Emeric Bouin
email: bouin@ceremade.dauphine.fr

Jérôme Coville
email: jerome.coville@inrae.fr

Xi Zhang

Acceleration or finite speed propagation in weakly monostable reaction-diffusion equations

Keywords: reaction-diffusion equations, propagation phenomena, weakly monostable equations

This paper focuses on propagation phenomena in reaction-diffusion equations with a weakly monostable nonlinearity. The reaction term can be seen as an intermediate between the classical logistic one (or Fisher-KPP) and the standard weak Allee effect one. We investigate the effect of the decay rate of the initial data on the propagation rate. When the right tail of the initial data is sub-exponential, finite speed propagation and acceleration may happen and we derive the exact separation between the two situations. When the initial data is sub-exponentially unbounded, acceleration unconditionally occurs. Estimates for the locations of the level sets are expressed in terms of the decay of the initial data. In addition, sharp exponents of acceleration for initial data with sub-exponential and algebraic tails are given. Numerical simulations are presented to illustrate the above findings.

Introduction

In this paper, we study rates of invasion in the following one-dimensional reaction-diffusion equations    ut(t, x) = uxx(t, x) + f (u(t, x)), t > 0, x ∈ R, u(0, x) = u0(x) ≥ 0,

x ∈ R.

(1.1)

Hypothesis 1.1. The non-linearity f ∈ C 1 ([0, 1], R) is of the weakly monostable type, in the sense that f (0) = f (1) = 0, f (s) > 0 for any s ∈ (0, 1), f (1) < 0, and there exists s0 ∈ (0, 1), K ≥ 0, α > 0 and r > 0 such that f (s) ≤ r s (1 + | ln s|) α for all s ∈ (0, 1), (1.2) and f (s) ≥ r s (1 + | ln s|) α (1 -Ks) for all s ∈ (0, s0].

(1.3) After Kolmogorov, Petrovskii and Piskunov [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologigue[END_REF], and Fisher [START_REF] Ronald | The wave of advance of advantageous genes[END_REF], the classical monostable equation is equation (1.1) with Fisher-KPP type nonlinearity, that is, f (0) = f (1) = 0 and 0 < f (s) ≤ f (0)s for all s ∈ (0, 1). (1.4) In population dynamics, this type of non-linearity is commonly used to model the situation where growth per capita is maximal at low densities. The decay rate of the initial data at infinity is crucially important for the propagation problem. For the Fisher-KPP equation with front-like initial data, initial data u0 with exponentially bounded decay, that is, lim x→+∞ u0(x)e εx < ∞ for some ε, (1.5) lead to finite propagation speed [12,[START_REF] Mallordy | A parabolic equation of the kpp type in higher dimensions[END_REF]. On the other hand, for an exponentially unbounded initial data, meaning that condition (1.5) is not met, or lim x→+∞ u0(x)e εx > +∞ for any ε,

Hamel and Roques [20] have presented evidence of acceleration of the solution to the Fisher-KPP equation. They also provided an expression of the locations of level sets based on the decay of the initial data. We refer to references [3, 7-9, 13, 15, 17, 21, 22, 25] for the further results about propagation in KPP equations.

When an Allee effect occurs, meaning that the per capita growth is no longer maximal at low densities, the KPP assumption (1.4) becomes unrealistic. Hence, incorporating the Allee effect into models becomes necessary.

An acceleration phenomenon may take place in the degenerate situation f (0) = 0. Indeed, when the initial data is front-like and the nonlinearity f (s) ∼ rs α+1 with α > 0 as s → 0 + , Alfaro [2] has studied the balance between the decay rate of the initial data at infinity and the weak Allee effect and found that for exponentially unbounded tails but lighter than algebraic acceleration does not occur in the presence of the Allee effect, which is in contrast with the KPP equation. Similarly to the KPP situation, the initial data with exponentially bounded decay lead to a finite propagation speed [START_REF] Kay | Comparison theorems and variable speed waves for a scalar reaction-diffusion equation[END_REF]28]. On the other hand, algebraic decay leads to acceleration despite the Allee effect and the position of the level sets of u(t, •) as t → ∞ propagates polynomially fast [2,26]. We refer to references [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal-diffusion[END_REF]6,[START_REF] Ascher | Implicit-explicit runge-kutta methods for timedependent partial differential equations[END_REF]19] for other kinds of Allee effect.

It is worth mentioning that these results about propagation phenomena in degenerate monostable equations are based on the assumption f (s) ∼ rs α+1 with some α > 0 and r > 0 as s → 0 + . This assumption is used to quantify the degeneracy. In this paper, we also take into account that the growth per capita is small at small densities, but we quantify the degeneracy by a weakly monostable type nonlinearity [START_REF] Achleitner | Traveling waves for a bistable equation with nonlocal-diffusion[END_REF]. Notice that such nonlinearity is between the KPP type and the Allee effect type near the right side of zero point, see Figure 1. Thus, this type of nonlinear term fill an existing gap between two classical nonlinearities.

f satisfying f (s) ∼ r s | ln s| α with α > 0 and r > 0 as s → 0 + , like f (s) = r s (1+| ln s|) α (1 -s) for s ∈ (0,
To describe the propagation speed, we introduce three notations. For any λ ∈ (0, 1), the (upper) level set of u(t, x) is defined by

E λ (t) := {x ∈ R : u(t, x) ≥ λ}.
Let x λ (t) be the largest element of level set of u(t, x) defined by

x λ (t) := sup E λ (t).
For any subset Λ ⊂ (0, 1], we set

u -1 0 {Λ} := {x ∈ R : u0(x) ∈ Λ} the inverse image of Λ by u0. 2 0 s f (s) s r r | ln s| α rs α 0 1 s f (s) rs(1 -s) r s(1-s) (1+| ln s|) α rs α+1 (1 -s)
u0 > 0 in R, lim inf x→-∞ u0 > 0, lim x→+∞ u0 = 0.
In this paper, we always denote by u(t, x) the solution to (1.1) with initial data u0. We mainly consider the following types of initial data:

• Sub-exponentially bounded for large x, that is, there exist x0 > 0 such that, for any x > x0, u0(x) e -µx β , with β ∈ (0, 1) and µ > 0.

• Sub-exponential decay for large x, that is, there exist x0 > 0 such that, for any x > x0, u0(x) e -µx β , with β ∈ (0, 1) and µ > 01 .

• Algebraic decay for large x, that is, there exists x0 > 1 such that, for any x > x0, u0(x)

1 x β ,
with β > 0.

• Initial data u0 that decay as a negative power of ln x for large x, that is, there exists x0 > e such that, for

any x > x0, u0(x) (ln x) -β , with β > 0.
Our first result shows that for sub-exponentially bounded initial data, acceleration does not happen.

Theorem 1.3. Let α > 0 and β > 0 be such that

β ≥ 1 α + 1 .
Assume that the non-linearity f and the initial data u0 satisfy Hypotheses 1.1 and 1.2, respectively. Assume that there exist x0 > 0 and µ > 0 such that u0(x) e -µx β for any x ≥ x0.

(1.7)

Then, for any λ ∈ (0, 1), there exist some positive constants c and a time T λ such that

Γ < x λ (t) t < c for any t > T λ . (1.8)
Now, we turn to cases where it is assumed that the initial data u0 decay more slowly than e -εx 1 α+1 as x → +∞ for any ε > 0, that is,

∀ε > 0, ∃xε ∈ R, u0(x) ≥ e -εx 1 α+1 in [xε, +∞). (1.9) Let us denote ϕ0(x) := -ln u0(x) ≥ 0. (1.10) Notice that if u0 is C 2 , then we can get ϕ 0 (x) = - u 0 u0 (x) and ϕ 0 (x) = - u 0 u0 (x).
Observe that if we assume that ϕ 0 (x) = o(ϕ -α 0 (x)) as x → +∞, then condition (1.9) is fulfilled. For such initial data, we have the following result. Lemma 1.4. Assume that the non-linearity f and the initial data u0 satisfy Hypotheses 1.1 and 1.2, respectively.

Assume that u0 is of class C 2 and non-increasing on [ξ0, +∞) for some ξ0 > 0, and ϕ 0 (x) = o(ϕ -α 0 (x)) and ϕ 0 (x) = o(ϕ 0 (x)) as x → +∞.

(1.11)

Then, for any fixed λ ∈ (0, 1) and small ε > 0, there is a time T λ,ε such that

E λ (t) ⊂ u -1 0 e -[(r+ε)(α+1)t] 1 α+1 , e -[(r-ε)(α+1)t] 1 α+1
for any t > T λ,ε .

It is easy to check that initial data u0(x) e -µx β satisfy (1.11) in the regime β < 1 α+1 . Thus, according to the above lemma, we obtain the following theorem.

Theorem 1.5. Let α > 0 and β > 0 be such that

β < 1 α + 1 .
Assume that the nonlinearity f and the initial data u0 satisfy Hypotheses 1.1 and 1.2, respectively. Assume that there exists x0 > 0 and µ > 0 such that u0(x) e -µx β for any x ≥ x0.

Then, for any λ ∈ (0, 1) and ε > 0, there exists a time T λ,ε such that 2

x λ (t) λ,ε,µ t

1 β(α+1) for any t > T λ,ε . 1 x λ (t) t 1 β(1+α) x λ (t) t α = 1 β -1 β α Figure 2:
The separation for sub-exponential decay initial data case.

For initial data with algebraic tails, that is u0(x)

x -β for β > 0, by Lemma 1.4, we just obtain a rough estimate:

C1e 1 β ((r-ε)(α+1)t) 1 α+1 ≤ x λ (t) ≤ C2e 1 β ((r+ε)(α+1)t) 1 α+1
for some constants C1 and C2. Notice that the position of the level set depends strongly on the constant ε. Hence the estimate is not enough for such initial data.

To get an exact estimate of the position of the level sets, we add a concavity assumption, that we believe not to be a huge restriction given that classical sub-exponentials are usually log-concave functions.

Lemma 1.6. Assume that the nonlinearity f and the initial data satisfy Hypotheses 1.1 and 1.2, respectively.

Assume that u0 is of class C2 and nonincreasing on [ξ0, +∞) for some ξ0 > 0, and

ϕ 0 (x) = o(ϕ -α 0 (x)) and ϕ 0 (x) = o(ϕ 0 (x)) as x → +∞.
Assume that ϕ 0 (x) ≤ 0 for large x.

(1.12)

Then, for any λ ∈ (0, 1), there are two constants C λ > 0 and Cλ > 0 and a time T λ such that

E λ (t) ⊂ u -1 0 Cλ e -[r(α+1)t] 1 α+1 , C λ e -[r(α+1)t] 1 α+1
for any t > T λ .

(1.13)

We point out that (1.12) is used only in the proof of the lower bound. Our approach can be used to prove the exact result in the KPP situation of [20].

Equipped with the above lemma, we can get exact estimates for the level sets of the solution to equation (1.1)

with the algebraic decay initial data. We check the assumptions in Lemma 1.6 and obtain the following theorem.

Theorem 1.7. Assume that the nonlinearity f and the initial data satisfy Hypotheses 1.1 and 1.2, respectively.

Assume that there exist x0 > 1 and β > 0 such that u0(x) 1

x β for any x ≥ x0.

Then, for any λ ∈ (0, 1), there exists a time T λ such that

x λ (t) λ e 1 β [r(α+1)t] 1 α+1
for any t > T λ .

Observe that when α = 0, one recovers the rate of the KPP situation [20,22]. For the degenerate monostable case, Alfaro [2] shows that if

f (s) = s 1+α (1 -s) then x λ (t) t 1 αβ
for t large enough, where 0 < α < 1 β . Thanks to Lemma 1.6, we can also get the following theorem for the initial data u0(x) (ln x) -β . Theorem 1.8. Assume that the nonlinearity f and the initial data u0 satisfy Hypotheses 1.1 and 1.2, respectively.

Assume that there exists x0 > e and β > 0 such that u0(x) (ln x) -β for any x ≥ x0.

(1.14)

Then, for any x(t) ∈ E λ (t), there exists a time T λ , such that

ln x λ (t) λ e 1 β [r(α+1)t] 1 α+1
for any t > T λ .

Remark 1.9. One can obtain Lemma 1.4 and Theorem 1.5 under the weaker hypothesis

f (s) ∼ r s (1 + | ln s|) α as s → 0 + ,
for some r > 0 and α > 0. However, Lemma 1.6 and Theorems 1.7 and 1.8 need crucially Hypothesis 1.1 with both precise bounds (1.2) and (1.3) for f . An insight can be easily seen in the proofs in Sections 3 and 4.

Nevertheless, Lemmas 1.4 and 1.6 and Theorems 1.5, 1.7 and 1.8, are true under Hypothesis 1.1 where (1.3) is replaced by

f (s) ≥ r s (1 + | ln s|) α (1 -Ks δ ) for any s ∈ (0, s0],
for some δ > 0, s0 ∈ (0, 1) and K ≥ 0. Their proofs are similar to the one we provide in Sections 3 and 4 but a bit messier so we have chosen to stick to δ = 1 for the sake of readability.

The rest of this paper is organized as follows. In Section 2, we shall prove that the solution to equation (1.1), starting from an exponentially unbounded initial data, propagates at constant speed. In Section 3 and Section 4, we provide the proof of the main results, Lemma 1.4 and Lemma 1.6, respectively. In Section 5, some numerical simulations shall be given to illustrate our main results. This paper is the first part of our work on weakly monostable equations; a companion paper [11] with nonlocal dispersal follows. In this latter paper, we have proved the existence and nonexistence of traveling waves, and studied the effect of the tails of the dispersal kernel on the propagation rate. Exact rates of invasion have been provided for the sub-exponential and algebraic tails.

2 Finite speed propagation: Proof of Theorem 1.3

In this section, we prove Theorem 1.3: the level sets of the solution to (1.1) moves at a constant speed.

As in [2, Theorem 2.3], we can also obtain that, for any λ ∈ (0, 1), there is a time t λ > 0 and Γ > 0 such that ∅ = E λ (t) ⊂ (Γt, +∞) for any t > t λ .

(2.1) Indeed, we consider the equation

   vt -vxx -r1v 2 (1 -v) = 0, t > 0, x ∈ R, v(0, x) = v0 ≥ 0, (2.2)
where the initial data v0(x) = inf y≤0 u0(y)1 (-∞,0) (x) and r1 > 0 small enough so that r1s 2 (1 -s) ≤ f (s) for all s ∈ (0, 1). According to [29], the solution v(t, x) to (2.2) satisfies limt→∞ inf x≤Γt v(t, x) = 1 for some Γ > 0. It follows from the comparison principle that propagation of u(t, x) is at least linear, that is,

lim inf t→∞ x≤Γt u(t, x) = 1. (2.3)
On the other hand, we can reproduce the proof of [20, Theorem 1.1 part a], which does not require the KPP assumption, and get lim x→+∞ u(t, x) = 0 for any t ≥ 0.

(2.4) Thus, combining (2.3) and (2.4), we can conclude (2.1).

Inspired by [2], for the initial data with sub-exponential decay, we use a suitable shifted profile which construction now follows. Take α > 0 and β > 0 such that

β ≥ 1 α + 1 .
Let us define

w(z) := M e -µz p (≤ 1) for z ≥ z0 := ln M µ 1 p , (2.5)
where p := 1 α+1 < 1 and M > e.

Lemma 2.1. Assume that f satisfies Hypothesis 1.1. Then, for any M > e, there is c > 0 such that

w (z) + cw (z) + f (w(z)) ≤ 0, ∀z ≥ z0.
Proof. By definition of w, we have, for z ≥ z0,

w (z) = -µpz p-1 w(z) and w = µp(1 -p)z p-2 + µ 2 p 2 z 2(p-1) w(z). Since 1 -ln M + µz p ≥ µ 2 z p for any z ≥ ( 2(ln M -1) µ ) 1
p , then we have, for all z ≥ z1 := max{z0, ( 2(ln M -1)

µ ) 1 p }, w (z) + cw (z) + f (w(z)) ≤ µw(z) µp 2 z 2(1-p) + p(1 -p) z 2-p - cp z 1-p + r µ(1 -ln M + µz p ) α ≤ µw(z) µp 2 z 2(1-p) + p(1 -p) z 2-p - cp -2 α r µ α+1 z 1-p .
Choosing c > 2 α r µ α+1 p , the above is nonpositive for z large enough, say z ≥ z2. On the other hand, for the remaining region z0 ≤ z ≤ z2, we have

w (z) + cw (z) + f (w(z)) ≤ µw(z) µp 2 z 2(1-p) + p(1 -p) z 2-p - cp z 1-p + r µ(1 -ln M + µz p ) α ≤ µw(z) µp 2 z 2(1-p) 0 + p(1 -p) z 2-p 0 - cp z 1-p 2 + r µ(1 -ln M + µz p 0 ) α ,
by taking c large enough so that the above is nonpositive.

Equipped with the above lemma, we can construct a supersolution to (1.1). Let M = max{e, e x β 0 ||u0||∞} and define

v(t, x) :=    w(x -x0 -ct), x > ct + x0 + z0, 1, x ≤ ct + x0 + z0,
where c and x0 is from the above lemma and (1.7) respectively. We claim that v(t, x) is a supersolution for (1.1)

for any x ∈ R and t > 0. Indeed, it is enough to check it when v(t, x) < 1, that is, x > ct + x0 + z0. It follows from the above lemma that

vt -vxx -f (v) ≥ -(cw -w + r w (1 -ln w) α ) ≥ 0. For x > x0 + z0, since p = 1
α+1 ≤ β and (1.7), then we have

v(0, x) = M e -µ(x-x 0 ) p ≥ e x β 0 ||u0||∞e -µx β ≥ u0(x).
On the other hand, since u0 ≤ 1, for x ≤ x0 + z0, we have v(0, x) = 1 ≥ u0(x).

In the regime β ≥ 1 α+1 , the comparison principle then implies that for all t > 0 and x ∈ R, we have

u(t, x) ≤ v(t, x) ≤ w(x -x0 -ct).
Therefore, for any λ ∈ (0, 1), there is T λ large enough such that for all t > T λ , we have

x λ (t) ≤ x0 + 1 µ ln M λ 1 p + ct ≤ (c + 1)t,
which gives the upper bound of (1.8). Together with (2.1), the proof of Theorem 1.3 is complete.

3 The acceleration regime: Proof of Lemma 1.4

In this section, we prove Lemma 1.4: the level sets of solution to the equation (1.1) with front-like initial data that is sub-exponentially unbounded move by accelerating, and the locations of the level sets are expressed in terms of the decay of the initial data.

The long-time behaviour of the solution to the Cauchy problem (1.1) is captured approximately by the ODE

     wt = ρ w (1 -ln w) α , t > 0, x ∈ R, w(0, x) = u0(x) ≥ 0, x ∈ R, (3.1)
where ρ > 0 is to be determined. We solve the above ODE and obtain

w(t, x) = exp 1 -(1 + ϕ0(x)) α+1 -ρ(α + 1)t 1 α+1 , (3.2)
where ϕ0 is defined by (1.10). Notice that w(t, x) ≥ w(0, x) = u0(x) since w(x, •) is increasing for each x ∈ R.

Let us define

x0(t) := sup x ∈ R : u0(x) = exp 1 -ρ(α + 1)t + 1 1 α+1 . (3.3)
Observe that w(t, x0(t)) = 1 and 0 < w(t, x) ≤ 1 for x ≥ x0(t). For any x ≥ x0(t) and t > 0, we have

wx = - w (1 -ln w) α ϕ 0 (1 + ϕ0) α , (3.4)
and

wxx = w (1 -ln w) α (ϕ 0 ) 2 (1 + ϕ0) 2α (1 -ln w) -α + α(1 -ln w) -(α+1) -α(1 + ϕ0) -(α+1) -ϕ 0 (1 + ϕ0) α . (3.5)
For wxx, we have the following estimate. Lemma 3.1. Let u0 such that ϕ0 = -ln u0 satisfies ϕ 0 = o(ϕ -α 0 ) and ϕ 0 = o(ϕ 0 ) as x → +∞. Then, for any small ε > 0, there exists t # > 0, depending on ε, such that |wxx| < ε w (1 -ln w) α for any x ≥ x0(t) and t ≥ t # .

(3.6)

Proof. Since 0 < u0(x) ≤ w(t, x) ≤ 1, we have

0 < (1 + ϕ0) -(α+1) ≤ (1 -ln w) -(α+1) ≤ 1.
It follows from 0 < w ≤ 1 for all x ≥ x0(t) and t > 0 that, for any x ≥ x0(t) and t > 0, we have

0 < (1 -ln w) -α + α(1 -ln w) -(α+1) -α(1 + ϕ0) -(α+1) < 2. (3.7)
In view of the definition (3.3) of x0(t), since u0 is nonincreasing and limx→+∞ u0 = 0, we have x0(t) → +∞ as t → ∞. For any small ε > 0, it follows from the assumption on ϕ 0 (x) that there exists t # > 0 such that for

x > x0(t) and t ≥ t # , we have

|ϕ 0 (x)(1 + ϕ0(x)) α | < ε 4 .
On the other hand, it follows from ϕ 0 (x) = o(ϕ 0 (x)) that there exists X such that for x > x0(t) ≥ X and t ≥ t # , up to enlarge t # if necessary, we have

|ϕ 0 (x)(1 + ϕ0(x)) α | < ε/2.
Therefore, by collecting the above estimates, we have, for any x ≥ x0(t) and t ≥ t # ,

|wxx(t, x)| ≤ w(t, x) (1 -ln w(t, x)) α (ϕ 0 (x)(1 + ϕ0(x)) α ) 2 (1 -ln w(t, x)) -α + α 1 -ln w(t, x) -(α+1) -α(1 + ϕ0(x)) -(α+1) + |ϕ 0 (x)(1 + ϕ0(x)) α | ≤ w(t, x) (1 -ln w(t, x)) α ( ε 4 × 2 + ε 2 ) = ε w(t, x) (1 -ln w(t, x)) α ,
which gives the estimate (3.6). This completes the proof.

Here we present a lemma, which will play a key role in the proof of Lemma 1.6. Lemma 3.2. Let u0 such that ϕ0 = -ln u0 satisfies ϕ 0 = o(ϕ -α 0 ) and ϕ 0 = o(ϕ 0 ) as x → +∞. Then there is t 1 > 0 such that, for x ≥ x0(t) and t ≥ t 1 , we have wx + wxx ≤ 0.

(3.8)

Proof. By the assumptions ϕ 0 (x) = o(ϕ -α 0 (x)) and ϕ 0 (x) = o(ϕ 0 (x)), there exists X0 such that for all x ≥ X0, we have

ϕ 0 (1 + ϕ0) α ≤ 1 4 and |ϕ 0 | ≤ 1 2 ϕ 0 .
Since x0(t) → +∞ as t → ∞, there is t 1 > 0 such that x0(t) > X0 for all t ≥ t 1 . In view of the definition (1.10) of ϕ, since u0 is a nonincreasing function, then ϕ ≥ 0. It then follows from (3.4), (3.5) and (3.7) that we have, for all x ≥ x0(t) and t ≥ t 1 ,

wx + wxx ≤ w (1 -ln w) α (1 + ϕ0) α ϕ 0 (-1 + 2ϕ 0 (1 + ϕ0) α ) + |ϕ 0 | ≤ w (1 -ln w) α (1 + ϕ0) α - 1 2 ϕ 0 + 1 2 ϕ 0 = 0.
This completes the proof.

The upper bound

In this subsection, we prove the upper bound of the level sets in Lemma 1.4 by constructing an accurate supersolution.

We define

m(t, x) =    w(t + t # , x), x ≥ x0(t + t # ), 1, x < x0(t + t # ),
where t # is defined in Lemma 3.1. Observe that m(t, x) is well defined for all t ≥ 0 and all x ∈ R, and 0 < m(t, x) ≤ 1.

Let ε > 0 be given and define

ρ = r + ε 2 . (3.9)
Now, we prove that m is a supersolution of equation (1.1).

Lemma 3.3. Let u0 such that ϕ0 = -ln u0 satisfies ϕ 0 = o(ϕ -α 0 ) and ϕ 0 = o(ϕ 0 ) as x → +∞. Then m(t, x) is a supersolution to equation (1.1) for all t > 0 and x ∈ R.

Proof. To prove m is a supersolution, we need to check that mt -mxx -f (m) ≥ 0 for all t > 0 and x ∈ R. For x < x0(t + t # ) and t > 0, since mt = mxx = f (m) = 0, we have

mt(t, x) -mxx(t, x) -f (m(t, x)) = 0.
On the other hand, for all x ≥ x0(t + t # ) and t > 0, by the definitions of m and w, we have

mt(t, x) = wt(t + t # , x) = ρ w(t + t # , x) (1 -ln w(t + t # , x)) α = r + ε 2 w(t + t # , x) (1 -ln w(t + t # , x)) α .
Thus, by Lemma 3.1 and Hypothesis 1.1, for all x ≥ x0(t + t # ) > x0(t) and t > 0, we obtain

mt(t, x) -mxx(t, x) -f (m(t, x)) = wt(t + t # , x) -wxx(t + t # , x) -f (w(t + t # , x)) ≥ r + ε 2 w(t + t # , x) (1 -ln w(t + t # , x)) α -2 w(t + t # , x) (1 -ln w(t + t # , x)) α -r w(t + t # , x) (1 -ln w(t + t # , x)) α = 0.
This completes the proof.

In view of the definition of m, for x < x0(t + t # ), since u0 ≤ 1, we have m(0, x) = 1 ≥ u0(x). For x ≥ x0(t + t # ), since w(•, x) is nondecreasing for each x ∈ R, we have m(0, x) = w(t # , x) ≥ w(0, x) = u0(x).

Thus, m(0, x) ≥ u0(x) = u(0, x) for all x ∈ R. Equipped with Lemma 3.3, it then follows from the comparison principle that m(t, x) ≥ u(t, x) for all t > 0 and x ∈ R.

(3.10) Now, we prove the upper bound in Lemma 1.4.

The proof of the upper bound. We need to prove that, for any λ ∈ (0, 1) and large time t, we have Let us pick a y ∈ E λ (t), then w(t + t # , y) ≥ λ. It follows that, by the definitions of w and ϕ0, we have

E λ (t) ⊂ u -1 0 e -[(r+ε)(α+1)t]
w(t + t # , y) = exp 1 -[(1 -ln u0(y)) α+1 -ρ(α + 1)(t + t # )] 1 α+1 ≥ λ, whence u0(y) ≥ exp 1 -ρ(α + 1)(t + t # ) + (1 -ln λ) α+1 1 α+1 . Since ρ = r + ε 2 , there is a time tλ,ε > 0 such that u0(y) ≥ e -[(r+ε)(α+1)t] 1 α+1
for any t ≥ tλ,ε ,

which gives (3.11). This completes the proof.

The lower bound

In this subsection, we explore the lower bound of the level sets of a solution to (1.1) by constructing an adequate subsolution.

Let ε > 0 be given. We take

max r - ε 2 , 3 4 r < ρ < r. (3.13)
Let us define the function g(y)

:= y(1 -M y) with M > 0. Notice that 0 ≤ g(y) ≤ g 1 2M = 1 4M , ∀y ∈ 0, 1 2M .
We define

xM (t) := sup x ∈ R : u0(x) = exp 1 -(1 + ln (2M )) α+1 + ρ(α + 1)t 1 α+1 ≥ x0(t),
where x0(t) is defined by (3.3). Observe that w(t, xM (t)) = 1 2M and w(t, x) < 1 2M for all x > xM (t). Let us define

ζ := inf x∈(-∞,ξ 1 ) u0(x),
where ξ1 := max{ξ0, x0(0)}. Notice that ζ ∈ (0, 1] according to Hypothesis 1.2, and that u0 is non-increasing on [ξ1, +∞). We select large enough M > 0 so that

M ≥ M0 := max 1 2ζ , 1 4s0
.

Then, by u0(xM

(0)) = 1 2M < ζ ≤ u0(ξ1), we have xM (0) > ξ1. Let us define v(t, x) :=      1 4M , x ≤ xM (t), g(w(t, x)), x > xM (t). (3.14)
Since M ≥ 1 4s 0 , then we have 0 < v(t, x) ≤ s0 for all t ≥ 0 and x ∈ R.

Lemma 3.4. Let u0 such that ϕ0 = -ln u0 satisfies ϕ 0 = o(ϕ -α 0 ) and ϕ 0 = o(ϕ 0 ) as x → +∞. Then there exists large enough M > 0 such that v(t, x) is a subsolution to equation (1.1) for all t > 0 and x ∈ R.

Proof. In view of the definition (3.14) of v, we obtain

vt(t, x) = ρ w(t, x) (1 -ln w(t, x)) α 1 -2M w(t, x) + . It then follows that vt(t, x) ≤      0, x ≤ xM (t), ρ w(t, x) (1 -ln w(t, x)) α 1 -2M w(t, x) , x > xM (t).
(3.15) Since 1 (1+y) α ≥ 1 -αy for any y ≥ 0, then, for any x > xM (t), we have

1 ≥ 1 -ln w 1 -ln v α = 1 1 -ln(1-M w) 1-ln w α ≥ 1 + α ln(1 -M w) 1 -ln w .
It follows from the inequality ln(1 -y) ≥ -cy for any y ∈ (0, 1 2 ), where c = 2 ln 2, that, for any x > xM (t), we have

1 ≥ 1 -ln w 1 -ln v α ≥ 1 -αcM w 1 -ln w , from 0 < M w < 1 2 . Thus, since f (s) ≥ r s (1-ln s) α (1 -Ks)
for s ∈ (0, s0] and 0 < v(t, x) ≤ s0 for all t ≥ 0 and x ∈ R, when x > xM (t), we have

f (v(t, x)) ≥ r v(t, x) (1 -ln v(t, x)) α (1 -Kv(t, x)) ≥ r w(t, x)(1 -M w(t, x)) (1 -ln w(t, x)) α 1 -ln w(t, x) 1 -ln v(t, x) α 1 -Kw(t, x) ≥ r w(t, x) (1 -ln w(t, x)) α 1 -(M + K)w(t, x) -αcM w(t, x) 1 -ln w(t, x)
-αcKM 2 w 3 (t, x) 1 -ln w(t, x) .

(3.16)

On the other hand, when x ≤ xM (t), we have and

f (v(t, x)) ≥ 0, (3.17
vxx(t, x) = (1 -2M w(t, x)) wxx(t, x) -2M w 2 x (t, x) for x > xM (t).
In view of (3.4) and (3.5), we get

vxx(t, x) ≥ -ϕ 0 (x)(1 + ϕ0(x)) α (1 -2M w(t, x)) w(t, x) (1 -ln w(t, x)) α -2M ϕ 0 (x)(1 + ϕ0(x)) α 2 w 2 (t, x) (1 -ln w(t, x)) 2α .
Since ϕ 0 (x) = o(ϕ (x)) and ϕ 0 (x) = o(ϕ -α 0 (x)) as x → +∞, there exists X1 > ξ0 such that

ϕ 0 (x)(1 + ϕ0(x)) α ≤ r -ρ 4 and ϕ 0 (x)(1 + ϕ0(x)) α ≤ √ r -ρ 2 , (3.18)
as x → +∞. In view of the definition of xM , we take M large enough, say M > M1 ≥ M0, such that xM (t) > X1 for all t > 0. Thus, for x > xM (t), we have (3.16), (3.17) and (3.19), for x ≤ xM (t), we obtain

vxx ≥ - r -ρ 4 w (1 -ln w) α - r -ρ 2 M w 2 (1 -ln w) 2α ≥ - r -ρ 2 w (1 -ln w) α , (3.19) thanks to 0 < w ≤ 1 2M . Collecting (3.15),
(vt -vxx -f (v))(t, x) ≤ 0, (3.20) whereas, for x > xM (t), (vt -vxx -f (v))(t, x) ≤ w(t, x) (1 -ln w(t, x)) α - 1 2 (r -ρ) + (-2ρ + r)M + rK w(t, x) + rαcM w(t, x) 1 -ln w(t, x) + αcKM 2 w 3 (t, x) 1 -ln w(t, x) . (1) {t θ } × (-∞, y θ (t + θ )]; (2) (t, x)|t > t θ and x ∈ [y θ (t), y θ (t + )] .
For the first part, u(t θ , •) is continuous, positive, and lim infx→-∞ u(t θ , x) > 0. Thus, we have inf

x∈(-∞,y θ (t + θ )] u(t θ , x) > 0.
For the second part, if t > t θ and x ∈ [y θ (t), y θ (t + )], then w(t, x) = θ, whence

u(t, x) ≥ θ -M θ 2 > 0.
As a consequence, Θ := inf ∂Ω u > 0. Since Θ > 0 is a subsolution of equation (1.1), the comparison principle yields

u(t, x) ≥ Θ for all x ∈ Ω. (3.23)
Let us pick any x ∈ E λ (t) for any λ ∈ (0, Θ). Then

x > y θ (t + ) ≥ y θ (t) for any t ≥ t θ .

Since ρ > r -ε 2 , then there exists a time t λ,ε > t θ such that

u0(x) ≤ u0(y θ (t)) = exp 1 -ρ(α + 1)t + (1 -ln θ) α+1 1 α+1 ≤ e -[(r-ε)(α+1)t] 1 α+1 (3.24)
for any t > t λ,ε , which gives the lower bound for small λ.

Let us prove the lower bound for any λ ∈ (0, 1). Let λ ∈ [Θ, 1) be given. Denote by u θ the solution to 1.1 with initial data

u θ,0 :=          Θ, x ≥ -1, -Θx, -1 < x < 0, 0, x ≥ 0. (3.25)
In view of (2.3), we can also obtain that, for some γ1 > 0, we have limt→∞ inf

x≤γ 1 t u θ (t, x) = 1.
There exists a time t λ > 0 such that u θ (t λ , x) > λ for all x ≤ 0.

(3.26) Furthermore, by (3.23) and (3.25), we have u(t, x) ≥ u θ,0 (x -y θ (T )) for any x ∈ R and T ≥ 0.

It follows from the comparison principle that

u(T + t, x) ≥ u θ (t, x -y θ (T )).
By (3.26), we obtain u(T + t λ , x) > λ, for all x ≤ y θ (T ) and T ≥ 0.

Therefore there exists a time

t λ,ε > max(t λ,ε , t λ ) such that u0(x λ (t)) < u0(y θ (t -t λ )) = exp 1 -ρ(α + 1)(t -t λ ) + (1 -ln λ) α+1 1 α+1 < e -[(r-ε)(α+1)t] 1 α+1 (3.27)
for any t > t λ,ε , which gives the lower bound. This completes the proof.

Let T λ,ε = max{ tλ,ε , t λ,ε }. Thus, combining (3.12), (3.24) and (3.27), the proof of Lemma 1.4 is complete.

4 A more precise bound: Proof of Lemma 1.6

In this section, we give a more precise bound for the level sets of the solution to the equation (1.1).

The upper bound

We derive a more precise upper bound by translating the spatial variables in w, so that the supersolution can approximate the solution to (1.1) more accurately.

Let us define

m(t, x) =    w(t + t 1 , x -t), x ≥ x0(t + t 1 ) + t, 1, x < x0(t + t 1 ) + t,
where w(t, x) is defined by (3.2) with ρ = r and t 1 is from (3.8).

We claim that m is a supersolution to equation 1.1 for any t > 0 and x ∈ R.

To prove m is a supersolution, we need to check that mt -mxx -f (m) ≥ 0 for all t > 0 and x ∈ R. For

x < x0(t + t 1 ) + t and t > 0, since mt(t, x) = mxx(t, x) = f (m(t, x)) = 0, we have

mt(t, x) -mxx(t, x) -f (m(t, x)) = 0.
For x ≥ x0(t + t 1 ) + t and t > 0, by the definitions of m and w and Hypothesis 1.1, we have

mt(t, x)-f (m(t, x)) ≥ r w(t + t 1 , x -t) (1 -ln w(t + t 1 , x -t)) α -wx(t+t 1 , x-t)-r w(t + t 1 , x -t) (1 -ln w(t + t 1 , x -t)) α = -wx(t+t 1 , x-t).
Therefore, for all x ≥ x0(t + t 1 ) + t and t > 0, since x -t ≥ x0(t + t 1 ) > x0(t) and t + t 1 > t 1 for t > 0, by Lemma 3.2,we have mt(t, x) -mxx(t, x) -f (m(t, x)) ≥ -wx(t + t 1 , x -t) -wxx(t + t 1 , x -t) ≥ 0.

When t = 0, since u0 ≤ 1 and w(•, x) is nondecreasing for each x ∈ R, we have that m(0, x) = 1 ≥ u0(x) = u(0, x) for x < x0(t 1 ), and that m(0, x) = w(t 1 , x) ≥ w(0, x) = u0(x) = u(0, x) for x ≥ x0(t 1 ). The comparison principle then yields that u(t, x) ≤ m(t, x) for all t > 0 and x ∈ R.

(4.1)

The proof of the upper bound. Now, we prove that, for any λ ∈ (0, 1) and large time t, there are a constant Cλ > 0

we have

E λ (t) ⊂ u -1 0 [ Cλ e -(r(α+1)t) 1 α+1 , 1] . (4.2)
It follows from (4.1) that u(t, x) ≤ m(t, x) ≤ w(t + t 1 , x -t) for all t > 0 and x ∈ R.

If we pick y ∈ E λ (t), then w(t + t 1 , y -t) ≥ λ. Thus, by the definitions of w and ϕ0, we have

w(t + t 1 , y -t) = exp 1 -[(1 -ln u0(y -t)) α+1 -ρ(α + 1)(t + t 1 )] 1 α+1 ≥ λ, whence u0(y -t) ≥ exp 1 -r(α + 1)(t + t 1 ) + (1 -ln λ) α+1 1 α+1 .
By the assumption ϕ 0 (x) = o(ϕ -α 0 (x)) as x → +∞, we have, for some C > 0,

U0(Ce -(r(α+1)t) 1 α+1 ) t → +∞ as t → ∞,
where U0(z) := sup{x ∈ R, u0(x) = z}. Therefore, there is a time tλ and a constant Cλ such that

E λ (t) ⊂ u -1 0 [ Cλ e -(r(α+1)t) 1 α+1 , 1] ∀t ≥ tλ , (4.3)
proving (4.2).

The lower bound

Assuming additionally that ϕ 0 (x) ≤ 0 for large x, we derive a more precise lower bound. To do so, we take ρ = r and recall

v(t, x) =      1 4M , x ≤ xM (t), g(w(t, x)), x > xM (t), (4.4)
where M ≥ M0 = max 1 2ζ , 1 4s 0 . We claim that

E λ (t) ⊂ u -1 0 0, Ce -[r(α+1)t] 1 α+1
for t large enough.

(4.5)

In view of (3.5), since ϕ 0 (x) ≤ 0 for large x, we obtain wxx ≥ 0 for any x ≥ xM (t) with large enough M , say M > M * ≥ M0. For x > xM (t), it follows from the assumption ϕ 0 (x) = o(ϕ -α 0 (x)) as x → +∞, the definition (4.4) of v and (3.4) that we have for M > M * , up to enlarge M * ,

vxx(t, x) = (1 -2M w)wxx -2M w 2 x ≥ -2M ϕ 0 (1 + ϕ0) α 2 w 2 (1 -ln w) 2α ≥ - 1 4 rM w 2 (1 -ln w) 2α .
On the other hand, for x ≤ xM (t), we have vxx(t, x) = 0.

In view of (3.16) and (3.17), similar to (3.20), we obtain, for x ≤ xM (t), (vt -vxx -f (v))(t, x) ≤ 0, whereas, for x > xM (t), we have

(vt -vxx -f (v))(t, x) ≤ w(t, x) (1 -ln w(t, x)) α (- 3 4 rM + rK)w(t, x) + αcM w(t, x) 1 -ln w(t, x) + αcKM 2 w 3 1 -ln w(t, x) . For 0 < w ≤ 1 2M , when M ≥ M * * := max 1 2 exp 4αc(1 + 1 r ) -1 , K 4
, we have

r + KM w 2 1 -ln w ≤ r + 1 1 + ln(2M ) ≤ r 4αc .
Thus, for any x > xM (t), we take M large enough, say

M ≥ M := max 2rK, M * , M * * , so that rαcM w(t, x) 1 -ln w(t, x) + αcKM 2 w 3 (t, x) 1 -ln w(t, x) = αcM r + KM w 2 (t, x) 1 -ln w(t, x) w(t, x) ≤ 1 4 rM w(t, x), whence (vt -vxx -f (v))(t, x) ≤ w 2 (t, x) (1 -ln w(t, x)) α - 1 2 M + rK ≤ 0.
In view of (4.4), we notice that:

• when x > xM (0), we have v(0, x) = g(w(0, x)) ≤ w(0, x) = u0(x);

• when ξ1 ≤ x ≤ xM (0), since u0 is nonincreasing on [ξ1, +∞), we have v(0, x) = 1 4M < 1 2M ≤ u0(x); • when x < ξ1, since M > 1 2ζ , we have v(0, x) = 1 4M ≤ 1 2M < ζ ≤ u0(x)
. Thus, we obtain v(0, x) ≤ u0(x) = u(0, x) for all x ∈ R. Therefore, the comparison principle implies that v(t, x) ≤ u(t, x) for all x ∈ R, t > 0.

It follows from the proof of the lower bound of Lemma 1.4 that, for λ ∈ (0, Θ), if x ∈ E λ (t), then there exists

a time t λ > t θ such that u0(x) ≤ u0(y θ (t)) = exp 1 -r(α + 1)t + (1 -ln θ) α+1 1 α+1 ≤ C 1 e -[r(α+1)t] 1 α+1
for any t > t λ , (4.6)

and, for λ ∈ (Θ, 1), there exists a time t λ > max{t λ , t λ } such that

u0(x) < u0(y θ (t -t λ )) = exp 1 -r(α + 1)(t -t λ ) + (1 -ln λ) α+1 1 α+1 < C 2 e -[r(α+1)t] 1 α+1 (4.7)
for any t > t λ which gives (4.5). Let T λ = max{t λ , tλ } and C λ = max{C 1 , C 2 }. Thus, combining (4.3) (4.6) and (4.7), the proof of Lemma 1.6 is complete.

Numerical simulations

In this section, we provide some numerical simulations to illustrate the previous results.

To get an approximate solution for equation (1.1), we discretize the equation in space by the finite difference method and then use Implicit-Explicit scheme (IMEX) [4, 5] to integrate it in time, where the implicit scheme handles the diffusion term while the explicit handles the reaction term. The influence of the initial data u0 on the propagation speed is illustrated under the some fixed α in Figure 345678. We mainly consider the initial data with two kinds of decay: sub-exponential decay and algebraic decay. In the following simulations, we all take f (u) = u (1-ln u) α (1 -u) for u ∈ (0, 1). For the initial data with sub-exponential decay, we take the initial data to be u0 = min{e -5x β , 1} and α = 0.2, 0.4, 0.6. Figure 3, Figure 4 and Figure 5 show that the acceleration can be observed over a small time range when β is small. This is consistent with our theoretical results, that is, x λ (t) t 1 β(1+α) tends to infinite as β → 0 + . When β is large enough, as we show in Theorem 1.3, for initial data u0 e -µx β with β > 1 α+1 and µ > 0, the solution propagates at a finite rate. Notice that the width of the solution becomes larger and larger as β gets smaller and smaller. This is because the flattening effect [10,[START_REF] Berestycki | The speed of propagation for kpp type problems. ii: General domains[END_REF].

For initial data with algebraic decay, we take u0 = min{ 1 1+100x β , 1} and α = 0.2, 0.4, 0.6. In Figure 6, Figure 7, and Figure 8, we observe that decreasing the parameter β leads to an increase of the propagation speed. Our theoretical findings support this observation, as demonstrated by the fact that x λ (t) exp (r(α+1)t) 1 α+1 β tends to infinity as β → 0 + . We can also observe the flattening effect. Therefore, the decay of the initial data is the key to the propagation of solution to equation (1.1). When the initial data increases, meaning β decreases, the propagation speed also increases.

In Figure 9, we provide a comparison between the largest element x λ (t) of level sets E λ (t) of the solution with three different types of initial data. Observe that the slope of curve for the algebraic decay case is maximum, followed by the sub-exponential decay, and the sub-exponentially bounded case show a straight line. This is consistent with our theoretical results. We fit the corresponding theoretical results for each cases, as shown in the thick continuous curves in the figure 9. Notice that in each pair of curves when time t is large enough, our

Figure 1 :

 1 Figure 1: Comparison of the size of three kinds of nonlinearities near zero, where the parameter r and α are positive.

1 α+1 , 1 .

 11 10) and the definition of m, we have u(t, x) ≤ m(t, x) ≤ w(t + t # , x) for all t > 0 and x ∈ R.

  ) thanks to v ∈ (0, 1). Now, let us estimate the value vxx. In view of the definition (3.14) of v, we have vxx(t, x) = 0 for x ≤ xM (t)

Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 3456 Figure 3: Numerical approximations of the solution to (1.1) with the initial data u 0 (x) = min{e -5x β , 1} at different times for α = 0.2 and different values of β. The threshold for acceleration is β = 1 α+1 = 5 6 .

Figure 7 :Figure 8 :

 78 Figure 7: Numerical approximations of the solution to (1.1) with the initial data u 0 (x) = min{ 1 1+100x β , 1} at different times for α = 0.4 and different values of β.

Figure 9 :

 9 Figure9: Comparison between the largest element x λ (t) of level sets E λ (t) of the solution starting from three types of initial data: sub-exponentially bounded u 0 (x) = min{e -5x , 1}, sub-exponential decay u 0 (x) = min{e -5x 0.2 , 1} and algebraic decay u 0 (x) = min{ 1 1+100x , 1}. In this figure, the thick continuous curves are theoretical results. Here, we choose α = 0.4 and λ = 1 2 .

The notation a b means that there exists a constant C such that Cb ≤ a ≤ C -1 b.

The notation a Λ 1 ,Λ 2 ,... b means that there exists a constant C Λ 1 ,Λ 2 ,... , depending on some constants Λ 1 , Λ 2 ,..., suchthat C Λ 1 ,Λ 2 ,... b ≤ a ≤ C -1Λ 1 ,Λ 2 ,... b.

For 0 < w ≤ 1 2M , when M ≥ M2 := max 1 2 exp 2αc(1 + 1 r ) -1 , K

4

, we have

.

Thus, for any x > xM (t), we take M large enough, say

thanks to ρ > 3 4 r. This completes the construction of the subsolution v(t, x).

In view of (3.14), we notice that:

• when x > xM (0), we have v(0, x) = g(w(0, x)) ≤ w(0, x) = u0(x);

. Thus, we obtain v(0, x) ≤ u0(x) = u(0, x) for all x ∈ R. As a consequence, the maximum principle yields v(t, x) ≤ u(t, x) for all t ≥ 0 and x ∈ R.

(

Now, we prove the lower bound in Lemma 1.5.

The proof of the lower bound. Firstly, we prove it for small θ. Let us fix 0 < θ < 1 4M .

We define the level set of w(t, x) as

Recall that E θ (t) is not empty for t > t θ . It follows from Hypothesis 1.2 that there exists a time t θ > t θ such that, for any t > t θ , the closed set F θ (t) is nonempty. For any t ≥ t θ , denote

Then the function y θ : [t θ , +∞) → R is nondecreasing and left-continuous. In addition, since u0 is nonincreasing, for all points t ≥ t θ where the function y θ is discontinuous, there exist a < b such that

denotes the largest such interval, then a = y θ (t) and b = y θ (t + ) = lims→t,s>t y θ (s).

We claim that inf Ω u > 0, where Ω is an open set defined by Ω := {(t, x), t > 0, x < y θ (t)} .

Let us evaluate u(t, x) on the boundary ∂Ω. ∂Ω consists in two parts:


Maher Mylène Boulay

Al Haddad

Françoise Rul
email: francoise.rul@inrae.fr

Streptococcus thermophilus growth in soya milk: Sucrose consumption, nitrogen metabolism, soya protein hydrolysis and role of the cell-wall protease PrtS

Keywords:

Societal demand for plant-based foods is increasing. In this context, soya products fermented using lactic acid bacteria (LAB) are appealing because of their potential health and nutritional benefits. The thermophilic LAB Streptococcus thermophilus is an essential starter species in the dairy industry. However, while its physiology is well characterized, little is known about its general metabolic activity or its techno-functional properties when it is grown in soya milk. In this study, S. thermophilus LMD-9 growth, sugar production, and lactic acid production in soya milk versus cow's milk were measured. Additionally, the main metabolic pathways used by the bacterium when growing in soya milk were characterized using a proteomic approach. Streptococcus thermophilus LMD-9 growth decreased soya milk pH, from 7.5 to 4.9, in 5 h. During fermentation, acidification thus occurred in tandem with lactate production and increasing population size (final population: 1.0 × 10 9 CFU/ml). As growth proceeded, sucrose was consumed, and fructose was produced. The proteomic analysis (LC-MS/MS) of the strain's cytosolic and cell envelope-associated proteins revealed that proteins related to amino acid transport and nitrogen metabolism were the most common among the 328 proteins identified (63/328 = 19.2% of total proteins). The cell-wall protease PrtS was present, and an LMD-9 deletion mutant was constructed by interrupting the prtS gene (STER_RS04165 locus). Acidification levels, growth levels, and final population size were lower in the soya milk cultures when the ΔprtS strain versus the wild-type (wt) strain was used. The SDS-PAGE profile of the soluble proteins in the supernatant indicated that soya milk proteins were less hydrolyzed by the ΔprtS strain than by the wt strain. It was discovered that S. thermophilus can grow in soya milk by consuming sucrose, can hydrolyze soya proteins, and can produce acidification levels comparable to those in cow's milk. This study comprehensively examined the proteomics of S. thermophilus grown in soya milk and demonstrated that the cell-wall protease PrtS is involved in the LAB's growth in soya milk and in the proteolysis of soya proteins, which are two novel findings. These results clarify how S. thermophilus adapts to soya milk and can help inform efforts to develop new fermented plant-based foods with better-characterized biochemical and microbiological traits.

Introduction

By 2030, the world population will reach 8.4 billion, and providing enough food for everyone will be challenging. However, one step towards this goal is incorporating more vegetables into our diets. The development of plant-based foods is driven by consumer concerns about health, ethics, lifestyle choices, dietary issues, and sustainability. Societal demand for new fermented plant-based products has increased; in Europe in particular, efforts are directed towards creating and diversifying alternatives to meat and dairy products (e.g., as part of the growing flexitarianism and veganism movements). Legumes could contribute to this process because of their agroecological and nutritional properties. Because of their symbiotic relationships with nitrogen-fixing bacteria (i.e., rhizobia), legumes can produce high-quality biomass without the need for large amounts of nitrogen fertilizer. Soya is a legume that has long been consumed in Asia and Africa and that has become a major plant-based alternative to meat and dairy in the Western world [START_REF] Mäkinen | Foods for special dietary needs: non-dairy plant-based milk substitutes and fermented dairy-type products[END_REF], notably because of its nutritional properties. It is rich in high-quality proteins that are easily digested and that represent up to 40% of total dry matter [START_REF] Mäkinen | Foods for special dietary needs: non-dairy plant-based milk substitutes and fermented dairy-type products[END_REF]. It also has high levels of unsaturated fatty acids [START_REF] Sarkar | Potential of soyoghurt as a dietetic food[END_REF], dietary fibers, bioactive compounds (e.g., peptides, vitamins), and certain minerals (Ca, Fe, Mg, Zn;[START_REF] Sandberg | Bioavailability of minerals in legumes[END_REF]; at the same time, its levels of saturated fatty acids are low [START_REF] Sarkar | Potential of soyoghurt as a dietetic food[END_REF]. In addition, because they contain no lactose, soya-based products can serve as alternatives to dairy-based products for people who suffer from lactose intolerance [START_REF] Hajirostamloo | Comparaison of nutritional and chemical parameters of soymilk and cow milk[END_REF]. However, like other legumes, soya has some disadvantages. It is rich in poorly digested or undigestible oligosaccharides (mainly stachyose and raffinose), whose fermentation can lead to digestive discomfort (e.g., flatulence, diarrhea). People from Western populations often perceive soya as having undesirable off-flavors because of its beany/green/grassy notes [START_REF] Buono | Soymilk yogurt: sensory evaluation and chemical measurement[END_REF][START_REF] Favaro Trindade | Development and sensory evaluation of soy milk based yoghurt[END_REF][START_REF] Lee | Comparison of milk-based and soymilk-based yogurt[END_REF]. Soya also contains some anti-nutritional factors (e.g., anti-trypsic factors, phytic acid, polyphenols) as well as isoflavones, which are phytoestrogens whose effect on human health is still a subject of debate [START_REF] Patisaul | The pros and cons of phytoestrogens[END_REF].

Among the numerous plant-based fermented products that exist worldwide (> 5000; [START_REF] Tamang | Fermented Foods and Beverages of the World[END_REF], fermented soya products predominate. Soya products fermented by lactic acid bacteria (LAB) are appealing because of their potential health and nutritional benefits [START_REF] Mäkinen | Foods for special dietary needs: non-dairy plant-based milk substitutes and fermented dairy-type products[END_REF][START_REF] Tangyu | Fermentation of plant-based milk alternatives for improved flavour and nutritional value[END_REF][START_REF] Vij | Biofunctionality of probiotic soy yoghurt[END_REF]. The technical, nutritional, and functional properties of plant matrices following fermentation are determined by the growth and metabolic activities of the microorganisms present. When growing in soya, LABs consume carbohydrates and subsequently produce acid, which leads to gel formation and, in turn, contributes to overall product aroma. Another result is that the fermented product is protected from the development of undesirable or pathogenic microorganisms. The bacteria's proteolytic enzymes hydrolyze proteins in the soya matrix, leading to changes in texture and to the production of free amino acids [START_REF] Ghosh | Studies on changes in microstructure and proteolysis in cow and soy milk curd during fermentation using lactic cultures for improving protein bioavailability[END_REF]. Proteolysis also improves protein digestibility and bioaccessibility, creating a potential source of bioactive peptides (with immunomodulatory, antihypertensive properties mediated via ACE inhibition). LAB can degrade the indigestible oligo/α-galactosides found in soya milk [START_REF] Donkor | α-Galactosidase and proteolytic activities of selected probiotic and dairy cultures in fermented soymilk[END_REF][START_REF] Garro | Growth characteristics and fermentation products of Streptococcus salivarius subsp. thermophilus, Lactobacillus casei and L. fermentum in soymilk. Zeitsch Lebensmitt Untersuch Forsch[END_REF][START_REF] Leblanc | Reduction of non-digestible oligosaccharides in soymilk: application of engineered lactic acid bacteria that produce alpha-galactosidase[END_REF][START_REF] Mital | Utilization of oligosaccharides by lactic acid bacerai during fermentation of soy milk[END_REF][START_REF] Wang | Sugar and acid contents in soymilk fermented with lactic acid bacteria alone or simultaneously with bifidobacteria[END_REF], and some LAB can transform isoflavone glucosides (e.g., soya phytoestrogens) into their respective bioactive isoflavone aglycones [START_REF] Chien | Transformation of isoflavone phytoestrogens during the fermentation of soymilk with lactic acid bacteria and bifidobacteria[END_REF][START_REF] Hati | β-Glucosidase activity and bioconversion of isoflavones during fermentation of soymilk[END_REF][START_REF] Kano | Bioavailability of isoflavones after ingestion of soy beverages in healthy adults[END_REF][START_REF] Pham | Performance of starter in yogurt supplemented with soy protein isolate and biotransformation of isoflavones during storage period[END_REF][START_REF] Wei | Using of Lactobacillus and Bifidobacterium to product the isoflavone aglycones in fermented soymilk[END_REF][START_REF] Zhao | Changes in antioxidant capacity, isoflavone profile, phenolic and vitamin contents in soymilks during extended fermentation[END_REF], which are better absorbed by the gut. LAB fermentation of soya can also reduce the abundance of some antinutritional factors such as phytic acids, which chelate divalent metals and thus limit their bioavailability [START_REF] Lai | Effect of lactic fermentation on the total phenolic, saponin and phytic acid contents as well as anti-colon cancer cell proliferation activity of soymilk[END_REF][START_REF] Rekha | Isoflavone phytoestrogens in soymilk fermented with β-glucosidase producing probiotic lactic acid bacteria[END_REF][START_REF] Rui | Protein bioaccessibility of soymilk and soymilk curd prepared with two Lactobacillus plantarum strains as assessed by in vitro gastrointestinal digestion[END_REF]. The consumption of fermented soya milk has also been found to have a beneficial influence on fecal microbiota in animals [START_REF] Ara | Dietary effect of fermented soymilk on lipid metabolism and intestinal flora[END_REF][START_REF] Butteiger | Soy protein compared with milk protein in a Western diet increases gut microbial diversity and reduces serum lipids in Golden Syrian hamsters[END_REF][START_REF] Liu | The anti-allergenic properties of milk kefir and soylmilk kefir and their beneficcial effects in the intestinal microflora[END_REF] and humans [START_REF] Cheng | Effect of fermented soy milk on the intestinal bacterial ecosystem[END_REF]. Finally, LAB fermentation can improve the flavor of soya milk [START_REF] Pinthong | The development of a soya-based yoghurt II. Sensory evaluation and analysis of volatiles[END_REF], notably by reducing the beany note [START_REF] Favaro Trindade | Development and sensory evaluation of soy milk based yoghurt[END_REF][START_REF] Mital | Flavor acceptability of unfermented and lactic-fermented soy milks[END_REF][START_REF] Wang | Lactic acid fermentation of soybean milk[END_REF].

The lactic acid bacterium Streptococcus thermophilus (S. thermophilus) is an essential starter species in the dairy industry and is notably important in the production of yogurt, mozzarella, and hard cheeses. Its behavior and physiology in cow's milk are well characterized, as are its techno-functional properties (acidification, effect on texture and flavor) in dairy products. Soya milk (i.e., the liquid obtained after soaking and grinding up soybeans) is inexpensive and has been used for years as a culture medium for LAB. A recent study showed that, among the 276 LAB strains tested, S. thermophilus most effectively acidified soya milk [START_REF] Harlé | Diversity of the metabolic profiles of a broad range of lactic acid bacteria in soy juice fermentation[END_REF].

Here, we characterized the growth and production of certain sugars and organic acids by S. thermophilus cultures grown in soya milk; cultures grown in cow's milk were used as a standard of reference. The objective was to better understand the behavior and physiology of S. thermophilus in soya milk. This study also identified the strain's important/limiting metabolites in soya milk and established the first overall proteomic profile for S. thermophilus grown in soya milk.

Materials and methods

Bacterial strains and culture conditions

Two bacterial strains were used: the S. thermophilus wild-type (wt) strain LMD-9 and its negative mutant for the cell-wall protease PrtS (ΔprtS, EryR; see below). The strains were grown in soya milk (Sojasun containing calcium and vitamin D; Triballat-Noyal, Noyal-sur-Vilaine, France) and in reconstituted cow's milk (nonfat dried milk, M7409; Sigma Aldrich).

Frozen stock culture of S. thermophilus LMD-9 was grown in the two media as follows. The first pre-culture was created over the course of a day: an isolated colony of an overnight culture grown on M17Lac was then grown at 37 °C in M17 broth (Difco) supplemented with lactose (10 g/l). The second pre-culture was created by inoculating 5 ml of M17 sucrose (10 g/l) with the first pre-culture (1%) and carrying out incubation at 37 °C overnight. The final cultures were generated by inoculating soya milk or cow's milk with the second preculture (OD 600nm 0.05) and carrying out incubation at 37 °C. Culturing was ended when the pH reached 5.8 for the soya milk (2.5 h) and 5.3 for the cow's milk (3 h); the cultures were then frozen in liquid nitrogen and stored at -80 °C. Chain disruption in the stock cultures was performed using a mechanical blender (Turax X620, Labo-Moderne, France) that was run for 40 s, and the resulting dilutions were then plated on M17Lac (1%) agar. Enumeration took place after 16 h of incubation at 42 °C under anaerobic conditions (Anaerocult A, Merck, Darmstadt, Germany). Despite the 10 months of storage at -80 °C, the matrices protected S. thermophilus against freezing, and the cultures were 100% viable in both media (reaching 2 × 10 8 CFU/ml in soya milk and 8 × 10 8 CFU/ ml in cow's milk).

Experimental cultures were created, in triplicate, by inoculating soya milk or cow's milk with the appropriate frozen stock culture (1 × 10 6 CFU/ml), and incubation was carried out at 37 °C. Acidification was tracked by measuring pH, and growth was quantified via enumeration following culturing on M17 Lac agar, as described above. The apparent growth rate (μ max ) was defined as the maximum slope of the semi-logarithmic growth curves established from population size (the CFU/ml counts).

Construction of the ΔprtS mutant

The negative mutant of the S. thermophilus LMD-9 strain, ΔprtS, was constructed via gene interruption using an erythromycin (Ery) resistance cassette and two additional PCR fragments. These three fragments were generated using PCR (TaqPhusion polymerase, 1 min elongation, 56 °C) and the primers ErmF (CCAAGGAGCTAAAGAGGT CCC) and ErmR (GGAGATAAGACGGTTCGTGTTCG) for the Ery resistance cassette; RG254 (TGGTAAGCACGTAGACC) and RG259 (CTA CTGACAGCTTCCAAGGAGCTAAAGAGGTCCAGGCTTGTCAATTCAT CTG) for the upstream fragment; and RG257 (CGTATGCTTACCAACA GAG) and RG260 (GCAAGTCAGCACGAACACGAACCGTCTTATCTCCG AAAGCCAACTTAGATGG) for the downstream fragment. The 3′ end of the upstream fragment contained a sequence complementary to the 5′ end of the Ery resistance cassette, whereas the 5′ end of the downstream fragment contained a sequence complementary to the 3′ end of the cassette. The three fragments (upstream, resistance cassette, and downstream) were joined via PCR (3 min elongation) using the primers RG254 and RG257 (TaqPhusion polymerase, activation at 98 °C for 3 min, 35 cycles of 30 s at 97 °C, 30 s at 55 °C, and 4 min at 72 °C). After purification with a Promega column, 10 μl of the resulting combination of fragments was used to transform natural competent cells of the wt S. thermophilus LMD-9 strain, as described in [START_REF] Gardan | The oligopeptide transport system is essential for the development of natural competence in Streptococcus thermophilus strain LMD-9[END_REF]. Transformants were selected on M17Lac plates containing erythromycin (5 μg/ml); they were verified using PCR, and the flanking regions were sequenced to ensure that no unwanted mutations had been introduced.

Quantification of sugar and lactate concentrations

Bacteria were removed from the soya milk and cow's milk cultures at 0, 2, 2.5, and 3 h after culturing was initiated. Samples were centrifuged at 5000 ×g at 4 °C for 10 min to obtain the supernatants. The supernatants were then clarified with Carrez reagent: 100 μl of culture supernatant was mixed with 50 μl of Carrez I reagent (3.6% tetrapotassium hexacyanoferrate, w/v), 50 μl of Carrez II reagent (7.2% zinc sulfate, w/v), 100 μl of NaOH (0.1 N), and 1 ml of Milli-Q® water q.s. The mixture was centrifuged for 10 min at 35,000 ×g and filtered (0.45 μm, 4 mm PVDF membrane). The supernatants were then injected into an Aminex HPX-87H column (Biorad), which was heated to 40 °C. Sugars and organic acids were separated out during 60-min runs (rate of 0.6 ml/min); H 2 SO 4 buffer (5 mM) was employed. The compounds were detected with a UV 2996 detector and a 2410 differential refractometer. Quantities were determined via comparisons with standard curves that were established using lactate, lactose, glucose, galactose, fructose, sucrose, stachyose, and raffinose.

Scanning electron microscopy

Microscopy analyses were performed at the Microscopy and Imaging Platform MIMA2 (INRA, Jouy-en-Josas, France) (INRA, http:// www6.jouy.inra.fr/mima2/). Bacteria were recovered from 1-ml samples of cow's milk and soya milk cultures during two growth phases -the exponential phase (at 2 h of culture) and the stationary phase (at 3.5 h of culture)-by centrifugation (5000 ×g at 4 °C for 10 min). The bacteria were then washed twice with 3.5 ml of Tris 0.1 M (pH 7.5). One 40-μL drop of washed bacteria was deposited on a sterile cover glass (12 mm in diameter; Marienfeld, VWR, France). Each biofilm coupon was placed into one well of a 24-well polystyrene plate. Samples were fixed via careful immersion in a solution of 2.5% glutaraldehyde containing 0.1 M sodium cacodylate buffer (pH of 7.4) kept at room temperature. The samples remained immersed overnight at 4 °C. They were then rinsed three times for 10 min in sodium cacodylate solution (pH 7.4) and underwent progressive dehydration by being soaked in a graded series of ethanol solutions (50 to 100%) before being subjected to critical point drying using CO 2 . The coupons were subsequently mounted on aluminium stubs (10 mm in diameter) with carbon adhesive disks (Agar Scientific; Oxford Instruments SAS, Gometz-la-Ville, France) and sputter coated with gold/palladium (Polaron SC7640; Elexience, Verrières-le-Buisson, France) for 200 s at 10 mA. The samples were viewed as secondary electron images (2 kV) using a field emission gun scanning electron microscope (SEM) (Hitachi S-4500; Elexience, Verrières-le-Buisson, France).

SDS-PAGE electrophoresis

Supernatant was obtained from the soya milk and cow's milk cultures by centrifugation (5000 ×g for 10 min), and levels of soluble proteins were quantified using Coomassie protein assay reagent (Pierce Biotechnology Inc., Rockford, Illinois, USA); bovine serum albumin served as the standard. A 10-μg sample of proteins was then mixed with 4 μl of a denaturation solution (4× Laemmli sample buffer; Biorad) and 20 μl of Milli-Q H 2 O q.s.; the mixture was heated to 96 °C and kept there for 2 min before being placed in NuPage™ 4-12% Bis-Tris Gel (Invitrogen) containing 10 μl of SeeBlue™ Plus2 marker (Invitrogen). SDS-PAGE electrophoresis was then carried out. Migration occurred at 220 V-110 mA for 45 min. The gels were colored for 1 h with Coomassie Blue. Then, the gels were washed four times for 15 min and scanned to digitize the results.

Proteomic analysis 2.6.1. Protein extraction

Samples (100 ml) of the soya milk and cow's milk cultures collected during the exponential phase of growth were mixed with 100 ml of sodium citrate buffer (1 M) and centrifuged (32,000 ×g at 4 °C for 10 min). The resulting bacterial pellets were washed twice with Tris-EDTA buffer; they were then frozen and stored at -80 °C. Bacteria were broken up using a cell disruptor (one passage at 2.5 Kbar/sample; AZIC Z cell disruptor, Celld, Warwickshire, UK). Unbroken bacteria and large cellular debris were eliminated using centrifugation (20,000 ×g); the supernatant was recovered and ultracentrifuged (200,000 ×g) to separate the cell-envelope proteins from the cytosolic proteins. Proteins in the cell-envelope fractions and the cytosolic fractions were quantified as described above and stored at -80 °C.

2.6.2. Sample preparation, mass spectrometry analysis Samples (10 μg) of the cell-envelope proteins and the cytosolic proteins were separated using one-dimensional electrophoresis (4-12% acrylamide gel, SDS-PAGE) with short migration times. Biological triplicates were loaded onto the gel, and the protein bands were cut and digested in gel. The resulting peptides were extracted as previously described [START_REF] Gardan | The oligopeptide transport system is essential for the development of natural competence in Streptococcus thermophilus strain LMD-9[END_REF]. Liquid chromatography coupled with tandem mass spectrometry (LC-MS/MS) was carried out using an Ulti-Mate™ 3000 RSLCnano system (Thermo Fisher Scientific) with a 50-cm column (PepMap 100, 3 μm; column: 75 μm × 500 mm; one run = 210 min) coupled to a LTQ-Orbitrap Discovery™ mass spectrometer (Thermo Fisher Scientific) in CID mode, as described in [START_REF] Gardan | The oligopeptide transport system is essential for the development of natural competence in Streptococcus thermophilus strain LMD-9[END_REF]. Protein identification was performed with X! Tandem software by comparing the experimental results against known proteins of S. thermophilus LMD-9 (GenBank v. 2013, 1710 entries), soya (Uniprot v. 16-11-14; 150,681 entries), and potential contaminants. Results were filtered using the built-in X! Tandem Pipeline (http://pappso.inra. fr/bioinfo/); the requirements were a peptide E-value of 0.01, a protein E-value of -4, and the presence of at least 3 specific peptides per protein. The proteins identified from the cell-envelope and cytosolic fractions are presented together (Tables S1 andS2).

The analysis of the soya milk samples was less extensive than the analysis of the cow's milk samples because the former contained a larger percentage of soya proteins (57% of soya milk proteins identified/total vs. 4.2% of cow's milk proteins identified/total). Because differences in the relative levels of protein identification in the two sample types could stem from detection differences, as opposed to true protein presence/absence, the two sets of data were not compared.

Results and discussion

This study examined growth, sugar production, and lactic acid production by an S. thermophilus strain (LMD-9) cultured in soya milk and cow's milk. It also characterized the main metabolic pathways that the bacterium employs in soya milk using a proteomic approach, which highlighted the important role of nitrogen metabolism.

1. Streptococcus thermophilus LMD-9 is metabolically active in soya milk.

First, an assessment was performed of the overall morphology of S. thermophilus LMD-9 cultures grown in the soya milk and cow's milk via SEM observations conducted during two growth phases: the exponential phase and the stationary phase. During both phases in both media, S. thermophilus exhibited ovococcus-shaped bacteria and constructed chains of > 10 cells (Fig. 1), traits typical of this bacterium. Consequently, S. thermophilus LMD-9 displayed no morphological differences when grown in soya milk. Furthermore, dividing cells had visible septa, which indicates that the bacteria were metabolically active in soya milk. In both media, bacteria were sticky in the matrix.

2. Streptococcus thermophilus LMD-9 rapidly drives down soya milk pH.

The growth of S. thermophilus LMD-9 resulted in the acidification of the cow's milk, with pH dropping from 6.5 to 4.7 in 5 h (Fig. 2); the acidification rate peaked between 2 and 4 h. Curding occurred 4 h after fermentation, at a pH of around 5.1 in the cow's milk cultures and 5.3-5.4 in the soya milk cultures.

Acidification kinetics in the soya milk cultures paralleled those in the cow's milk cultures in terms of the number and duration of the different phases: acidification was absent or very low during the first hour, accelerated and then peaked between 2 and 3.5 h, and finally decelerated. The pH decreased from 7.5 to 4.9 in 5 h (Fig. 2). The major difference between the two media was that the maximum acidification rate was 1.6 higher in the soya milk cultures (ΔpH/h = 0.64) than in the cow's milk cultures (ΔpH/h = 0.41); this result explains why final pH was similar for both media even though initial pH was higher in the soya milk cultures (pH = 7.5) than in the cow's milk cultures (pH = 6.5) (Fig. 2).

When S. thermophilus LMD-9 was grown in soya milk, it generated a higher level of acidification (ΔpH of 2.6 in 5 h) than other S. thermophilus strains studied up until now (ΔpH of 1.6-2.3; [START_REF] Champagne | Selection of probiotic bacteria for the fermentation of a soy beverage in combination with Streptococcus thermophilus[END_REF][START_REF] Chumchere | Selection of stater cultures for the fermentation of soya milk[END_REF][START_REF] Garro | Growth characteristics and fermentation products of Streptococcus salivarius subsp. thermophilus, Lactobacillus casei and L. fermentum in soymilk. Zeitsch Lebensmitt Untersuch Forsch[END_REF]Mital andSteinkraus, 1975, 1976;[START_REF] Wang | Sugar and acid contents in soymilk fermented with lactic acid bacteria alone or simultaneously with bifidobacteria[END_REF]. These differences could result from the strains themselves and/or from differences in soya milk composition or preparation. A previous study found that the ST5 strain acidified laboratory-prepared soya milk in 8 h (ΔpH of 2) and a commercial soya milk in 12 h [START_REF] Champagne | Selection of probiotic bacteria for the fermentation of a soy beverage in combination with Streptococcus thermophilus[END_REF].

As expected for a LAB, S. thermophilus LMD-9 produced lactate as it grew, acidifying both the cow's milk and the soya milk (Fig. 2). While the lactate concentration increased steadily in both media during initial culturing, it was lower in the soya milk than in the cow's milk. Levels of lactate production during the first 3 h of culture were 0.82 g/l ( ± 0.076) and 1.74 g/l ( ± 0.013) in the soya milk and cow's milk, respectively.

3. Streptococcus thermophilus LMD-9 grows in soya milk via sucrose consumption.

Growth kinetics

Streptococcus thermophilus LMD-9 displayed two growth phases in the soya milk and cow's milk: an exponential phase, where growth accelerated after 1 h, and a stationary phase, which began after 4 h of culture (Fig. 3). In both media, growth stopped when the pH reached around 5.2, which is expected for this bacterium. The growth rate was higher in the cow's milk than in the soya milk (2.5 h -1 vs 1.8 h -1 ); after 5 h of culture, population sizes were higher in the cow's milk than in the soya milk (2.3 × 10 9 CFU/ml [ ± 0.4 × 10 9 ] vs. 1.0 × 10 9 CFU/ ml [ ± 0.15 × 10 9 ]), which correlated well with the levels of lactate production in the two media (Fig. 2). [START_REF] Mital | Utilization of oligosaccharides by lactic acid bacerai during fermentation of soy milk[END_REF] observed a similar difference in S. thermophilus growth: populations were 3-4 larger in cow's milk than in soya milk. Such differences could be related to the contrasted buffering capacities of the two media [START_REF] Champagne | Selection of probiotic bacteria for the fermentation of a soy beverage in combination with Streptococcus thermophilus[END_REF][START_REF] Lutchman | Evaluation of the buffering capacity of powdered cow's, goat's and soy milk and non-prescription antacids in the treatment of non-ulcer dyspepsia[END_REF]. Therefore, β-glycerophosphate (19 g/l) was added to the soya milk as a buffer in an effort to reduce acidification, but the strain's growth did not improve (data not shown). Then, lactose was added as it is the preferred sugar of S. thermophilus LMD-9 [START_REF] Thomas | Carbohydrate metabolism is essential for the colonization of Streptococcus thermophilus in the digestive tract of gnotobiotic rats[END_REF]. This supplementation (10 g/l) also failed to increase the strain's growth (data not shown). Since large amounts of usable sugars were still present at the end of the culturing period, both in the cow's milk and the soya milk, it was concluded that carbohydrate availability was not limiting the growth of S. thermophilus LMD-9 in the soya milk.

Streptococcus thermophilus strains vary dramatically in their ability to grow and/or reach large population sizes in soya milk. For example, in our study, the population size increased 170 fold during the stationary phase (from 5.8 × 10 6 to 1.0 × 10 9 CFU/ml); in other studies, this increase ranged between 36 and 10,000 fold [START_REF] Champagne | Selection of probiotic bacteria for the fermentation of a soy beverage in combination with Streptococcus thermophilus[END_REF][START_REF] Chumchere | Selection of stater cultures for the fermentation of soya milk[END_REF][START_REF] Garro | Growth characteristics and fermentation products of Streptococcus salivarius subsp. thermophilus, Lactobacillus casei and L. fermentum in soymilk. Zeitsch Lebensmitt Untersuch Forsch[END_REF][START_REF] Wang | Growth and survival of bifidobacteria and lactic acid bacteria during the fermentation and storage of cultured soymilk drinks[END_REF]. Like the results for acidification, the results for population size in soya milk could be related to strain traits (i.e., the capacity to use soya sugars and proteins), properties of the soya milk matrices, the soya milk preparation method (e.g., the sterilization temperature), and/ or the soya milk's composition (e.g., sugars and yeast extract are sometimes added to soya milk).

Sugar profiles

The concentrations of certain types of sugars were quantified in the cow's milk and soya milk cultures of S. thermophilus LMD-9 during the first 3 h of growth. Lactose and sucrose are the main sugars in cow's milk (170 mM = 58.2 g/l) and soya milk (72 mM = 24.6 g/l), respectively.

In the cow's milk cultures, the lactose concentration declined steadily (Δ of 8.64 mM [= 2.96 g/l]), while the galactose concentration increased (Δ of 8.6 mM [= 1.55 g/l]) (Fig. 4A); no glucose was detected in the medium. [START_REF] Sørensen | Enhancing the sweetness of yoghurt through metabolic remodeling of carbohydrate metabolism in Streptococcus thermophilus and Lactobacillus delbrueckii subsp. bulgaricus[END_REF] observed similar dynamics of lactose consumption (44 mM) and galactose production (36 mM) over a 30-h culture period. This result fits with what is known about lactose transport in S. thermophilus: lactose is hydrolyzed to produce galactose, which is most commonly released into the medium, and glucose, which is metabolized via glycolysis to produce ATP. The kinetics of lactose consumption and galactose production are thus well correlated with growth kinetics (Fig. 3).

In the soya milk cultures, the sucrose concentration decreased (Δ of 4 mM [= 1.36 g/l]), while the fructose concentration increased (Δ of 2.1 mM [= 0.38 g/l]) (Fig. 4B). Similar sucrose consumption dynamics by S. thermophilus LMD-9 in soya milk have previously been observed (0.4 g/L; [START_REF] Chumchere | Selection of stater cultures for the fermentation of soya milk[END_REF][START_REF] Wang | Sugar and acid contents in soymilk fermented with lactic acid bacteria alone or simultaneously with bifidobacteria[END_REF]. The growth of S. thermophilus LMD-9 in soya milk thus appears to be fueled by the consumption of sucrose, the main sugar in the soya milk used here and a good carbohydrate source for this particular strain [START_REF] Thomas | Carbohydrate metabolism is essential for the colonization of Streptococcus thermophilus in the digestive tract of gnotobiotic rats[END_REF]. The ability to exploit sucrose is shared by most S. thermophilus strains [START_REF] Harlé | Diversity of the metabolic profiles of a broad range of lactic acid bacteria in soy juice fermentation[END_REF][START_REF] Mital | Utilization of oligosaccharides by lactic acid bacerai during fermentation of soy milk[END_REF][START_REF] Van Den Bogaard | Sugar utilisation and conservation of the gal-lac gene cluster in Streptococcus thermophilus[END_REF]. In addition, genes related to sucrose transport and the production of fructose from sucrose are common in S. thermophilus species [START_REF] Alexandraki | Comparative genomics of Streptococcus thermophilus support important traits concerning the evolution, biology and technological properties of the species[END_REF][START_REF] Bolotin | Complete sequence and comparative genome analysis of the dairy bacterium Streptococcus thermophilus[END_REF][START_REF] Goh | Specialized adaptation of a lactic acid bacterium to the milk environment: the comparative genomics of Streptococcus thermophilus LMD-9[END_REF][START_REF] Hols | New insights in the molecular biology and physiology of Streptococcus thermophilus revealed by comparative genomics[END_REF].

Predominance of proteins related to nitrogen metabolism in soya milk

To obtain a more holistic view of the metabolism and physiology of S. thermophilus LMD-9 during growth in soya milk, a proteomics analysis of the strain's cell-envelope proteins and cytosolic proteins was carried out using LC-MS/MS. Indeed, when proteins linked to certain metabolic pathways are present under certain growth conditions, it can be inferred that particular pathways and nutrient sources are being used. A similar analysis was performed on the samples of S. thermophilus LMD-9 grown in cow's milk to obtain an appropriate reference for this study, although it should be noted that proteomic data already exist for another strain of S. thermophilus grown in cow's milk [START_REF] Derzelle | Proteome analysis of Streptococcus thermophilus grown in milk reveals pyruvate formate-lyase as the major upregulated protein[END_REF][START_REF] Herve-Jimenez | Physiology of Streptococcus thermophilus during the late stage of milk fermentation with special regard to sulfur amino-acid metabolism[END_REF].

Proteomic profile in cow's milk

In the proteomic profile of S. thermophilus LMD-9 grown in cow's milk, there were 515 proteins that most commonly fell into the following clusters of orthologous groups (COGs): translation, ribosomal structure, and biogenesis ([J], 17.9%); amino acid transport and metabolism ([E], 17.3%); and cell wall/membrane/envelope biogenesis ([M], 8.7%) (Table S1, Fig. S1). The COGs [J] and [E] were thus predominant (around 35%), a result previously observed for another strain [START_REF] Herve-Jimenez | Physiology of Streptococcus thermophilus during the late stage of milk fermentation with special regard to sulfur amino-acid metabolism[END_REF].

Streptococcus thermophilus needs amino acids (AAs) for protein synthesis and ensures its supply in two main ways: by creating AAs via intracellular synthesis and/or by transporting proteins and peptides in from the extracellular medium and hydrolyzing them with amino-and oligopeptidases. This study identified the presence of the cell-wall protease PrtS (one of the first enzymes to act in the proteolytic cascade); 8 of the 12 intracellular aminopeptidases (i.e., general aminopeptidases, proline-specific peptidases, dipeptidase, oligopeptidases) present in the LMD-9 genome (https://www.ncbi.nlm.nih.gov/bioproject/ PRJNA13773); and most of the AA, di/tripeptide, and oligopeptide transport systems (14 out of 20). These results suggest that all the different AAs required by S. thermophilus are likely provided by cow's milk thanks to the activities of these diverse proteins, which target all types of peptides (of varying lengths and AA composition). In addition, proteins from all the complete AA biosynthetic pathways (19) of LMD-9 were identified.

The profile also revealed the presence of 4 of the 7 transporters involved in carbohydrate metabolism that are predicted to occur in the genome (3 phosphotransferase system [PTS] transporters, 2 ATPbinding cassette [ABC]-type transporters, and 2 permeases), including lactose permease (LacS). All 9 of the glycolytic enzymes were identified as previously observed. This result is unsurprising given that glycolysis is the main process by which S. thermophilus forms ATP.

Proteomic profile in soya milk

In the proteomic profile of S. thermophilus LMD-9 grown in soya milk, there were 328 bacterial proteins (Table S2), which is fewer than the number in the cow's milk cultures. This difference was caused by the higher level of soya protein contamination in soya milk cultures. When classified into COGs, half the proteins were identified as being involved in basic cell functions: amino acid transport and metabolism (19.2%); translation, ribosomal structure, and biogenesis (15.2%); carbohydrate transport and metabolism (8.8%); and cell wall/membrane/envelope biogenesis (8.5%) (Fig. 5). Fig. 3. Growth curve of the wild-type S. thermophilus LMD-9 cultures grown in cow's milk (○) and soya milk (▲). The curve was constructed from three independent replicates.

Nitrogen metabolism seems to play an important role in S. thermophilus LMD-9 growth in soya milk based on the number of proteins identified that were involved in nitrogen metabolism (63/328), which probably reflects the bacterium's significant need for nitrogen. The profile revealed the presence of the extracellular cell-wall protease PrtS, 11 amino acid transporters (for polar AAs, branched-chain AAs, Met, Ser/Thr, Ala, and Gln), and 1 oligopeptide transporter (Ami) out of the 20 transporters present in the genome. Ami transporters are capable of transporting peptides containing up to 23 AAs [START_REF] Juille | The specificity of oligopeptide transport by Streptococcus thermophilus resembles that of Lactococcus lactis and not that of pathogenic streptococci[END_REF]. Five of the 12 intracellular peptidases found in the genome were also present in the profile: the 2 oligopeptidases PepB/F and PepO (that hydrolyze peptides between 5 and 30 AAs in length; [START_REF] Chavagnat | Purification, characterisation, cloning and sequencing of the gene encoding oligopeptidase PepO from Streptococcus thermophilus A[END_REF]; the general aminopeptidase PepN that liberates all types of AAs; and 2 dipeptidases-the general dipeptidase PepV and the proline-specific prolidase PepQ. These results suggest that peptides of up to 23 AAs in length can potentially be transported and hydrolyzed intracellularly by S. thermophilus grown in soya milk. This study also identified the presence of biosynthesis pathways for all the AAs except histidine; evidence was particularly pronounced for the pathways for branchedchain AAs (Leu, Ile, Val), sulfur AAs (Met, Cys, Ser), and glutamic acid. From a qualitative point of view, these results generally suggest that S. thermophilus LMD-9 grown in soya milk can produce peptides from soya proteins via the action of PrtS, peptide transport, the hydrolysis of certain peptides, and the synthesis of most of the different amino acids. Carbohydrate metabolism was of particular interest in this study, and the proteomic profile revealed the presence of proteins potentially involved in the utilization of glucose polymers (AmyL, GlgP, MalQ), sucrose, lactose (LacS-Z), and possibly glucose (GlcK) and fructose (ManMNL PTS transporters). More specifically, the following proteins were observed: ScrA, PtsI, and PtsH-which have been implicated in sucrose transport in other bacteria-and ScrB hydrolase (β-fructofuranosidase = sucrose-6 phosphate hydrolase). Once imported into the cell, sucrose is likely hydrolyzed into glucose-6-phosphate and fructose by ScrB; the glucose-6-phosphate then enters the glycolysis pathway, and part of the fructose is released into the extracellular medium. Previous studies have shown that the growth of S. thermophilus in soya milk can lead to a similar accumulation of fructose in cultures (Fig. 4B; [START_REF] Champagne | Selection of probiotic bacteria for the fermentation of a soy beverage in combination with Streptococcus thermophilus[END_REF][START_REF] Garro | Growth characteristics and fermentation products of Streptococcus salivarius subsp. thermophilus, Lactobacillus casei and L. fermentum in soymilk. Zeitsch Lebensmitt Untersuch Forsch[END_REF][START_REF] Wang | Sugar and acid contents in soymilk fermented with lactic acid bacteria alone or simultaneously with bifidobacteria[END_REF]. All 9 glycolytic enzymes were present, as was lactate dehydrogenase (Ldh), which converts the end product of glycolysis, pyruvate, into lactate. The latter is toxic to the bacterium and is thus released into the extracellular medium, as was observed here (see above, Section 2.2).

The cell-wall protease PrtS contributes to S. thermophilus LMD-9 growth via its role in soya protein hydrolysis

As discussed above, the proteomic profile showed that nitrogen metabolism seems to play an important role in S. thermophilus LMD-9 growth in soya milk. We therefore explored this issue further using two approaches: 1) soya milk was supplemented with free amino acids and peptides and 2) an S. thermophilus LMD-9 mutant was used whose cellwall protease PrtS had been deleted. When S. thermophilus is cultured in cow's milk, PrtS helps the bacterium grow and is responsible for casein hydrolysis [START_REF] Courtin | Cell-wall proteinases PrtS and PrtB have a different role in Streptococcus thermophilus/Lactobacillus bulgaricus mixed cultures in milk[END_REF][START_REF] Dandoy | The fast milk acidifying phenotype of Streptococcus thermophilus can be acquired by natural transformation of the genomic island encoding the cellenvelope proteinase PrtS[END_REF].

Effects of amino acid and peptide supplementation on S. thermophilus LMD-9 growth in soya milk

To test the effects of nitrogen supplementation on S. thermophilus LMD-9 growth in soya milk, we used a casein hydrolysate (mostly composed of free amino acids/dipeptides; Bacto™ Casamino Acids) and either pure or mixed amino acids. In particular the bacterium was supplemented with lysine because, like other S. thermophilus strains, LMD-9 lacks the complete biosynthesis pathway for this AA [START_REF] Alexandraki | Comparative genomics of Streptococcus thermophilus support important traits concerning the evolution, biology and technological properties of the species[END_REF], suggesting it might be a lysine auxotroph.

Supplementation with casein hydrolysate did not modify the growth or acidification patterns of S. thermophilus LMD-9 in soya milk over the first 6 h of growth (with supplementation: ΔpH of 2.33 [ ± 0.17] and Δpopulation size of 1.64 × 10 9 CFU/ml [ ± 0.92 × 10 9 ]; without supplementation: ΔpH of 2.34 [ ± 0.15] and Δpopulation size of 1.6 × 10 9 CFU/ml [ ± 1.26 × 10 9 ]). The addition to the soya milk of lysine, various branched-chain AAs (Ile, Leu, Val), and various sulfur AAs (Met, Cys, Ser) at concentrations of 1 or 5 mM did not significantly increase the growth of S. thermophilus LMD-9 (data not shown). These results suggest that neither the AAs/dipeptides in the casein hydrolysate nor the previously mentioned free AAs limited the bacterium's growth in soya milk.

Effect of the cell-wall protease PrtS on S. thermophilus LMD-9 growth in soya milk

The proteomic profile revealed the presence of the cell-wall protease PrtS in S. thermophilus LMD-9 grown in soya milk, the next step was to explore the role of this protease in the acidification of soya milk. A deletion mutant of the LMD-9 strain (ΔprtS strain) was constructed by interrupting the prtS gene (locus: STER_RS04165) with an antibiotic gene cassette (see Materials and methods section). Acidification was less pronounced in the ΔprtS strain cultures (ΔpH of 1.77 [ ± 0.036]) than in the wild-type (wt) strain cultures (ΔpH of 2.22 [ ± 0.01]), underscoring the involvement of PrtS in the process. A similar pattern was seen for population size, which was 3 fold lower for the ΔprtS strain (2.7 × 10 8 CFU/ml [ ± 1.24 × 10 8 ]) than for the wt strain (7.6 × 10 8 CFU/ml [ ± 2.5 10 8 ]) after 6 h of culture. We thus concluded that the cell-wall protease PrtS is involved in S. thermophilus LMD-9 growth in soya milk. This finding fits with the results of a previous study, which suggested a link between the bacterium's proteolysis capacity and growth in soya milk [START_REF] Donkor | α-Galactosidase and proteolytic activities of selected probiotic and dairy cultures in fermented soymilk[END_REF].

There was no effect of nitrogen supplementation (bactocasitone peptone, 1 g/L) on the growth and acidification dynamics of the wt strain grown in soya milk. In contrast, there was an effect on the ΔprtS strain: over the first 4 h of culture, pH and population size were higher with supplementation (pH = 2.15 [ ± 0.21], population size: 1.22 × 10 8 CFU/ml [ ± 0.44]) than without supplementation (pH = 1.7 [ ± 0.17], population size: 2.7 × 10 8 CFU/ml [ ± 0.7]). This result suggests that PrtS was active when the bacterium was grown in soya milk; it was then studied if PrtS was able to degrade soya milk proteins. Using SDS-PAGE, the soluble proteins in the cultures of the wt and ΔprtS strains grown in soya milk were separated out (Fig. 6). Based on the bands observed, the most common proteins present are probably β-conglycinin and glycinin, which are the main storage proteins in soya seed [START_REF] Hsieh | Proteomic profiling of the coagulation of soymilk proteins induced by magnesium chloride[END_REF][START_REF] Krishnan | A rapid and simple procedure for the depletion of abundant storage proteins from legume seeds to advance proteome analysis: a case study using Glycine max[END_REF].

After 6 h of culture, the protein profile of the wt strain (lanes 2 & 3) was markedly different: bands associated with heavier proteins (> 62 kDa) had completely disappeared, while bands associated with lighter proteins (49-62 kDa and < 38 kDa) had appeared. This result indicates that some of the proteins had been proteolyzed during the bacterium's growth. In contrast, the protein profile of the ΔprtS strain suggested that less hydrolysis had taken place: the two bands associated with heavier proteins were more pronounced. However, the bands associated with the lighter proteins were similar to those of the wt strain. These findings show that the wt strain was more proteolytic than the ΔprtS strain and thus probably generated more peptides, which could explain its better growth. Just as in the cow's milk cultures, the hydrolytic activity of PrtS in the soya milk cultures could have promoted the growth of wt S. thermophilus LMD-9 via the production of peptides that could serve as an exogenous supply of nitrogen.

Conclusion

We showed that S. thermophilus LMD-9 could grow in and rapidly acidify soya milk. Furthermore, the bacterium displayed similar growth and acidification dynamics in soya milk and cow's milk. It consumes sucrose to grow, which leads to lactate production and, in turn, acidification, a process that can preserve soya milk from microbial contamination. This work also established the first proteomic map for S. thermophilus grown in soya milk, which highlighted the importance of nitrogen metabolism. Additionally, it was discovered that the LAB cellwall protease PrtS is involved in the bacterium's growth in soya milk and its proteolysis of soya proteins. The ability of S. thermophilus to hydrolyze soya proteins could help improve the digestibility of soyabased products and provide health benefits, such as reducing the allergenicity of soya proteins or modulating the production of potentially bioactive peptides.

The identification of the peptides and, more generally, the metabolites present after fermentation should enhance understanding of the organoleptic and probiotic properties of soya-based products fermented with S. thermophilus or with other LAB.

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ijfoodmicro.2020.108903.

CRediT authorship contribution statement

Boulay M.: conceptualization, formal analysis, investigation, methodology, validation, visualization. Al Haddad M.: conceptualization, formal analysis, investigation, validation. Rul F.: conceptualization, funding acquisition, investigation, project administration, supervision, visualization, writing -original draft

Declaration of competing interest

None.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Scanning electron microscope images of wild-type S. thermophilus LMD-9 cultures in cow's milk and soya milk illustrating the morphological similarities of the bacterium in the two growth media during its two growth phases: the exponential phase (2 h) and the stationary phase (4 h). The arrows indicate the bacterial septa.

Fig. 4 .

 4 Fig. 4. Sugar concentrations in the wild-type S. thermophilus LMD-9 cultures: A) lactose and galactose concentrations in cow's milk, B) sucrose and fructose concentrations in soya milk.

Fig. 5 .

 5 Fig. 5. Percentage of proteins in different COG categories found in S. thermophilus LMD-9 cultures after 3 h of growth in soya milk.

Fig. 6 .

 6 Fig.6. SDS-PAGE results for the proteins found in cultures of the S. thermophilus LMD-9 wt strain and ΔprtS strain grown in soya milk. The numbers on the left are the reference molecular masses. Lanes 1 and 6: molecular mass standard (SeeBlue™ Plus2 marker); Lane 2: wt strain at the start of culturing (T0); Lane 3: wt strain after 6 h of culture (T6); Lane 4: ΔprtS strain at the start of culturing (T0); Lane 5: ΔprtS strain after 6 h of culture (T6).

Acknowledgments

We thank K. Alves Silva and C. Besset for their technical assistance; A. Aubert and E. Haudebourg for their help with the LC-MS/MS analysis; and T. Meylheuc for the aid with the SEM analysis. We are grateful to M. Thomas for fruitful discussions, R. Gardan and C. Delorme for their constructive criticism of the manuscript and to J. Pearce-Duvet for the English-language editing. Financial support for this research mainly came from the OOlga Triballat Institute (Noyal sur Vilaine, France) (2016-2017).