Multiplexing Biochemical Signals

In this paper we show that living cells can multiplex biochemical signals, i.e. transmit multiple signals through the same signaling pathway simultaneously, and yet respond to them very specifically. We demonstrate how two binary input signals can be encoded in the concentration of a common signaling protein, which is then decoded such that each of the two output signals provides reliable information about one corresponding input. Under biologically relevant conditions the network can reach the maximum amount of information that can be transmitted, which is 2 bits.

The question of how cells can transduce multiple signals via pathways that share components is a key question in biology, since sharing components may lead to unwanted crosstalk between the different signals: from the perspective of one signal, the presence of additional signals constitutes noise. In recent years, several mechanisms for ensuring signaling specificity have been proposed. One is spatial insulation, where the shared components are incorporated into distinct macromolecular complexes on scaffold proteins [1]. Other proposals are based on the temporal dynamics of the system, such as cross-pathway inhibition [3][4][5] and kinetic insulation [6]. However, these studies only considered scenarios in which the system is stimulated with one signal at the time. Rensing and Ruoff studied what happens when two or three MAPK pathways that share components are stimulated simultaneously [7], but found that one pathway tends to dominate the response, suggesting that multiple messages cannot be transmitted simultaneously. Here we demonstrate that cells can truly multiplex signals: we show that they can transmit at least two signals simultaneously through a common pathway, and yet respond specifically to each of them.
We first have to understand how multiple signals can be encoded in the dynamics of a signaling pathway. Cells employ a number of coding strategies for transducing signals. One is to encode stimuli in the temporal dynamics, such as the duration [2] or frequency [8], of an intracellular signal. In principle, these coding strategies could be used to multiplex signals. Here, we consider what is arguably the simplest and most generic coding strategy cells could choose, namely one in which the signals are encoded in the concentrations of the signaling proteins. We will call this strategy AM multiplexing.
We will consider the biochemical network shown in Fig. 1A. It consists of N input species S 1 , . . . , S N with copy numbers S 1 , . . . , S N , a signal transduction pathway V consisting of M species V 1 , . . . , V M , and N output species X 1 , . . . , X N . The copy number of each input species S i can be in one of K states, s i = 0, . . . , K − 1, which are labelled in order of increasing copy number, S The input pattern is denoted by the vector s = (s 1 , . . . , s N ). Similarly, the copy number of each output species X i can be in one of L states x i = 0, . . . , L − 1 ordered by increasing copy number X i , and the output pattern is denoted by the vector x = (x 0 , . . . , x N ). A necessary condition for multiplexing is that the state space of V is large enough that it is possible to encode the total number of input patterns, We imagine that the N input signals are independent, and that the signal transduction network V replaces N independent signaling pathways. We therefore require that X i should provide reliable information about the state s i , but not necessarily about s j =i ; the N different input signals s simply have to be transduced to x, not necessarily integrated. In general, however, the state x i will be a function of the states of all the input species: x i = f (s). This reflects the fact that inevitably there is cross-talk between the different signals because they are transmitted via the same pathway. However, this crosstalk is not detrimental as long as it does not compromise the cell's ability to infer from x i what s i was.
Another key point is that while the precise mapping from s to x may not be critical for the amount of information transmitted per se, this is likely to be important for whether or not this information can be exploited. Let's imagine that the system contains three input species, say three sugars, and that each of these can be in one of only two states, s i = 0 or 1, corresponding to the absence or presence of the sugar; let's further assume that X i is an enzyme needed to consume sugar S i . With 8 input patterns X i can, in the absence of noise, take 8 values, identified as states x i = 0, . . . , 7. Now, it seems natural to demand that when the sugar S i is absent (s i = 0), the copy number of enzyme X i is low, while when S i is present, the copy number of X i is high; this means that the four lowest levels of X i (x i = 0, 1, 2, 3) should correspond to s i = 0, while the four highest levels of X i should correspond to s i = 1. We therefore require that the mapping from s to x is such that the output states {x i } corresponding to input s i = j are grouped into sets that are contiguous and either increase or decrease monotonically with j, for each signal i. This leads to a monotonic input-output relation between S i and X i for each i. We call this requirement the multiplexing requirement.
In the rest of the manuscript, we make these ideas concrete for a network in steady state with two input species, S 1 and S 2 , each of which has either a low (s i = 0) or a high concentration (s i = 1). We take a signaling pathway V consisting of only one species, V. Multiplexing requires that, in the absence of noise, the four input patterns s can be mapped onto four distinct states of V , v = 0, . . . , 3, again labelled in order of increasing copy number. These four levels of V lead to four states for each of the two output species X 1 and X 2 (Fig. 1B). As explained above, we require that we can group these four states into two sets, called LOW and HIGH, such that the LOW set, containing x i = 0, 1, corresponds to s i = 0 and the HIGH set, containing x i = 2, 3, corresponds to s i = 1 (or vice versa, leading to an inverse input-output relation). To elucidate which mechanisms make it possible to multiplex S 1 and S 2 , we note that there exists different ways of mapping s to v, but, as we will explain shortly, not all of these mappings can be decoded into x in a manner that satisfies the multiplexing requirement. We therefore first address the question which combinations of mapping from s to v and decoding from v to x fulfill the multiplexing requirement, and then we will discuss what encoding mechanisms actually allow for the required mapping from s to v.
Due to the symmetry in the problem, there are 3 unique ways of mapping the four input patterns s to v (Fig. 1C). To determine whether there exists a scheme for decoding the signals from v to x that satisfies the multiplexing requirement, we examine for each mapping all possible network topologies between V, X 1 and X 2 , except those that involve autoregulation or mutual repression/activation since these may lead to bistability. In particular, we allow not only for activation and repression of X 1 and X 2 by V, but also for activation and repression of X 2 by X 1 , leading to feedforward loops, a common motif in signal transduction pathways and gene networks [9]. In the deterministic mean-field limit the steady-state values of X 1 and X 2 are thus given by where k is the maximum activation/production rate, µ is the degradation/deactivation rate, and each regulation function is either an activating or repressing Hill function, f (V ; K, n) = V n /(V n + K n ) or f (V ; K, n) = K n /(V n + K n ). The multiplication in Eq. 2 indicates that we assume that at X 2 , X 1 and V are integrated according to AND logic [9]. We performed extensive sampling of the space of parameters k 1 , k 2 , K α , n α , K β , n β , K γ , n γ for each of the mappings in Fig. 1C.
Only for mapping C do we find decoding schemes that satisfy the multiplexing requirement [16]. Interestingly, all valid decoding networks are incoherent feedforward loops [9]. Figure 2 illustrates the principle for one such motif. Panel B shows for each of the four input patterns s the copy number V together with the threshold copy numbers, K α , K β and K γ , while panels C and D show X 1 and X 2 respectively as a function of V . X 1 (V ) is a simple activation curve with activation threshold K α . In contrast, X 2 (V ) starts low and rises around K β , but then decreases again due to repression by X 1 . This nonmonotonicity, which is a result of the incoherent character of the feedforward loop, is critical, since this makes The key parameters are the activation/repression thresholds K, since they determine where in the state space of V the outputs switch between high and low levels. The precise values of k and n are of less importance, although n should not become so large that X 1 (V ) becomes Boolean: it is critical that X 1 , which needs to be activated by V around K α to transmit S 1 , is not fully activated at K α : to multiplex S 2 , X 1 should reach the threshold K γ for repressing X 2 only when V has become significantly larger than K α . Indeed, if X 1 can only take two states, then only three states of V could be decoded, and not the required four. AM multiplexing thus relies on the fact that signals can be encoded over a range of concentrations.
We can now also understand why mappings A and B are difficult to decode: they would require an inputoutput relation between X 2 and V that rises more than once. This is difficult to achieve in a feedforward loop without mutual repression or activation.
The above analysis shows that it is possible to decode multiple signals simultaneously, provided that the input s can be encoded in v according to mapping C. The next question is how these mappings, which correspond to particular input-output relations V (S 1 , S 2 ), can be generated. Experiments [10] and modelling [11,12] have shown that transcriptional regulation can be very sophisticated, allowing for complex logical operations [12]. We indeed find that a simple scheme for transcriptional regulation based on the mechanism of 'regulated recruitment' [11] can generate the required input-output relation V (S 1 , S 2 ), where S 1 and S 2 are now transcription factors that regulate the expression of the protein V. In this scheme, S 1 and S 2 independently activate gene ex- 3: (a) An input-output relation V (S1, S2) consistent with mapping C (Fig. 1C); symbols , , •, correspond to states in Fig. 2. (b) Architecture of the promoter. (c) A feedforward loop that can generate mapping C.
pression by binding next to the core promoter, thus recruiting the RNA polymerase (RNAp), while S 1 and S 2 together repress gene expression by cooperative binding to the core promoter, thereby blocking the binding of RNAp (see Fig. 3). This yields where β is the maximum expression rate and µ is the degradation rate of V, q p = c p /K p is the concentration of RNAp c p scaled with its dissociation constant K p , i are the dissociation constants for the binding of S i to the promoter sites where the RNAp is recruited or blocked, respectively; ω and ω ′ are factors reflecting cooperative interactions between the respective molecules [11,12]. We thus conclude that gene regulation networks have the capacity to multiplex signals.
While it is clear that signaling pathways often share common components [1,2], the logic of signal integration in these pathways has been characterized in much less detail than for gene regulatory networks. It is conceivable that the desired input-output function V (S 1 , S 2 ) could be implemented at the level of a single protein V, using competitive and/or cooperative binding between the three molecules S 1 , S 2 , V. Alternatively, the required encoding could also be implemented at a higher level of network interactions. For instance, a network in which S 1 and S 2 regulate V via two additional components, Q 1 and Q 2 , in an incoherent feedforward loop (Fig. 3C), could achieve the required encoding V (S 1 , S 2 ). In essence, the feedforward loop between Q 1 , Q 2 and V can be used to control the ordering of V in the encoding process, just as the feedforward loop between V, X 1 and X 2 can be used to regulate the ordering of X 2 in the decoding step. Since feedforward loops are common motifs in signal transduction pathways [9], we argue that multiplexing can also be implemented in these networks.
The analysis above shows that in principle biochemical networks can multiplex signals in the mean-field, deterministic limit. However, there remains the question of whether signals can be multiplexed reliably in the presence of inevitable biochemical noise. To address this, we 100 1000 10000 FIG. 4: The transmitted information I as a function of V max for four different values of X max . I can reach the maximum value of 2 bits provided V max and X max are large enough.
estimate a lower bound on the information about two binary signals S 1 and S 2 that are transmitted through the network studied above (Eqs. [1][2][3]. We define the total information I ≡ I(S 1 , X 1 ) + I(S 2 , X 2 ) as the sum of the mutual information for each of the individual signals, [13], where p(s i ) and p(x i ) are respectively the probabilities of S i being in state s i and X i being in state x i , and p(x i , s i ) is the joint probability of input s i and output x i . Note that in the presence of noise X i is not limited to 4 states but can in principle take any value. This definition of I makes it straightforward to directly compare the performance of this network with that of two independent pathways. If each of the two input states for each S i is equally likely then the maximum value of I(S i , X i ) is 1 bit for each transmitted signal i; the maximum value of I is thus 2 bits.
To maximize the lower bound on I we optimize the network parameters using a simulated-annealing algorithm; we have verified that the final results are robust by varying the initial conditions, and by also using an evolutionary algorithm. We fix the degradation rate of all proteins to be µ = 1hr −1 and vary n α , n β and n γ between 1 and 4. Values of k 1 , k 2 are set such that the maximum mean value of each X i is X max ; similarly, β is set such that the maximum mean value of V is V max . X max and V max are varied systematically (see Fig. 4). The threshold parameters K α , K β and K γ are varied over the range [0, V max ] or [0, X max ] as appropriate. We vary q p , q i from 10 −2 to 10 2 and ω, ω ′ between 1 and 10. For each parameter set we compute p(x i , s i ) using the linear-noise approximation [14]. Its accuracy was verified by performing Gillespie simulations of the optimized networks [15]. Figure 4 shows that below a threshold copy number V max c ≈ 50 the total information is low regardless of X max because four distinct states of V cannot be generated. Above V max c , for large X max the information I reaches 2 bits, the maximum information about the two signals S 1 and S 2 that could be transmitted via two inde-pendent channels. For lower values of X max , I saturates at a value lower than 2 bits, limited by the intrinsic noise in the production and decay of X i . Importantly, I reaches 2 bits for V max ≈ X max ≈ 500, which is well within the range of typical protein copy numbers inside living cells. This shows that biochemical networks can multiplex two signals reliably in the presence of biochemical noise under biologically relevant conditions.
In summary, our results suggest that cells can transmit at least two binary signals through one and the same pathway, and yet respond specifically and reliably to each of them. The proposed mechanism for biochemical multiplexing is based on swapping the order of states during the encoding and decoding steps using incoherent feedforward loops. It is clear that the principle is generic, and could be implemented in signal transduction pathways and gene networks -indeed incoherent feedforward loops are commonly found in these networks [9]. Our predictions could be tested experimentally by simultaneously stimulating two MAPK pathways that share components [1], although perhaps a more controlled experiment would be one using synthetic gene networks. In future work, we will address how more than two input signals can be transduced simultaneously, and how cells can multiplex signals by encoding them into the temporal dynamics of the signaling pathway.
We thank Tom Shimizu for a critical reading of the manuscript. This work is supported by FOM/NWO.