Hans Van Ditmarsch

Davide Grossi

Andreas Herzig

Wiebe Van Der Hoek

Louwe B Kuijer

Parameters for Epistemic Gossip Problems

We introduce a framework that can model different kinds of epistemic gossip problems. We also formalize some parameters that distinguish the different types of gossip problem. Finally, we present a few results that show the effect of the parameters.

Introduction

The so-called gossip problem [START_REF] Tijdeman | On a telephone problem[END_REF][START_REF] Hajnal | A cure for the telephone disease[END_REF][START_REF] Hedetniemi | A survey of gossiping and broadcasting in communication networks[END_REF] can be described as follows: there are n agents and each of them knows a secret. In the beginning, each agent knows only their own secret. The agents then start making phone calls to each other. Whenever one agent calls another, they tell each other all the secrets they know-including their own secret as well as all secrets that they learned prior to the current call. The goal is to make sure that all agents learn all secrets, and to use as few calls as possible in the process. The gossip problem is of interest as an abstract puzzle, but it also has implications for more practical problems related to information gathering.

Originally, the gossip problem was studied mostly from a combinatorial and graph-theoretical point of view. See [START_REF] Hedetniemi | A survey of gossiping and broadcasting in communication networks[END_REF] for a survey of the combinatorial results about the gossip problem. For our current purpose, the most important combinatorial result is that, if n ≥ 4, the optimal solutions to the gossip problem require 2n -4 calls. More recently, people have started to study the gossip problem from a knowledge based or epistemic point of view [START_REF] Attamah | Knowledge and gossip[END_REF][START_REF] Attamah | The pleasure of gossip[END_REF][START_REF] Krzyzstof | Epistemic protocols for distributed gossiping[END_REF].

The difference between the two approaches lies in the perspective they take. In the combinatorial approach, we look at protocols through the eyes of an allknowing scheduler who tells the agents whom they have to call, and when they should do so. In the epistemic approach, we look at protocols through the eyes of the individual agents, who must make a decision about which call to make based on the information that they have available.

Here we follow the epistemic approach, so the agents base their decision whom to call on what they know. This, of course, makes it important to specify

The Formal Framework

The choice of framework requires a balance between generality and usefulness. The more general we make the framework, the more parameters we can model. But this generality comes at the cost of a more unwieldy and complicated system. In order to achieve the desired balance there are a few important things that we will treat as invariant truths, instead of parameters. Firstly, we model the gossip problem on complete graphs, instead of the problem on general graphs or the dynamic gossip problem [START_REF] Hans Van Ditmarsch | Dynamic gossip[END_REF]. This means that we assume that for every two agents a and b, it is always possible for a to call b. So a "knows b's phone number." Secondly, we assume that agents have perfect memory. This does not mean that once an agent knows ϕ it will always know ϕ in the future. It does mean that if a used to know ϕ, then it knows that it used to know ϕ. Finally, we assume that calls are made sequentially. These assumptions can be weakened by making small changes to the framework, but for reasons of simplicity we will not make these changes in this paper.

In order to reason about knowledge and protocols, we use a variant of Propositional Dynamic Logic [START_REF] Vaughan | Application of modal logic to programming[END_REF][START_REF] Michael | Propositional dynamic logic of regular programs[END_REF] that includes a few epistemic operators. Specifically, the language we use is defined as follows.

Definition 1. Let A be a finite set of agents and let P be a countable set of propositional variables such that for every a, b ∈ A there is an element k ab ∈ P. Furthermore, let C := {c ab | a, b ∈ A, a = b}. The sets L of formulas ϕ and Π of protocols π are given by the following normal forms:

ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | K a ϕ | Cϕ | [π]ϕ π ::= c ab | ϕ? | π π | π; π | π * | π -1
where p ∈ P, a, b ∈ A and c ab ∈ C.

We use ∧, →, ↔, Ka , π , , and in the usual way as abbreviations. Furthermore, we also define Eϕ := a∈A K a ϕ and Init := a =b ¬k ab ∧ a k aa as abbreviations. Finally, by convention we let the empty sequence of calls represent ?, so in the degenerate case n = 0 the sequence c a1b1 ; • • • ; c anbn represents ?.

We use c ab to represent the action "agent a calls agent b." We don't allow an agent to call itself, so C = {c a,b | a, b ∈ A : a = b}. Furthermore, if a, b ∈ A we use k ab ∈ P to represent "agent a knows agent b's secret." The reason we use k ab instead of, say, K a s b is that K a ϕ means that a knows that ϕ is true. Secrets are neither true nor false, so they cannot be known in the K a sense.

Note that while P is required to include k ab for every a, b ∈ A it also contains other propositional variables. These other variables can be used to encode relevant information. For example, if an agent bases its next call on a coin toss, we can use p ∈ P to represent the outcome of the toss.

The agents' goal is to have everyone know all secrets. We define the formula Goal as a representation of this goal in the object language, i.e. Goal := a,b k ab .

This language can be evaluated on Kripke models with two sets of relations: an indistinguishability relation R(a) for every agent a and an outcome relation O(c ab ) for every distinct a, b.

Definition 2. A model M is a tuple M = (S, R, O, V )
where S is a set of states, R : A → 2 S×S assigns to each agent an equivalence relation on S, O : C → 2 S×S assigns to each call a relation on S and V : P → 2 S is a valuation.

The language L is evaluated on these models in the usual way.

M, s |= p ⇔ s ∈ V (p), for p ∈ P M, s |= ¬ϕ ⇔ M, s |= ϕ M, s |= ϕ 1 ∨ ϕ 2 ⇔ M, s |= ϕ 1 or M, s |= ϕ 2 M, s |= K a ϕ ⇔ M, s |= ϕ for all s such that (s, s ) ∈ R(a) M, s |= Cϕ ⇔ M, s |= ϕ for all s such that (s, s ) ∈ a∈A R(a) * ,
where * indicates the reflexive transitive closure

M, s |= [π]ϕ ⇔ M, s |= ϕ for all s such that (s, s ) ∈ O(π)
and

(s, s ) ∈ O(ϕ?) ⇔ s = s and M, s |= ϕ (s, s ) ∈ O(π 1 π 2 ) ⇔ (s, s ) ∈ O(π 1 ) or (s, s ) ∈ O(π 2 ) (s, s ) ∈ O(π 1 ; π 2 )
⇔ there is an s such that (s, s ) ∈ O(π 1 ) and (s , s

) ∈ O(π 2 ) (s, s ) ∈ O(π * ) ⇔ (s, s ) ∈ O(π) * , where * indicates the reflexive transitive closure (s, s ) ∈ O(π -1 ) ⇔ (s , s) ∈ O(π).
We write M |= ϕ if M, s |= ϕ for all s ∈ S.

We can coherently evaluate L on all models. In general, models do not accurately represent the gossip problem, however. For example, there is nothing in the definition of a model to guarantee that every agent starts out knowing its own secret, or that agents exchange secrets during calls. We therefore define the subclass of gossip models, that model the gossip problem. But before defining gossip models, we first need a few auxiliary definitions.

; • • • ; c a k b k of calls and s 1 , • • • , s k+1 of states such that • s k+1 = s and (s m , s m+1 ) ∈ O(c ambm ) for all 1 ≤ m ≤ k, • s 1 is an initial state then p(s) = c a1b1 ; • • • ; c a k b k is the past of s and h(s) = s 1 , • • • , s k+1 is the his- tory of s.
If there is no such unique sequence, then p(s) and h(s) are undefined.

If p(s) and s(s) exist, then the state s 1 is the origin state of s, and k is the length of p(s).

Intuitively, p(s) is the sequence of calls that have already taken place in s, and h(s) is the sequence of states that preceded s. Unfortunately, in general models such sequences are not guaranteed to be well-defined, because the relations O(c ab ) may be cyclical and because a world may have multiple predecessors. Gossip models are defined in such a way that every state s does have a (by definition unique) past p(s) and history h(s).

∈ A. Suppose h(s) = s 1 ; • • • ; s k , s k+1 and h(t) = t 1 ; • • • ; t m , t m+1 . Then h(s) and h(t) are a-similar, denoted h(s) ∼ =a h(t), if there are functions f 1 : {1, • • • , k + 1} → {1, • • • , m + 1} and f 2 : {1, • • • , k + 1} → {1, • • • , m + 1} such that 1. f 1 and f 2 are increasing, 2. for every 1 ≤ i ≤ k + 1, (s i , t f1(i) ) ∈ R(a), 3. for every 1 ≤ i ≤ m + 1, (t i , s f1(i) ) ∈ R(a),
If h(s) and h(t) exist, then h(s) ∼ =a h(t) exactly if a could use its memory to distinguish between s and t. For example, agent a could reason "I used to be in state s i , or some state indistinguishable from s i . But there is no state t j in the history of t that is indistinguishable from s i . So I cannot be in t."

Definition 8. A gossip model M is a model M = (S, R, O, V ) such that: 1. if s is an initial state then for all a, b ∈ A we have s ∈ V (k ab ) if and only if a = b; 2. if (s, s ) ∈ O(c a1a2 ), then s ∈ V (k a3a4 ) if and only if s ∈ V (k a3a4 ) or (a 3 ∈ {a 1 , a 2 } and (s ∈ V (k a1a4 ) or s ∈ V (k a2a4 )));
3. if s is a non-initial state, then it has exactly one predecessor s and there is exactly one pair (a, b) such that (s, s ) ∈ O(c ab );

4. for every s and every a = b there is exactly one s such that (s, s ) ∈ O(c ab );

5. if (s, s ) ∈ R(a) then s ∈ V (k ab ) if and only if s ∈ V (k ab ); 6. if (s, s ) ∈ R(a) then p(s)| a = p(s )| a ; 7. if (s, s ) ∈ R(a), then h(s) ∼ =a h(s ).
Given a class M of gossip models we write

M |= ϕ if M |= ϕ for all M ∈ M.
If M is the class of all gossip models we write |= ϕ for M |= ϕ.

Definition 8 is quite complicated, so let us explain which aspect of the gossip problem is encoded in which requirement in the definition.

In the gossip problem every agent starts out knowing its own secret, and only its own secret. This is encoded in condition 1. Agents only learn secrets by being involved in calls, and in a call they exchange all secrets they know. This is encoded in condition 2.

We assume that any uncertainty in a gossip setting is epistemic in nature. So in a given state s it is fully determined which calls have taken place in the past, and given an action c ab it is determined what the outcome of performing c ab in s would be. 1 In Definition 8 this is encoded as conditions 3 and 4.

The last three conditions constrain the indistinguishability relations of the agents. Recall that we modeled a knowing b's secret as the propositional variable k ab . This means that introspection about the knowledge of secrets is not guaranteed by the fact that R(a) is an equivalence relation. We use condition 5 to guarantee that agents are introspective about which secrets they know.

Condition 6 demands that the agents remember the calls that they themselves participated in. Finally, condition 7 makes sure that if an agent a is uncertain about whether it is in s 1 or in s 2 , then the histories of s 1 and s 2 are consistent, at least as far as a is concerned. So a cannot dispel its uncertainty simply by checking which of the two histories is consistent with its memory.

At this point, we should note that Definition 8 is stronger than what we need for the proofs presented here. We could have omitted conditions 6 and 7 and weakened some of the other conditions without invalidating any of the results that we present. Typically, we would use definitions that are as weak as possible, since this provides the most general results. In this case, however, one of our goals is to design a general framework to reason about different kinds of epistemic gossip problems. As such, we add not just the conditions that we need for our results, but also the other conditions that we believe to be reasonable for a model of the gossip problem.

Modeling the Parameters

Our main goal is to provide a framework that can be used to formalize the parameters that distinguish different kinds of epistemic gossip problem. We cannot discuss every parameter here, so we restrict ourselves to two important ones. The first of these parameters is the nature of the protocols that we consider: all protocols, epistemic protocols or symmetric epistemic protocols. The second parameter is the amount of knowledge agents have about calls that they are not directly involved in: is communication asynchronous, synchronous or observable? Synchrony and observability are not new concepts, of course. But their formalization in the gossip framework is new. Epistemic protocols have already been defined in the context of gossip protocols [START_REF] Attamah | The pleasure of gossip[END_REF][START_REF] Attamah | Knowledge and gossip[END_REF], but the distinction between epistemic and symmetric epistemic protocols is new.

Constraints on Protocols

In principle, every PDL protocol can be considered on a gossip model. But in general such protocols may require agent a to make a call c ab even if a doesn't know that this call should be made. We are therefore mostly interested in two subclasses of protocols, the epistemic and epistemic symmetric protocols. We follow [START_REF] Attamah | The pleasure of gossip[END_REF] and [START_REF] Attamah | Knowledge and gossip[END_REF] in the definition of epistemic protocols.

Definition 9. A protocol π is an epistemic protocol if it is of the form π =   a =b∈A K a ψ ab ?; c ab   * ;   ¬ a =b∈A K a ψ ab   ?
where ψ ab is a formula for every a = b ∈ A. The formula K a ψ ab is called the call condition for c ab in π.

Many constructions in PDL have abbreviations that resemble the way one would phrase a protocol in a programming language. Using such an abbreviation, an epistemic protocol π can be written as while a =b K a ψ ab do a =b (if K a ψ ab then make call c ab ). So as long as at least one of the call conditions is true, the protocol nondeterministically executes one of the actions c ab for which the call condition holds. We allow only one call condition per pair a, b, but we can always combine multiple call conditions into one. If, for example, we want to allow a call c ab if either K a ϕ ab or K a χ ab , this is equivalent to allowing the call c ab if K a (K a ϕ ab ∨K a χ ab ).

In some cases we want to restrict the available protocols even further, to the symmetric epistemic protocols. A symmetric protocol is one where all agents are treated in the same way. In other words, a protocol is symmetric if all agents "are running the same software." So, for example, a protocol is not symmetric if it assigns one pre-determined agent a the role of leader, and requires this a to call every other agent.

Definition 10. A permutation on A is a bijection σ : A → A. Given a permutation σ and a formula ϕ, the formula σ(ϕ) is obtained by simultaneous replacement of all occurrences in of a in ϕ by σ(a) for all a ∈ A. Definition 11. Let π be an epistemic protocol, and for every a = b ∈ A let K a ψ ab be the call condition for c ab . The protocol π is symmetric if for every permutation σ on A and every a = b ∈ A, we have σ(ψ ab ) = ψ σ(a)σ(b) .

Knowledge of Other Agents' Calls

In our gossip models we assume that agents are aware of all calls that they themselves participate in, so if a cannot distinguish between state s and s then p(s)| a = p(s )| a . In some cases we may also want to make assumptions about agents' knowledge of calls they do not participate in. We distinguish three levels of such knowledge.

Firstly, it is possible that the agents are simply unaware of calls that they do not participate in. In this case, the setting is asynchronous. Secondly, it is possible that the agents know how many calls take place, but don't know who is participating in the calls (unless they are one of the participating agents). In this case, the setting is synchronous. Finally, it is possible that the agents know exactly which calls take place, although they cannot observe the content of the call (so an agent that is not involved in the call will not learn the values of the secrets that are exchanged). In this case, the setting is observable.

The three different levels of knowledge can be represented by constraints on the knowledge relation in gossip models. For the asynchronous case we need not place any restrictions, we can consider the class of asynchronous models to be equal to the class of all gossip models. The classes of synchronous and observable models do require restrictions. The observable models can easily be defined as follows.

Definition 12. A gossip model M = (S, R, O, V ) is observable if for all s, s ∈ S and all a ∈ A such that if (s, s ) ∈ R(a), we have p(s) = p(s ). Let M Obs be the class of observable models.

Defining the synchronous models is slightly harder, and requires an auxiliary definition. The symbol is used as a placeholder for "a call took place, but I don't know which one".

Definition 14. A gossip model M = (S, R, O, V ) is synchronous if, for every s, s ∈ S and all a ∈ A such that (s, s ) ∈ R(a), we have p(s)| synch a = p(s )| synch a .
Let M Synch be the class of synchronous models.

Optimal Solution Length

Now that we have defined some of the parameters for gossip protocols, we can determine how the parameters affect the gossip problem. There are several aspects of the gossip problem that can change based on the parameters. For example, in some cases we can get common knowledge of Goal while in other cases this is impossible. The aspect of the gossip problem that we study here is closer to the traditional questions about gossip: we want to determine how many calls are required to achieve Goal in the different scenarios.

Definition 15. Let M be a class of models. A protocol π is executable on M if for all M ∈ M and every initial state s of M we have M, s |= π . A protocol π is effective on M if for all M ∈ M and every initial state s of M , we have M, s |= [π]Goal . Given m ∈ N, a protocol π is m-effective on M if it is effective on M and furthermore, for every M ∈ M and every initial state s of M , all traces of π in M, s are of length at most m. Definition 16. Let M be a class of models and Γ a set of protocols. The optimal solution length for Γ on M is the lowest m ∈ N such that there is a protocol π ∈ Γ that is executable and m-effective on M.

We have not yet fully determined the optimal solution length for all combinations of models and protocols discussed above. The ones that we do know, as well as bounds for the ones that we have not yet fully determined, are as follows (assuming n ≥ 4, and in the asynchronous/symmetric epistemic case that n ≥ 8 and n is even).

Observable Synchronous Asynchronous

All protocols 2n -4 2n -4 2n -4 Epistemic 2n -4 2n -4 ≤ 2n -3 Symmetric Epistemic 2n -4 2n -3 ≥ 2n
We do not yet have a proof for the exact optimal solution lengths on asynchronous models, but we conjecture that the optimal solution length on asynchronous models is 2n -3 for epistemic protocols and in O(n 2 ) for symmetric epistemic protocols. We do not provide full proofs for these optimal solution lengths here, but we do give proof sketches for all but one of the non-trivial cases. In the proof sketches we informally describe a number of protocols. These protocols can be formalized in a straightforward way, but such formalizations are harder to read than the informal description. We therefore omit the formalizations.

Observable models. In observable models all agents know the call history. Consider the protocol where a call c ab is allowed if and only if a knows that the current history followed by c ab is an initial segment of some effective sequence of calls with length 2n -4. This is an epistemic symmetric protocol, and it is executable and (2n -4)-effective on the class of observable models.

Synchronous models. In synchronous models all agents know how many calls have taken place. Fix any call sequence that causes all agents to know all secrets after 2n -4 calls. We can then require all agents to take turns creating this call sequence: if the k-th call in the sequence is c ab , then agent a should make that call after k -1 other calls have happened. This is an epistemic protocol, but not a symmetric one.

For the lower bound of the synchronous/symmetric case we need a combinatorial property that holds for any call sequence of 2n -4 calls that is effective: there are at least two calls c ab and c a b between "fresh" agents that had not participated in any calls before. After all, suppose towards a contradiction that there is only one call between fresh agents. Then in the first call the number of fresh agents decreases by 2, and every call afterwards it decreases by at most 1. So after n -2 calls there is at least one fresh agent left. Consider the number of agents that know this last fresh agent's secret. After n -2 calls, only the agent itself knows it. Every call afterwards can teach it to at most one more agent, so it takes at least n -1 more calls for every agent to know this secret. In total, the call sequence therefore has to contain at least 2n -3 calls.

This property of having at least two calls between fresh agents cannot be guaranteed by an epistemic symmetric protocol in a synchronous model; agents know how many calls have taken place but not who was involved in the calls so they cannot decide to call an agent that has not been called yet. As such, the optimal solution length for symmetric epistemic protocols on synchronous models is at least 2n -3.

To see that 2n -3 is also an upper bound, consider the following protocol: every agent starts out by trying to call every other agent. One of the agents, say agent a 1 , will be the first one to successfully place a call. On synchronous models, all agents know that the first call has taken place. While most agents do not know who placed the first call, they do know that it wasn't them. So after the first call, continue by having everyone but a 1 remain passive while a 1 calls all other agents. Then, after a 1 has called all agents and therefore knows all secrets, a 1 calls every agent again, except for the agent that was called last in the first round (because that agent already knows all secrets). This protocol is symmetric epistemic, and it is (2n -3)-effective on synchronous models.

Asynchronous models. We are not yet certain whether an epistemic (2n -4)-effective protocol exists on asynchronous models, although we suspect it does not. We do know that there is an epistemic (2n -3)-effective protocol: fix one agent a. Like in the symmetric epistemic/synchronous case, have a call all other agents and then all but one of the other agents again. We fixed one particular agent a, so this protocol is not symmetric. It is epistemic, however.

This leaves only the asynchronous/epistemic symmetric case. The proof for the 2n lower bound in this case is rather long and not very insightful. As such, we omit it here.

Conclusion

We introduced a framework that is capable of modeling many different kinds of epistemic gossip problem. We also formally defined some of the parameters for epistemic gossip protocols, and showed the effect of choosing different parameters by considering how they affect the optimal solution length.

One target for future research is the exact optimal solution length on asynchronous models. The case of symmetric epistemic protocols is especially interesting, since there is a relatively large difference between the bound of 2n that we can prove and the O(n 2 ) bound that we believe to hold.

Furthermore, although we have formalized some parameters, there are many more that we have not yet formalized. As such, another interesting target for future research is to treat more of these parameters. One interesting variant is to consider higher-order knowledge: in the variants of the gossip problem that we discussed here, the agents try to make sure that every agent knows all secrets. But there are other variants, where the agents want to make sure that every agent knows that every agent knows every secret, and so on. For the set of all protocols, it was proved in [START_REF] Herzig | How to share knowledge by gossiping[END_REF] that shared knowledge of depth d, for d ≥ 1 can be achieved in at most (d + 1) × (n -2) calls. For the smaller sets of protocols, it is not yet known how hard it is to achieve higher order knowledge of secrets.

Other interesting parameters include how much agents tell each other in their conversations (do they only tell each other their secrets, or also when and how they learned the secrets?) and how much the agents know about the protocol that they are collectively following (is the protocol common knowledge?).

Definition 3 .

 3 Let M = (S, R, O, V ) be a model and s ∈ S. The satisfaction relation |= is given inductively by

Definition 4 .

 4 Let M = (S, R, O, V ) be a model and let s, s ∈ S. The state s is a predecessor of s if there are a, b ∈ A such that (s, s ) ∈ O(c ab ). The state s is an initial state if it has no predecessors. Definition 5. Let M = (S, R, O, V ) be a model and let s ∈ S. If there are unique finite sequences c a1b1

Definition 6 .

 6 Let M = (S, R, O, V ) be a model and let s ∈ S. Suppose p(s) = c a1b1 ; • • • ; c a k b k and let a ∈ A. Then the a-reduction of p(s), denoted p(s)| a , is the subsequence of p(s) that contains exactly those terms c ambm where a m = a or b m = a. So p(s)| a is the sequence of calls that a participated in. Definition 7. Let M = (S, R, O, V ) be a model and let s, t ∈ S and a

Definition 13 .

 13 Let M = (S, R, O, V ) be a gossip model, s ∈ S and a ∈ A. Suppose p(s) = c a1b1 ; • • • ; c a k b k . The synchronous a-reduction of p(s), noted p(s)| synch a , is the sequence x 1 ; • • • ; x k where x m = c ambm if a m = a or b m = a and x m = otherwise.

Note that while the past and the outcomes of actions are fixed, this does not imply that the agents know the past or the outcome of actions.

Acknowledgments

Some of the parameters discussed here were inspired by discussion at the 2015 Lorentz Center workshop "To Be Announced!" We would like to thank the participants of the workshop for their input in the discussion. We would also like to thank the LOFT reviewers for their remarks and suggestions. Hans van Ditmarsch acknowledges support from ERC project EPS 313360. He is also affiliated to IMSc, Chennai, India, and Zhejiang University, China.


to water availability segregated in the progeny

test for the difference between RI and GW Maps of the experimental plot were different for δ 13 C and pruning weight Traits for plant vigour and response

Integrating spatial variations in the vineyard to enhance QTL detection Xavier de Badts de Cugnac (1) , Vincent Dumas (1) , Nathalie Jaegli (1) , Lionel Ley (2) , Didier Merdinoglu (1) , Eric Duchêne (1)

(1) SVQV, Université de Strasbourg, INRA, 68000 Colmar, France

(2) SEAV, INRA, 68000 Colmar, France

Conclusion

French vineyards are often located on the least fertile lands, generally more heterogeneous than lands devoted to food crops. The detection of quantitative trait loci (QTLs) is all the more precise as the number of genotypes studied is high, but increasing the size of experimental plots means taking the risk of exploring more spatial heterogeneity. Designing randomized blocks experiments is a classical way to overcome this difficulty. However, when several hundred of genotypes are studied, this not only requires more land but is also time consuming. Another approach is to take into account the spatial heterogeneity when searching for QTLs.

Plant material and phenotyping

Three hundred and eighty-three genotypes from a Riesling x Gewurztraminer progeny are planted in elementary plot (EP) of three plants, without replicates. The parents, Riesling (RI) and Gewurztraminer (GW), also planted in EPs of three plants but these EPs are replicated 12 and 13 times, respectively, with a regular spatial distribution (Figure 2). Carbon isotope composition of the berries at harvest (12-18/10/2016), expressed as δ 13 C, was measured in 2016 on 117 genotypes and the parents.

Data analysis

The broad sense heritability was calculated as H 2 = (σ progeny ²-σ e ²) / σ progeny ². σ e ² was extracted from analysis of variance (lm and anova commands under R software) testing the "genotype" effect with RI and GW.

A map of the experimental plot was built using the raw measures ("raw data") obtained on the parent varieties using the ArcGIS 10.6 Geostatistical Analyst extension.

For each trait, the kriging method providing the lowest root mean square error with a leave-out-one procedure was chosen. In a second step, the expected value of the trait of interest for each single elementary plot of 3 plants was calculated, as if a unique variety was covering all the experiment (Figure 2). Finally, for each elementary plot of the experiment, the difference between the observed value and the expected value was calculated ("differential data").

Genetic data

Genetic data were obtained the "Genotyping by sequencing" technology. Keygene N.V. owns patents and patent applications protecting its Sequence Based Genotyping technologies. Markers segregating in both parents at positions inside sliding windows of 30 kb in the reference genome were encoded as <abxcd>. Phases were determined with JoinMap 2.0 (Van Ooijen and Voorrips, 2001). JoinMap format was transformed into R/qtl format (Broman et al., 2003) with the phasedJoinMapCP2qtl command of the rutilstimflutre R package. The genetic map was then calculated with R/ qtl and finally contained 748 markers.

QTL detection

QTL detection was performed with R/qtl on raw values recorded for each genotype but also on the differences between the raw values and the parental expected values for each EP ("differential data"). Statistical significance of LOD values was assessed after 1000 permutations. LOD scores were also calculated after building a QTL model that included QTLs with a LOD score significant at p = 0.20 (makeqtl, refineqtl and fitqtl commands).

Acknowledgments to Timothée Flutre for initiating and supervising the acquisition of "genotyping by sequencing" data.

A B

Variable Raw data Differential data δ 13 C 17@32 (3.90) 17@21 (5.13) Pruning weight (2017/2018) 1@50 (5.86) 13@52.7 (5.66) 14@39.4 (3.80) 1@40.5 (3.7) 13@46 (4.34) 17@28 (4.31) NBI_R 5@0 (3.62) 15@11.9 (3.63) 5@0 (3.64) NFI 3@84 (3.62) 8@66.9 (4.49) 13@63 (4.40) 17@32 (4.39) 8@66.9 (4.39) 13@72.5 (4.10) 17@30 (4.45) BINBI 5@0 (3.89) 13@90 (3.73) 15@11.9 (4.11) 5@0 (3.95) 13@87 (4.47) 17@3 (3.73) Using differences between measured and expected values reduced the environmental variance and increased the heritability of the traits.

Raw data Differential data

QTLs significant at p = 0.05 genomewide were detected for all the traits (Table 2). Taking into account spatial information was interesting for δ 13 C because a QTL on chromosome 17 reached a significance at p = 0.05 and a new QTL on chromosome 18 reached a significance at p = 0.20 when included in a QTL model. Figure 3 illustrates the improvement of LOD scores for δ 13 C. For the pruning weight, a new QTL is detected on chromosome 17 but the level of significance of QTLs in chromosome 1 and 13 decreased. Differential data did not improved QTL detection for NBI_R and NFI Multiplex ® indices but for BINBI, a QTL on chromosome 13 became significant at p = 0.05.

  δ 13 C is related to stomatal closure and response to drought stress. Features of the canopy were evaluated on August, 2nd, 2017, around veraison, by Multiplex® Research (Force A, Orsay, France), a field fluorometer providing the following indices: NBI_R (nitrogen balance index) is an indicator of the nitrogen status of the plants (Blank et al., 2018), NFI (Normalized Fluorescence Index) estimates the foliage density, BINBI is the product of NBI_R and NFI, giving an assessment of global foliar activity. Finally, pruning weight was measured on February 21st, 2018 on 249 genotypes as well as on the parents.

Figure 1 :

 1 Figure 1: Segregation of NFI, BINBI, pruning weight and δ 13 C. Red arrow: Gewurztraminer, green arrow: Riesling.

Figure 2 :

 2 Figure 2: Maps of the expected values calculated with the data from the parent varieties. A : δ 13 C (2016), B: pruning weight (2017/18 season). Rectangles represent the elementary plots of the parents (green for RI, red for GW).

Figure 3 :

 3 Figure3: LOD scores profiles for δ 13 C, blue line for raw data, red line for differential data. Note that the LOD scores for raw data on chromosome 17 are not the same as in table3because introduction of the QTL on chromosome 18 was forced in the model, even though not significant at p=0.05 at the first round of detection. Horizontal lines: LOD threshold determined after 1000 permutations. Plain and dotted lines: at p = 0.05 and p = 0.20, respectively. QTLs related to response to drought were also detected by Coupel-Ledru et al (2017) on chromosome 17 was on a phenotyping platform.

Table 1 :

 1 Statistical criteria and heritabilities


Hedi Karoui

Marianne J Seck

Nicolas Martin
email: nicolas.martin@crpp.cnrs.fr

Self-Programmed Enzyme Phase Separation and Multiphase Coacervate Droplet Organization

Membraneless organelles are phase-separated droplets that are dynamically assembled and dissolved in response to biochemical reactions in cells. Complex coacervate droplets produced by associative liquid-liquid phase separation offer a promising approach to mimic such dynamic compartmentalization. Here, we present a model for membraneless organelles based on enzyme/polyelectrolyte complex coacervates able to induce their own condensation and dissolution. We show that glucose oxidase forms coacervate droplets with a cationic polysaccharide on a narrow pH range, so that enzyme-driven monotonic pH changes regulate the emergence, growth, decay and dissolution of the droplets depending on the substrate concentration. Significantly, we demonstrate that time-programmed coacervate assembly and dissolution can be achieved in a single-enzyme system. We further exploit this self-driven enzyme phase separation to produce multiphase droplets via dynamic polyion self-sorting in the presence of a secondary coacervate phase. Taken together, our results open perspectives for the realization of programmable synthetic membraneless organelles based on self-regulated enzyme/polyelectrolyte complex coacervation.

Introduction

Liquid-liquid phase separation (LLPS) of biological polymers has emerged as a ubiquitous phenomenon in the formation of membraneless organelles in living cells. [1][2][3] These biomolecular condensates participate in the organization of intracellular contents, 2 favor dynamic molecular exchanges with their environment, 3 and can exhibit a multi-layered structure that contributes to the spatiotemporal regulation of biochemical reactions. 4,[START_REF] Boeynaems | Proc. Natl. Acad. Sci[END_REF] A critical feature of these biological assemblies is their ability to reversibly form and dissolve in response to biochemical reactions, such as post-translational modifications or DNA transcription, 3,[START_REF] Nott | [END_REF][7][8][9] which enables spatiotemporal control over the compartmentalization of biomolecules and reactions in the cytoplasm. Drawing inspiration from nature, in vitro LLPS steps into the spotlight as a viable strategy for the bottom-up construction of synthetic membraneless organelles. [10][11][12][13][14][15] Complex coacervate micro-droplets produced by associative LLPS between oppositely charged polyions recapitulate most of the features of biomolecular condensates: they exhibit selective solute uptake, [16][17][18][19] accelerate biochemical reactions, [20][START_REF] Sokolova | Proc. Natl. Acad. Sci[END_REF][START_REF] Koga | [END_REF][23] and readily form or dissolve in response to physicochemical stimuli, including changes in pH, [START_REF] Koga | [END_REF]23 temperature, [24][25][26] ionic strength, 27 and under light irradiation. 28,29 Yet, emulating both the spatial and temporal complexity of biomolecular condensates in synthetic coacervates remains challenging.

Recent directions have been geared towards increasing the spatial complexity of coacervates via the formation of multiphase droplets under thermodynamic equilibrium conditions, [30][31][32] including by programming molecular interactions. 33,34 On the other hand, active processes such as enzyme [35][36][37][38][39] and chemical reactions 40 have started being explored to increase the complexity of the temporal dynamics of coacervate droplets. Pioneering studies have demonstrated the use of two antagonistic enzymes acting as endogenous catalytic controllers to trigger either the condensation or dissolution of coacervates. 36,37,41 Yet, attempts to combine both an active regulation of coacervate droplets and their spatial organization into hierarchical droplets have not yet been reported.

Our strategy relies on the use of enzymes as "scaffold" macroions to assemble self-regulating complex coacervate droplets. Studies on such a protein/polyelectrolyte coacervation phenomenon have been limited so far to single-phase systems at thermodynamic equilibrium. 17,[42][43][44][45][46][47][48] Here, we show that an enzyme, glucose oxidase (GOx), acts as a catalytically active "scaffold" coacervate component able to self-modulate its phase separation with an oppositely charged polysaccharide, diethylaminoethyl-dextran (DEAE-dextran). Specifically, we show that GOx and DEAE-dextran form coacervate micro-droplets on a narrow pH range corresponding to conditions close to charge stoichiometry, and exploit this pH-responsive behavior to demonstrate programmed assembly and dissolution of coacervates based on GOx-driven pH decrease in the presence of glucose. Significantly, the amount of glucose fuel supplied to the system controls the amplitude of the pH decrease so that either stable or transient assembly of coacervate droplets with controllable lifetime, together with multiple cycles of transient coacervation, are achieved. We further exploit such enzymeresponsive dynamic coacervates to create non-equilibrium multiphase droplets in the presence of a secondary coacervating system, which, to the best of our knowledge, have not been reported yet. Overall, our results highlight opportunities for the realization of self-actuated enzyme/polyelectrolyte phase separation, together with enzyme-driven polyion self-sorting into multiphase complex coacervate droplets under non-equilibrium conditions, providing new approaches to the construction of programmable synthetic membraneless organelles with increased spatiotemporal complexity.

Results and discussion

Formation of pH-responsive enzyme-based coacervate droplets

Coacervate micro-droplets were first produced at equilibrium as a turbid aqueous suspension via liquid-liquid phase separation between DEAE-dextran and GOx at physiological pH (phosphate buffer, 2.5 mM, pH = 7.4; Fig. 1a). Turbidity measurements at varying polyion ratio (Fig. 1b), together with charge titration and calculation studies, revealed that phase separation occurred near charge neutrality (ESI Note 1 and Fig. S1 †), consistent with previous studies on protein/polyelectrolyte coacervates. 17,[42][43][44][45][46][47][48] Optical microscopy images of the suspension produced at equimolar charge ratio (corresponding to conditions leading to maximum turbidity) confirmed the presence of polydisperse spherical micro-droplets (Fig. 1d) that fused on contact (ESI Fig. S2 † Since both polyions are weak polyelectrolytes, their net charge and charge density strongly depends on the pH. Phase behaviour of GOx/DEAE-dextran mixtures was therefore examined over a broad range of pH values at fixed protein : polymer ratio ([GOx] = 0.25 mg mL -1 , [DEAE-dextran] = 0.04 mg mL -1 ) (Fig. 1c andd). We observed that phase separation occurred on a relatively narrow pH range (6.5 ≤ pH ≤ 8.5) corresponding to conditions close to charge neutralization between GOx (negative net charge) and DEAEdextran (positive net charge) (ESI Fig. S1 †). In comparison, coacervation was inhibited at high and low pH values due to charge mismatch between DEAE-dextran and GOx (Fig. 1c andd). These empirical observations correlated well with charge titration and calculation studies: notably, we observed that the optimal coacervation pH shifted to higher or lower values when we altered the protein : polycation ratio, an observation that could be well-predicted (ESI Note 1 and Fig. S1 †). Overall, these results establish that GOx and DEAEdextran form coacervate micro-droplets on a relatively narrow pH range and that the optimal pH for phase separation corresponds to charge neutralization conditions. Fig. 2 (a) Schematic representation of enzyme-mediated self-regulated complex coacervation of GOx with DEAE-dextran in the presence of glucose. GOx catalyzes the oxidation of glucose into gluconolactone that spontaneously hydrolyses into gluconic acid, producing a pH decrease that can drive coacervate formation and dissolution. (b) Time-dependent evolution of the absorbance at 700 nm of a solution of GOx (0.25 mg mL -1 ) and DEAE-dextran (0.04 mg mL -1 ) produced at pH 10.2 after the sequential addition of glucose (0.6 mM at each addition). The colored area represents error as the standard deviation of three independent repeats. (c) Time-dependent evolution of the absorbance at 700 nm of a solution of GOx (0.25 mg mL -1 ) and DEAE-dextran (0.04 mg mL -1 ) produced at pH 10.2 after the single-step addition of varying final glucose concentrations, as indicated. Above a certain glucose concentration, a bell-shape is observed, attributed to the nucleation, grow, decay and dissolution of coacervate droplets. In such conditions, τ1/2 denotes the full width at half maximum turbidity (here shown on the example of 1.4 mM glucose). The colored area represents error as the standard deviation of three independent repeats. (d and e) Optical microscopy snapshots of GOx/DEAE-dextran mixtures ([GOx] = 0.4 mg mL -1 , [DEAE-dextran] = 0.064 mg mL -1 ) prepared at pH 10.2 at different times after addition of 0.5 mM (d) or 25 mM (e) glucose, showing the formation of stable or transient coacervate droplets, respectively. Scale bars, 20 μm. (f) Evolution of τ1/2 as defined in c as a function of the final glucose concentration. The red line represents a mono-exponential fit of the data. Error bars represent standard deviations of three independent repeats. (g) Time-dependent evolution of the absorbance at 700 nm of a solution of GOx (0.25 mg mL -1 ) and DEAE-dextran (0.04 mg mL -1 ) produced at pH 10.2 after the single-step addition of 5 mM glucose and the repeated additions of 10 mM NaOH (black arrows). The dilution factor after the last NaOH addition was ∼1.05, so the final concentrations of components did not appreciably change.

Enzyme-driven reversible formation and dissolution of coacervate droplets

Given the above observations, we then sought to control coacervate formation in response to pH changes resulting from GOx's catalytic activity (Fig. 2a). GOx catalyzes the oxidation of glucose into gluconolactone, which spontaneously hydrolyses into gluconic acid in solution, thereby inducing a pH decrease in the absence of buffer. We first proceeded to investigate the programmed formation and disassembly of liquid droplets via the sequential addition of a fixed amount of glucose to mixtures of GOx (0.25 mg mL -1 ) and DEAE-dextran (0.04 mg mL -1 ) prepared at pH ∼ 10 in pure water. Addition of 0.6 mM glucose (final concentration) initiated the formation of coacervate micro-droplets, as observed by the gradual increase in the sample's turbidity, reaching a plateau after ∼40 min (Fig. 2b). This increase in turbidity was associated with a decrease in pH that stabilized around 7.4 (ESI Fig. S6 †), a value close to the optimal coacervation pH for this protein : polycation ratio. In comparison, addition of another 0.6 mM glucose to this turbid suspension caused a rapid turbidity decay to initial values, confirming the disassembly of the coacervate micro-droplets (Fig. 2b) due to a further decrease in pH. In control experiments performed in the absence of glucose, the pH remained almost constant (>9.5) and therefore no change in the solution's turbidity was observed (ESI Fig. S6 †).

Having established that the catalytic oxidation of glucose by GOx induced a sufficient pH decrease to sequentially trigger the condensation and dissolution of coacervate droplets, we then sought to demonstrate programmable behavior by investigating the outcome of a single-step addition of increasing amounts of glucose fuel to a clear GOx/DEAE-dextran solution prepared at pH ∼ 10. At low glucose concentrations (∼0.6 mM), the turbidity gradually increased until reaching a plateau value (Fig. 2c), indicating that stable coacervate micro-droplets had formed as the optimal coacervation pH was reached (ESI Fig. S6 †). Optical microscopy also revealed the gradual nucleation and growth of droplets that persisted for an extended period (Fig. 2d and ESI Movie 1 †). We observed that droplets' growth occurred by both fusion and gradual material uptake from the dilute continuous phase with an average area growth rate of 0.059 ± 0.01 μm 2 min -1 (ESI Fig. S7 †).

In comparison, higher glucose concentrations (>0.6 mM) resulted in the transient assembly then dissolution of coacervate micro-droplets, as suggested by the bell-shaped temporal evolution of the turbidity (Fig. 2c and ESI Fig. S8 †), indicative of the emergence, growth, decay and disassembly of coacervate droplets as the pH decreased from 10.5 down to <6.5 (ESI Fig. S6 †). Significantly, we also observed that the full width at half maximum turbidity, τ1/2, decreased mono-exponentially with the added glucose concentration (Fig. 2f), as expected from the kinetics of GOx-mediated pH decrease (ESI Fig. S9 †), indicating that the lifetime of coacervate droplets could be fine-tuned by the amount of added substrate. Optical microscopy further confirmed the transient assembly then dissolution of coacervate micro-droplets at high glucose concentration (Fig. 2e and ESI Movie 2 †). Interestingly, at even higher glucose concentrations (5 mM), transient cycles of enzyme-driven spontaneous coacervation could be established by repeated additions of NaOH after droplets had dissolved (to re-increase the pH above 9) until all glucose had been consumed (Fig. 2g). Taken together, these results demonstrate the self-induced biocatalytic condensation and dissolution of enzyme-rich coacervate droplets, together with temporal programmability depending on substrate turnover. Notably, we show that a single-enzyme system suffices to achieve reversible coacervate assembly, provided the enzymatic reaction allows to navigate across the coacervation phase diagram (here, by monotonically altering the net charge and charge density of the polyions).

Enzyme-driven multiphase droplet organization via dynamic polyion self-sorting

We last explored the possibility to use our self-triggered enzyme-rich coacervate platform in more complex environments. In vivo, intracellular membraneless organelles evolve in a crowded mixture of components. Studies have shown that these biomolecular condensates can organize into a hierarchical, multi-layered organization, which has been suggested to facilitate the coordination of biochemical reactions in cells. 4,[START_REF] Boeynaems | Proc. Natl. Acad. Sci[END_REF] The formation of multiphase complex coacervate droplets has also been recently reported in mixtures of polyelectrolytes at thermodynamic equilibrium in vitro. [30][31][32] We here sought to expand such hierarchical droplet organization and increase their functional complexity by demonstrating the formation of dynamic multiphase droplets under non-equilibrium enzymatic control.

We used our GOx/DEAE-dextran droplets in conjunction with coacervates assembled from adenosine triphosphate (ATP) and poly-l-lysine (pLL) as a basis for our dynamic multiphase complex coacervate microdroplets. Stable multiphase droplets were first successfully formed at equilibrium by mixing equal volumes of suspensions of each of the coacervate droplets prepared separately at their optimal ratio at pH 7.4 (phosphate buffer, 2.5 mM; see Methods). Optical microscopy images confirmed the formation of two-phase droplets showing a smooth interface (Fig. 3a), typical of coexisting liquid phases. [30][31][32]51 Both the outer and inner domains behaved as liquids, as demonstrated by their coalescence and ability to engulf other multiphase droplets (ESI Fig. S10 †). Confocal fluorescence microscopy further revealed that RITC-GOx and FITC-DEAEdextran co-localized in the outer phase (Fig. 3b andc), while FITC-labelled pLL selectively localized in the inner phase (ESI Fig. S11 †). The multiphase droplets could therefore be described as an outer GOx/DEAEdextran phase surrounding an inner droplet made from ATP and pLL. This hierarchical organization was likely driven by differences in the surface tension of the two phases, with the inner ATP/pLL coacervates presumably exhibiting a higher interfacial tension than the GOx/DEAE-dextran droplets. 30 In addition, ATP/pLL coacervates were more resistant to salt than GOx/DEAE-dextran droplets (ESI Fig. S4 †), which was consistent with previous studies showing that the inner phase of multiphase coacervates disassembled at higher salt concentrations compared to the outer phase. 30 Significantly, at high (pH > 8.5) and low (pH < 6.5) pH, or in the absence of DEAE-dextran, single-phase droplets were observed (ESI Fig. S12 †), and were attributed to ATP/pLL coacervates since these droplets are stable on a broader pH range (typically, 2 < pH < ∼10.5, ref. 22) compared to GOx/DEAE-dextran droplets. We finally proceeded to control and program the dynamics of these multiphase coacervate droplets in response to GOx's catalytic activity (Fig. 3d). We herein prepared mixtures of ATP, pLL, GOx and polysaccharide at pH ∼ 10.2, to which we added varying amounts of glucose. We initially observed the presence of single-phase ATP/pLL coacervate droplets (Fig. 3e andf, t = 0), as expected at these high pH values, then a second outer liquid phase gradually appeared, associated to the formation of GOx/DEAEdextran coacervate phase as the solution pH decreased upon GOx catalytic activity (Fig. 3e andf, t = 22 min, and ESI Movies 3 and 4 †). Depending on the amount of glucose added, we could observe the formation, growth and stabilization (low glucose, Fig. 3e and ESI Movie 3 †), or the formation, growth and decay (high glucose, Fig. 3f and ESI Movie 4 †) of multiphase coacervate micro-droplets, respectively. Taken together, these observations demonstrate the ability of mixtures of polyions to undergo dynamical self-sorting into multiphase coacervate droplets under non-equilibrium conditions powered by enzyme reactions. Significantly, the amount of fuel supplied to the system controls the formation of stable multiphase complex coacervate droplets or the transient assembly of hierarchical droplets, together with the displacement of GOx molecules between phases.

Conclusions

In conclusion, we report the self-induced phase separation of an enzyme in the presence of an oppositely charged polyelectrolyte based on enzyme-driven monotonic pH changes. We show that GOx catalytic activity can trigger the formation of stable enzyme-rich coacervate micro-droplets and generate transient coacervates with programmable lifetime depending on the glucose fuel added to the system. We further demonstrate enzyme-driven formation of multiphase complex coacervate droplets via spontaneous polyions self-sorting in the presence of a secondary coacervating system. The possibility to timely program such dynamical behavior opens perspectives for the realization of controllable synthetic membraneless organelles as it provides a simple self-mediated biochemical approach to control the compartmentalization of bio-catalytically active molecules.

Experimental Materials

The following chemicals were purchased from Sigma-Aldrich and used as received: glucose oxidase (GOx) from Aspergillus niger, diethylaminoethyl-dextran (DEAE-dextran, Mw = 500 kDa), rhodamine isothiocyanate (RITC), fluorescein isothiocyanate (FITC), fluorescein isothiocyanate labelled diethylaminoethyl-dextran (FITC-DEAE-dextran, Mw = 70 kDa), potassium carbonate, monopotassium and dipotassium phosphate, toluene, sodium hydroxide, hydrochloric acid (37%), bovine serum albumin (BSA, heat shock fraction, >98%), α-d-glucose (96%), and poly-l-lysine hydrobromide (pLL, (C6H12N2O)n, 4-15 kDa, monomer Mw = 208.1 g mol -1 ). Adenosine 5-triphosphate disodium salt hydrate (ATP, C10H14N5Na2O13P3, 551.1 g mol -1 ) was purchased from Carbosynth Limited and 3-[methoxy(polyethyleneoxy)propyl]trimethoxysilane (90%, 6-9 PE units) was purchased from abcr GmbH, Gute Chemie.

Preparation of stock solutions

Milli-Q water was used to prepare aqueous stock solutions of DEAE-dextran (1.8 mg mL -1 , corresponding to 8.9 mM average monomer concentration, pH = 7.4), FITC-DEAE-dextran (1.8 mg mL -1 , pH = 7.4), GOx (6 mg mL -1 , corresponding to 75 μM monomeric GOx concentration, pH = 7.4), glucose (1 M), pLL (100 mM monomer concentration, corresponding to 14.4 mg mL -1 , pH = 8.0), ATP (50 mM nucleotide concentration, corresponding to 50.7 mg mL -1 , pH = 8.0) and phosphate buffer (50 mM, pH = 7.4). The pH of all stocks was adjusted using either NaOH (0.1 M) or HCl (0.1 M). The concentration of GOx in the stock solution was checked by UV-vis spectroscopy (Varian CARY 100 Bio) at 280 nm using an extinction coefficient of 1.67 mL mg -1 cm -1 (ref. 49) and a molecular weight of 80 000 g mol -1 . All polymer, protein and mononucleotide stock solutions were stored at -20 °C until use.

Preparation of fluorescently labelled GOx

A GOx solution (4.0 mg mL -1 ) was prepared by dissolving the freeze-dried protein powder in 1 mL of 0.5 M carbonate buffer at pH = 9.0. An aliquot of a freshly-prepared anhydrous DMSO solution of RITC (10.0 mg mL -1 ) was added drop-wise to the protein solution at a final fluorophore : protein molar ratio of 10 : 1. The reaction mixture was kept at room temperature in the dark for 4 hours, then purified by size exclusion chromatography using a Sephadex G-25 resin (Sigma-Aldrich) eluted with Milli-Q water. The concentration of the fluorescently-labelled proteins in the collected fractions was determined by UV-visible spectrophotometry using the relationship: [protein] = (A280w × Amax,dye)/εprotein, where A280 and Amax,dye were the absorbances at 280 nm and at the maximum of absorption of the fluorophore respectively (552 nm for RITC), w the correction factor to account for the dye absorption at 280 nm (0.34 for RITC), and εprotein the extinction coefficient of the protein (1.67 mL mg -1 cm -1 for GOx). The dye : protein final molar ratio was determined from the ratio (Amax,dye/εdye)/([protein] (mg mL -1 )/MGOx), where εdye was the molar extinction coefficient of the dyes at their maximum of absorption (65 000 mol -1 L cm -1 for RITC), and MGOx the molar mass of the protein (80 000 g mol -1 for a GOx monomeric unit). Typically, the average dye : protein molar ratio was ca. 4 : 1. The RITC-GOx stock solution was split into aliquots and stored at -20 °C until use.

Preparation of fluorescently labelled poly-l-lysine

A pLL solution (4.0 mg mL -1 ) was prepared by dissolving the freeze-dried polypeptide powder in 1 mL of 0.1 M carbonate buffer at pH = 9.5. An aliquot of a freshly-prepared anhydrous DMSO solution of FITC (10.0 mg mL -1 ) was added drop-wise to the pLL solution at a final fluorophore : pLL chain molar ratio of 1 : 1. The reaction mixture was kept at room temperature in the dark for 4 hours, then purified by multiple washing steps with water using a centrifugal filter (Millipore, Amicon Ultra, MWCO 10 kDa) to remove any unreacted FITC. The FITC-pLL stock solution was split into aliquots and stored at -20 °C until use.

Passivation of glass coverslips

Glass coverslips were passivated to limit coacervate wetting. Ethanol-rinsed glass coverslips were first incubated for 48 hours in a toluene solution containing 5 wt% of 3-[methoxy(polyethyleneoxy)propyl]trimethoxysilane. Coverslips were subsequently rinsed with toluene, ethanol, and water, and then immersed in an aqueous BSA solution (10 wt%) for another 24 hours, washed with water, dried with compressed air, then assembled into a capillary chamber with a UV-curing glue.

Phase behaviour of GOx and DEAE-dextran at thermodynamic equilibrium

The influence of the protein and polysaccharide concentration, pH and ionic strength on the formation of coacervates micro-droplets was investigated by monitoring changes in the absorbance at λ = 700 nm. The optimal protein : polyelectrolyte ratio was determined at pH 7.4 on 25 μL of GOx/DEAE-dextran solutions produced at fixed GOx (0.25 mg mL -1 ) and varying DEAE-dextran (0-0.15 mg mL -1 ) concentrations in phosphate buffer (2.5 mM) by mixing in Milli-Q water aliquots of aqueous stock solutions of GOx (6 mg mL -1 ), phosphate buffer (50 mM), and DEAE-dextran (1.8 mg mL -1 ). GOx/DEAE-dextran solutions were prepared in a similar way in the presence of NaCl (10-100 mM final concentrations) to assess the influence of the ionic strength on phase separation. The absorbance of each sample was measured in a 384-well plate (Falcon, flat bottom) using a microplate spectrophotometer (Molecular Devices). All experiments were performed in triplicate and the average values and standard deviations reported.

The pH range of complex coacervation was determined at fixed GOx (0.20, 0.25 and 0.35 mg mL -1 ) and DEAE-dextran (0.03, 0.04, 0.05 mg mL -1 ) concentrations. 200 μL of GOx/DEAE-dextran solutions were prepared in a UV/vis plastic cuvette by mixing aliquots of stock solutions of GOx (6 mg mL -1 ) and DEAEdextran (1.8 mg mL -1 ) in Milli-Q water, then the pH was adjusted to pH = 10.0 using NaOH (0.1 M), and subsequently dropped by gradual additions of HCl (0.05 M). After each HCl addition, the pH was measured using a calibrated pH meter (Mettler Toledo) equipped with a microelectrode (SI Analytics), and the absorbance of the solution was monitored at 700 nm on a UV-vis spectrometer (Ocean Optics). DEAE-dextran titration was performed on a 3.6 mg mL -1 DEAE-dextran solution prepared in 4 mL Milli-Q water. The pH was initially adjusted to 11.0 using NaOH to ensure that the polysaccharide was fully deprotonated, then HCl (0.1 M) was added stepwise (2 μL steps) as titrant using a micro-pipette and the change in pH was measured using a calibrated pH meter (Mettler Toledo) equipped with a microelectrode (SI Analytics). The exact amount of positive charges was calculated using the volume and molarity of titrant, and the molar mass of each monomer, as detailed in ESI Note 1. † The charge of DEAE-dextran was then compared to that of GOx (see ESI Note 1 †).

The amount of GOx sequestered in the polyelectrolyte-rich phase at pH 7.4 was determined by preparing 200 μL of a GOx/DEAE-dextran solution (0.25 mg mL -1 GOx, 0.04 mg mL -1 DEAE-dextran, 2.5 mM phosphate buffer). The droplets suspension was incubated at room temperature for 15 min, then centrifuged at ∼20 000 × g for 15 min to separate the dense coacervate phase from the dilute supernatant. The supernatant solution was removed by pipetting, and the remaining dense coacervate phase was dissolved in 200 μL of a 1 M NaCl solution, then the concentration of GOx in each phase was measured by UV-vis spectroscopy.

Observation of GOx/DEAE-dextran coacervate micro-droplets

A freshly made GOx/DEAE-dextran solution (0.25 mg mL -1 GOx, 0.04 mg mL -1 DEAE-dextran) was adjusted to pH 5.5 or 9.5 using HCl (0.1 M) or NaOH (0.1 M) solutions; or prepared at pH 7.4 using phosphate buffer (2.5 mM). Samples were imaged ca. 15 minutes after incubation by loading an aliquot of the solution into a passivated capillary chamber (see above). When coacervates formed, the droplets were left to settle for ca. 2 minutes on the glass coverslip before imaging. Optical microscopy imaging was performed on a Leica DMI 4000B inverted microscope equipped with a ×63 oil immersion lens (HCX PL APO, 1.4 NA) using the MicroManager software. Images were processed using ImageJ.

To monitor the fusion behaviour of the droplets, a freshly prepared GOx/DEAE-dextran droplets suspension (0.25 mg mL -1 GOx, 0.04 mg mL -1 DEAE-dextran, 2.5 mM phosphate buffer, pH 7.4) was rapidly loaded into a custom-made passivated capillary chamber. Several coalescence events between contacting droplets could then be observed in real time by optical microscopy imaging. Movies of the coalescence process were acquired on the Leica DMI 4000B inverted microscope equipped with a ×63 oil immersion lens (HCX PL APO, 1.4 NA) using the MicroManager software. The aspect ratio (long axis, L, to short axis, l) of 10 individual droplets undergoing coalescence was measured as a function of time using ImageJ. The data was fitted to a monoexponential decay function using OriginLab to determine the relaxation time τ: L/l = a + b exp(-t/τ); and the characteristic length scale, R, of the droplets was the measured radius after coalescence. The relaxation time, τ, is expected to be directly proportional to the characteristic length scale, R, of the droplets according to the relation: τ ≈ (η/γ) × R (from ref. 50), which gives the inverse capillary viscosity as the ratio of the viscosity of the droplets, η, to surface tension, γ.

To determine the localization of coacervate components, GOx/DEAE-dextran micro-droplets were doped with RITC-GOx (0.1 μM final concentration) and FITC-DEAE-dextran (0.008 mg mL -1 final concentration) by first mixing an aliquot of the fluorescent GOx stock solution (10 μM) with GOx (0.25 mg mL -1 final GOx concentration), then adding DEAE-dextran mixed with FITC-DEAE-dextran (4 : 1 DEAE-dextran to FITC-DEAE-dextran molar ratio, 0.04 mg mL -1 final DEAE-dextran concentration). The droplets were loaded into a capillary slide and left to settle for 2 minutes before being imaged by confocal fluorescence microscopy on a Leica SP2 confocal laser scanning microscope attached to a Leica DMI RE2 inverted microscope using a ×63 oil immersion lens (HCX PL APO, 1.4 NA). Excitation (and emission) wavelengths were set to 488 nm (emission: 500-550 nm) and 543 nm (emission: 550-650 nm) to monitor FITC-DEAE-dextran and RITC-GOx fluorescence, respectively. Images were processed using ImageJ.

Enzyme-mediated condensation/dissolution of GOx/DEAE-dextran coacervate micro-droplets

The kinetics of GOx/DEAE-dextran coacervate droplets assembly/disassembly in response to GOx activity was first monitored by turbidity measurements at 25 °C after sequential or single-step addition of glucose. Turbidity measurements were carried out at 700 nm on a microplate spectrophotometer (Molecular Devices) using a 384-well plate (Falcon, flat bottom). A freshly made GOx/DEAE-dextran solution (0.25 mg mL -1 GOx, 0.04 mg mL -1 DEAE-dextran) was adjusted to pH 10.2 with NaOH (0.1 M) to produce a clear solution of disassembled droplets. For the sequential glucose addition experiment, 3 μL of a 10 mM glucose stock solution were first added to the GOx/DEAE-dextran solution (final glucose concentration of 0.6 mM; total final volume of 50 μL) and absorption values were recorded every minute for 80 min. At t = 80 min, another 3 μL of a 10 mM glucose stock solution were added and absorption values recorded every minute for another 80 min. For single-step glucose additions, a fixed amount of glucose (0 mM to 2 mM final concentration at 0.2 mM intervals) was added to the GOx/DEAE-dextran solution (total final volume of 50 μL). The absorbance of each sample was then measured every minute for 160 min. Cycles of transient coacervates assembly were monitored on 50 μL of GOx/DEAE-dextran solution adjusted to pH 10 and supplied with 5 mM glucose (final concentration). Absorption values were recorded every minute for 60 minutes, then 0.5 μL of 1 M NaOH was added (10 mM NaOH final concentration) and values recorded for another ∼45 min. The process was repeated until the absorption did not re-increase spontaneously after NaOH addition (indicating that all glucose had been consumed). All turbidity experiments were performed in triplicate and the average values and standard deviations reported.

We monitored the time-dependent evolution of the solution's pH after glucose addition as follows. A freshly made GOx/DEAE-dextran solution (0.25 mg mL -1 GOx, 0.04 mg mL -1 DEAE-dextran) was adjusted to pH 10.2 with NaOH (0.1 M) to produce a clear solution of disassembled droplets. To this mixture were added either 1.2 μL of a 100 mM glucose stock solution (total final volume of 200 μL; final glucose concentration of 0.6 mM), or 2.8 μL of a 100 mM glucose stock solution (total final volume of 200 μL; final glucose concentration of 1.4 mM), and the pH was monitored after ∼30 s of equilibration using a calibrated pH meter (Mettler Toledo) equipped with a microelectrode (SI Analytics) under continuous gentle magnetic stirring. pH values were recorded every minute from t = 0 to t = 90 min. Control experiment without any added glucose was also performed. Time-dependent optical microscopy of coacervate assembly/disassembly after glucose addition was performed as follows. A freshly prepared GOx/DEAE-dextran droplets suspension (0.40 mg mL -1 GOx, 0.064 mg mL -1 DEAE-dextran) was adjusted to pH 10.2 and supplied with glucose (0.5 mM or 25 mM final concentration; total final volume of 200 μL), then 5 μL of the suspension were rapidly loaded into a custommade passivated capillary chamber that was hermetically sealed with UV-curing glue. The sample preparation took ca. 2 min after glucose addition. We here used higher GOx (0.40 mg mL -1 ) and DEAE-dextran (0.064 mg mL -1 ) concentrations compared to turbidity measurements (0.25 mg mL -1 GOx, 0.04 mg mL -1 DEAE-dextran) to form larger droplets that were easier to observe, but we kept the GOx : DEAE-dextran ratio constant. To accelerate the transient coacervate formation process, we also used a higher final glucose concentration (25 mM) compared to turbidity experiments (2 mM). Optical microscopy images of the samples were then acquired every 10 s for 45 min on a Leica DMI 4000B inverted microscope equipped with a ×63 oil immersion lens (HCX PL APO, 1.4 NA) using the MicroManager software. Images were processed using ImageJ.

Multiphase coacervate preparation and characterization at thermodynamic equilibrium

Each coacervate phase was first prepared separately as follows. 10 μL of a GOx/DEAE-dextran coacervate droplets suspension was produced at 4.4 mg mL -1 GOx, 0.70 mg mL -1 DEAE-dextran and 2.5 mM phosphate buffer final concentrations by mixing in Milli-Q water aliquots of aqueous stock solutions of GOx (6 mg mL -1 ), phosphate buffer (50 mM), and DEAE-dextran (1.8 mg mL -1 ). Similarly, 10 μL of ATP/pLL coacervate droplets suspension was produced at 10 mM ATP (corresponding to 5.1 mg mL -1 ), 10 mM pLL (monomer concentration, corresponding to 1.4 mg mL -1 ) and 2.5 mM phosphate buffer final concentrations by mixing in Milli-Q water aliquots of aqueous stock solutions of ATP (50 mM), pLL (100 mM) and phosphate buffer (50 mM). The GOx/DEAE-dextran suspension was then added to the ATP/pLL suspension at 1 : 1 volume ratio (total final volume of 20 μL) to give final GOx and DEAE-dextran concentrations of 2.2 mg mL -1 and 0.35 mg mL -1 , respectively, and the obtained turbid suspension was gently mixed. These higher concentrations compared to single-phase droplets were required to observe a sufficiently thick outer coacervate layer in multiphase droplets, presumably due to the lower GOx/DEAE-dextran coacervation efficiency in the presence of ATP and pLL polyions that increased the ionic strength of the solution. Samples were imaged either rapidly (to observe fusion events of the inner droplets) or after ca. 5 minutes by loading an aliquot of the solution into a custom-made passivated capillary chamber (see above). Optical microscopy imaging was performed on a Leica DMI 4000B inverted microscope equipped with a ×63 oil immersion lens (HCX PL APO, 1.4 NA) using the MicroManager software. Images were processed using ImageJ.

To determine the localization of the coacervate components, GOx/DEAE-dextran micro-droplets were doped with RITC-GOx (0.25 μM final concentration) and FITC-DEAE-dextran (0.07 mg mL -1 final concentration) by first mixing an aliquot of the fluorescent GOx stock solution (10 μM) with GOx (2.2 mg mL -1 final GOx concentration), then adding DEAE-dextran mixed with FITC-DEAE-dextran (4 : 1 DEAE-dextran to FITC-DEAE-dextran molar ratio, 0.35 mg mL -1 final DEAE-dextran concentration). Control images were also acquired in the absence of DEAE-dextran. Alternatively, multiphase droplets were prepared using ATP/pLL droplets doped with FITC-pLL (2 mM, i.e. 0.25 mg mL -1 final concentration). Multiphase droplets were then formed as described above then loaded into a capillary slide and left to settle for 2 minutes before being imaged by confocal fluorescence microscopy on a Leica SP2 confocal laser scanning microscope attached to a Leica DMI RE2 inverted microscope using a ×63 oil immersion lens (HCX PL APO, 1.4 NA). Excitation (and emission) wavelengths were set to 488 nm (emission: 500-540 nm) and 543 nm (emission: 550-650 nm) to monitor FITC-DEAE-dextran and RITC-GOx fluorescence, respectively. Images were processed using ImageJ. We assessed the influence of the ionic strength on ATP/pLL coacervate microdroplets as for the GOx/DEAEdextran coacervates. Briefly, 50 μL of ATP/pLL coacervates were produced at pH 7.4 (10 mM ATP, 10 mM pLL, 2.5 mM phosphate buffer) in the presence of increasing amounts of NaCl (0-400 mM), and the turbidity monitored at 700 nm in a 384-well plate (Falcon, flat bottom) using a microplate spectrophotometer (Molecular Devices). Experiments were performed in triplicate and the average values and standard deviations reported.

Enzyme-regulated dynamic assembly/disassembly of multiphase droplets

GOx/DEAE-dextran (2.2 mg mL -1 GOx, 0.35 mg mL -1 DEAE-dextran) and ATP/pLL (10 mM ATP, 10 mM pLL) solutions were freshly prepared separately in Milli-Q water and mixed at 1 : 1 volume ratio. The pH was adjusted to ∼10.2 using NaOH and the solutions supplied with glucose (25 mM or 100 mM final concentration; total final volume of 20 μL), then 5 μL of the solution were rapidly loaded into a custom-made passivated capillary chamber that was hermetically sealed with UV-curing glue. The sample preparation took ca. 2 min after glucose addition. Optical microscopy images of the samples were then acquired every 10 s for 57 min on a Leica DMI 4000B inverted microscope equipped with a ×63 oil immersion lens (HCX PL APO, 1.4 NA) using the MicroManager software. Images were processed using ImageJ.

  ), as expected for a liquid-like state. The droplets contained ∼80% of GOx molecules (ESI Fig. S3 †) and readily disassembled upon increasing the ionic strength (ESI Fig. S4 †), which confirmed the central role of electrostatic interactions in the phase separation process. Confocal fluorescence microscopy of coacervate droplets doped with fluorescein isothiocyanate (FITC)-labelled DEAE-dextran and rhodamine isothiocyanate (RITC)-tagged GOx further showed that the two polyions distributed homogeneously throughout the droplets (ESI Fig. S5 †).

Fig 1

 1 Fig 1 (a) Scheme of the complex coacervation process between glucose oxidase (GOx) and diethylaminoethyl (DEAE)dextran. (b) Plot of the absorbance at 700 nm of solutions of GOx (0.25 mg mL -1 ) in the presence of varying concentrations of DEAE-dextran in phosphate buffer (2.5 mM, pH 7.4). The maximum turbidity (red dotted line) corresponds to the optimal ratio for coacervation. (c) Plot of the absorbance at 700 nm of a solution of GOx (0.25 mg mL -1 ) and DEAE-dextran (0.04 mg mL -1 ) as a function of the pH. The maximum turbidity (red dotted line) corresponds to the optimal pH for coacervation. On (b) and (c), the DEAE-dextran (positive) : GOx (negative) molar charge ratio is also reported (see ESI Note 1 †). Error bars represent the standard deviation of three independent repeats. (d) Optical microscopy images of GOx/DEAE-dextran mixtures ([GOx] = 0.25 mg mL -1 , [DEAE-dextran] = 0.04 mg mL -1 ) prepared at different pH, as indicated, and corresponding schematic representations of the charge conditions on both polyions. Scale bars, 10 μm.

Fig. 3

 3 Fig. 3 (a-c) Optical (a) and confocal fluorescence (b and c) microscopy images of multiphase ATP/pLL-in-GOx/DEAEdextran coacervate micro-droplets doped with RITC-GOx (b, red fluorescence) and FITC-DEAE-dextran (c, green fluorescence) in phosphate buffer (2.5 mM, pH 7.4). False coloring to magenta and cyan was used, respectively. Scale bars, 20 μm. (d) Schematic representation of GOx-mediated dynamic formation of multiphase coacervate droplets. At low glucose concentration, stable multiphase droplets are formed as the pH stabilizes to physiological values while higher glucose turnover gives rise to a transient multiphase droplet organization. (e and f) Optical microscopy snapshots of ATP/pLL/GOx/DEAE-dextran mixtures produced at pH 10.2 at different times after addition of 25 mM (e) or 100 mM (f) glucose, showing the formation of stable or transient multiphase coacervate droplets, respectively. Scale bars, 20 μm. Insets show zoomed areas (white box). Scale bars, 5 μm.

Acknowledgements

This work was supported by funding from IdEx Bordeaux (ANR-10-IDEX-03-02), a program of the French government managed by the Agence Nationale de la Recherche, and from Région Nouvelle-Aquitaine.

Author contributions

N. M. designed the project, N. M. and H. K. designed the experiments, H. K. and M. J. S. performed the experiments, N. M. and H. K. analysed the results and wrote the manuscript.

Conflicts of interest

The authors declare no conflict of interest.