because q = k A S − k M B is positive for increasing stimuli and negative for decreasing stimuli, response in A appears only for temporally increasing signal input and not for temporally decreasing input ( Supplementary Fig. 8o).
Supplementary Figure 8p shows results of the numerical simulations of the model in the wave stimulation. In the simulation, B is assumed to diffuse at D = 30 μm 2 sec -1 . The response A shows strong directional response only when S is increasing in time.
While these results indicate that the balanced-inactivation model 2 is capable of rectification ( Supplementary Fig. 8m-p), there were some disagreements with the experimental observations. For the spatially uniform concentrations of S that is decreasing in time, A always decline monotonically regardless of parameter values. Therefore, the model was not able to reproduce the pulsatile negative response to decreasing stimuli observed in weakly starved cells (Fig. 7b). Also, the response amplitude in the balanced-inactivation model shows strong dependence to the input level which is unlike the response observed for RBD localization.

Cell migration
By employing a phenomenological description of cell movement coupled to directional sensing, let us confirm how experimentally observed directionality of cell displacement could be brought about. Although vastly oversimplified, we shall assume the velocity of a cell, dx(t)/dt = v(t) correlates with the difference of the response R between both sides of the cell; namely, Here, μ defines the relaxation time of the cell motion and h(R) governs dependency of cell motion on R, for which we adopt a simple saturating function. For numerical simulations of equation (2) and (3) and Supplementary equations (1) and (2) (Fig. 1g). Moreover, as shown in Fig. 1g and Supplementary Fig. 6h, the total displacement at large transit time becomes negative, because the cells migrate in the waveback by following the trailing end. To be exact, relative speed between the cells and the signal propagating at the velocity V S is given by V S + v in the wavefront and V Sv in the waveback (Doppler shift), thus cells moving in the same direction as the signal wave spend longer time sensing the gradient than those moving against the wave ( Supplementary Fig. 6h, green).
When the wave is slow enough, i.e. in the stationary directional sensing scheme, this difference results in the negative net displacement (Supplementary Fig. 6h).