Said Kouachi

Global existence for reaction di¤usion systems with reactions changing sign

 where we aimed to prove global existence of solutions for coupled systems of reaction-di¤usion equations with nonlinearities of non-constant sign. We aim here to correct an error in the proof of Theorem 1 (see Remark 2). We note that there aren't many results in this …eld, and the question of the global existence of strong solutions, even weak solutions, is still unsolved, even when the mass control and the quasi-positivity proprieties on the nonlinearities are satis…ed. Based on re…ned Lyapunov techniques, proofs are provided.

Introduction

We consider the following reaction-di¤usion system 8 > > > < > > > :

@u @t a u = f (u; v) = c(t; x)'(u; v) in R + ; @v @t b v = g(u; v) = c(t; x)'(u; v) in R + ; (1)
with the boundary conditions

@u @ = @v @ = 0 on R + @ ; (2)
and the initial data

u(0; x) = u 0 (x); v(0; x) = v 0 (x) in ; (3)
where is an open bounded domain of class C 1 in R N , with boundary @ ; @ @ denotes the outward normal derivative on @ and a and b are positive constants.

The initial data are assumed to be bounded on and nonnegative. The function ' 2 C 1 (R + R + ; R) with at most polynomial growth j'(u; v)j C (u s + v s ) ; for a all u 0; v 0; [START_REF] Henry | Geometric Theory of Semi-linear Parabolic Equations[END_REF] for some Positive constants C and s and satisfy the quasi-positivity property '(0; v) = '(u; 0) = 0; for a all u 0; v 0; [START_REF] Hollis | On the Question of Global Existence for Reaction-Di¤usion Systems with Mixed Boundary Conditions[END_REF] which assures the positivity of the solution on at all time. The function c(t; x) 2 C(R + ; R) is not of constant sign. We suppose it possesses a …nite number of zeros independent of the time, that is there exist x 1 ; :::; x k 2 such that c (t; x 1 ) = ::: = c (t; x k ) = 0; for a all t 0; [START_REF] Hollis | Global Existence and Boundedness in Reaction Di¤usion Systems[END_REF] which is for example the case when c(t; x) = h(t)c 1 (x 1 ) :c 2 (x 2 ) :::c k (x k ) = 0, for all t 0. We denote by and + the following subsets of 8 < :

= fx 2 such that c (t; x) > 0; for a all t 0g ; + = fx 2 such that c (t; x) < 0; for a all t 0g ; [START_REF] Kirane | A strongly nonlinear reaction di¤usion model for a deterministic di¤usive epidemic[END_REF] which are not time-dependent. The existence of a positive regular solution locally in time is classical (see for example [START_REF] Henry | Geometric Theory of Semi-linear Parabolic Equations[END_REF]). The global existence in time of a regular solution is not so obvious. If c (t; x) is a constant or is of constant sign, then global existence is immediate by application of the maximum principle (see for example [START_REF] Kouachi | Global existence and asymptotics for a class of reaction di¤usion systems[END_REF]). The issue becomes problematic, though, if c changes sign. In [START_REF] Pierre | Global existence for a Reaction-Di¤usion System with a balance law[END_REF], the following case was investigated: c(t; x) > 0 in ( 1; 0) ; c(0) = 0; c(t; x) < 0 in (0; 1) ; where it is shown that the solutions are locally uniformly bounded in L 1 ([0; 1) ( 0; 1) and L 1 ([0; 1) ( 1; 0). The question with general c (t; x) is still an open question in all space dimensions (see [START_REF] Pierre | Global Existence in Reaction-Di¤usion Systems with Control of Mass: a Survey[END_REF]). In general, L 1 ( )-blow-up may occur in …nite time for quadratic systems with nonlinearities satisfying the quasipositivity f (0; v) 0; for all v 0 and g (u; 0) 0; for all u 0; [START_REF] Kirane | Asymptotic Behavior for a System Describing Epidemics with Migration and Spatial Spread of Infection[END_REF] and the control mass guarantied by the following condition f (u; v) + g(u; v) 0; for all u; v 0: Indeed, the authors in [START_REF] Pierre | Blowup in Reaction-Di¤usion Systems with Dissipation of Mass[END_REF] showed L 1 ( )-blow-up for system (1) with appropriate boundary conditions. Recently they ameliorate their results to more nonlinearities and homogenous Neumann boundary conditions. For this type of reaction di¤usion systems only two properties hold: the positivity of the solutions is preserved for all time and the total mass of the components is uniformly controlled in time

Z (u + v) dx Z (u 0 + v 0 ) dx; t 0: (9)
The uniform control on the mass (or, more mathematically, the solution's L1norm) suggests that no blow-up should occur in …nite time. It turns out that the situation is not so simple. This explains why there are so many incomplete …ndings in di¤erent directions are found in the literature on this topic, and why also the general question of global existence of strong solutions even weak solutions is still open, while numerous systems with these two natural features appear in applications. Here, we review the primary bene…ts and drawbacks for the global existence together with many references, a description of the still open problems and a few new results as well.

To show how the situation is di¢ cult, the authors in [START_REF] Desvillettes | About Global Existence for Quadratic Systems of Reaction-Di¤usion[END_REF] (see also [START_REF] Souplet | Global existence for reaction-di¤usion systems with dissipation of mass and quadratic growth[END_REF]) and forced these reactions to satisfy a Lyapunov structure of LogL entropy type

(log u) f + (log v) g 0; for a all u > 0; v > 0; (10)
to demonstrate that a class of quadratic reaction-di¤usion systems has weak solutions that are global in time for whom the reaction terms change sign. But this type of structure is satis…ed only for nonlinearities at most quadratic. The reaction terms have not a constant sign means that none of the equations is good in the sense that neither u nor v is a priori bounded or at least bounded in some Lp-space, allowing us to apply the well known regularizing e¤ect and deduce the global existence of strong solutions in time for problem (1)-(3). Or to get uniform integrability of the right-hand side of (1) to obtain a good approximation of system (1) and then apply the Dominated Convergence Theorem of integrals which with compactness properties of the heat operator give together global existence of a weak solution. In the case when the nonlinearities have a constant sign many results have been obtained. S. L. Hollis, R. H. Martin and M. Pierre [START_REF] Hollis | Global Existence and Boundedness in Reaction Di¤usion Systems[END_REF] established global existence of positive solutions for system (1) with appropriate boundary conditions, under the conditions of the uniform boundedness of u on [0; T max ] and f (r; s) + g(r; s) C(r; s) (r + s + 1) ; for all r 0 and s 0; [START_REF] Kouachi | Existence of global solutions to reaction-di¤usion systems via a Lyapunov functional[END_REF] where C(r; s) is positive and uniformly bounded function de…ned on R + R + . One notices that, to prove global existence for solutions to system (1), authors impose, in addition to [START_REF] Hollis | Global Existence and Boundedness in Reaction Di¤usion Systems[END_REF], to one of the components of the reaction term the same condition.

In the case when the nonlinearities have not a constant sign or for which one of the components u or v is a priori uniformly bounded. there is not many results: A. J. Morgan [START_REF] Morgan | Global Existence for Semilinear Parabolic Systems[END_REF] Generalized the results of S. L. Hollis, R. H. Martin and M. Pierre [START_REF] Hollis | Global Existence and Boundedness in Reaction Di¤usion Systems[END_REF] to show that solutions of the m-components reaction di¤usion systems exist globally (m 2) where also, he imposed to f and f + g conditions ( 6). In S. Kouachi [START_REF] Kouachi | Existence of global solutions to reaction di¤usion systems with no homogeneous boundary conditions via a Lyapunov functional[END_REF], we generalized the above results for two components reaction di¤usion systems under the unique condition

f + Dg C (1 + u + v) ; for a all u 0; v 0; (12)
for all positive constant D su¢ ciently large, where C is positive constant and we showed the global existence without imposing the boundedness of one of the components of the solution.

Notations and preliminary observations

It is well-known that to prove global existence of solutions to (1)-( 3) (see A.

Friedman [START_REF] Friedman | Partial Di¤erential Equations of Parabolic Type[END_REF], D. Henry [START_REF] Henry | Geometric Theory of Semi-linear Parabolic Equations[END_REF], A. Pazy [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] and F. Rothe [START_REF] Rothe | Global Solutions of Reaction-Di¤usion Systems[END_REF]), it su¢ ces to derive a uniform estimate on [0; T max [ of the reactions in the space L p ( ) for some p > N=2 where T max denotes the eventual bowing-up time in L 1 ( ). Our aim is to apply polynomial Lyapunov functional method (see M. Kirane and S. Kouachi [START_REF] Kirane | A strongly nonlinear reaction di¤usion model for a deterministic di¤usive epidemic[END_REF], [START_REF] Kirane | Global Solutions to a System of Strongly Coupled Reaction-Di¤usion Equations[END_REF] and [START_REF] Kirane | Asymptotic Behavior for a System Describing Epidemics with Migration and Spatial Spread of Infection[END_REF], S. Kouachi and A. Youkana [START_REF] Kouachi | Global existence and asymptotics for a class of reaction di¤usion systems[END_REF] and S. Kouachi [START_REF] Kouachi | Existence of global solutions to reaction-di¤usion systems via a Lyapunov functional[END_REF] and [START_REF] Kouachi | Existence of global solutions to reaction di¤usion systems with no homogeneous boundary conditions via a Lyapunov functional[END_REF]) according to the solutions (u; v) of system (1), to carry out their L p bounds and deduct their global existence. The nonnegativity of the solutions is preserved by application of classical results on invariant regions (see J. Smoller [START_REF] Smoller | Shock Waves and Reaction-Di¤usion Equations[END_REF]), since the reaction is quasi-positive. The usual norms in the spaces L p ( ), L 1 ( ) and C are respectively denoted by

kuk p p = 1 j j Z ju(x)j p dx;
and

kuk 1 = max x2 ju(x)j :
Since the nonlinear right hand side of ( 1) is continuously di¤erentiable, then for any initial data in C or L p ( ); p 2 (1; +1), it is easy to check directly its Lipschitz continuity on bounded subsets of the domain of a fractional power of the operator a 0 0 b :

Under these assumptions, the following local existence result is well-known (see A. Friedman [START_REF] Friedman | Partial Di¤erential Equations of Parabolic Type[END_REF], [START_REF] Henry | Geometric Theory of Semi-linear Parabolic Equations[END_REF], [START_REF] Smoller | Shock Waves and Reaction-Di¤usion Equations[END_REF] and [START_REF] Rothe | Global Solutions of Reaction-Di¤usion Systems[END_REF]). Before the statement of the results, let us do the following Remark 2 We recall that this manuscript is a correction of an error which I committed when I submitted it to a journal. I de…ned for a …xed integer p 1; the following polynomial functional which is of a great interest in the following

t 7 ! F (t) = Z G (u(t; x); v(t; x)) dx; (13)
where

G (u; v) = p X i=0 C i p i u i v p i ; (14)
and the coe¢ cient C i p is given by the formula

C i p = p! i!:(p i)! ;
which is de…ned in terms of the factorial function n!, and

i (x) = 8 < : i (x) = c i+1 K i 2 ; x 2 + ; + i (x) = C i+1 K i 2 ; x 2 ; ( 15
)
where the constants c and C are chosen such that the …nite sequence f i (x)g is decreasing for x 2 + and increasing for x 2 : That is

cK 2p+1 < 1 < CK; (16)
and where K is any positive constant satisfying

K 2 > (a + b) 2 4ab : (17)
Clearly, from [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] we have

i i+2 2 i+1
= K 2 ; for all x 2 ; i = 0; 1; :::p 2:

It was clear at …rst glance in order to apply below the Green's formula, I should solve the problem of discontinuity of the sequence f i g at the origin. I multiplied it by a functions n as follows I didn't take attention that the functions n i (x) wasn't C 1 but after a private communication with [START_REF] Souplet | A private communication[END_REF] who remarked to me that they become continues on but the discontinuity persists for its …rst derivative and then we can't apply the well known "Green's formula". I asked myself what operation makes disappearing this discontinuity and the …rst idea coming to me was the course on the "Pseudo-Di¤ erential Operators" presented at Paris VII university in 1982 by my teacher the celebrated mathematician Laurent Schwartz who used forever the convolution to regularize functions. By this operation functions with only …rst L 1 derivatives can be regularized by themselves and become of second order L 1 derivatives, noteworthy if the regularization is used by functions more regular. This is because the convolution is a commutative operation, we have

n i (x) = n (x) i (x) ;
d 2 (f g) dx 2 = f 0 g 0 :
De…nition 3 The convolution of two functions f and g with compact supports in R is the function f g given by

(f g) (x) = Z R f (x y) g (y) dx:
Remark 4 f g is also called the generalized product of f and g and it also holds in the case that one of the function is a generalized function, like Dirac's delta.

Remark 5

The convolution operation for L 1 functions is commutative, associative, distributive with respect to the addition and if f is C n (R) for some positive integer n with compact support in R and g is L 1 (R), then f g is C n (R) and we have (f g)

(n) = f (n) g.
Proposition 6 For all positive , there exists a nonnegative function 2 C 1 (R) with compact support in the interval

( 1; 1) satisfying Z R (x) dx = Z jx j (x) dx = 1: (20)
Proof. We use the following function

(x) = 8 > < > : exp 1 1 x ; jxj < 1; 0; jxj 1;
then we dilate to obtain

(x) = 1 R R (x) dx (x)
and a simple dilation-renormalization gives

(x) = 1 x :
3 Statement and proof of the main results

The main result of the paper is the following Theorem 7 Suppose that conditions (4) and ( 5) are satis…ed, then for any solution (u (t; :) ; v (t; :)) of the problem ( 1)-( 3) with positive initial data, the functional de…ned by (21a)-( 24) below is uniformly bounded on the interval (0; T max ).

Proof. We treat the special case when = ] 1; 1[ and the function c(t; x) = c (x) independent of the time and possess a unique zero, for example c(0) = 0. Suppose c(t; x) > 0 for x 2 ] 1; 0[ and c(t; x) < 0 for x 2 ]0; 1[. The general case can be deduced easily. The i 's given by ( 15)-( 18) can be regularized as follows i (x) = ( i ) (x) ; x 2 R; i = 0; 1; :::p;

= 1 2 + + ; (21a) where
and and + are two C 1 (R) nonnegative functions given by proposition 6 with Supp( ) ( 1; 0) and Supp( + ) (0; 1) : Then the function is C 1 (R) with Supp( ) ( 1; 1) and satis…es [START_REF] Rothe | Global Solutions of Reaction-Di¤usion Systems[END_REF]. Then, we easily check that the functions i (x) are C 1 (R) with Supp i ( 2; 2). Let

t 7 ! L(t) = Z H (u(t; x); v(t; x)) dx; (23)
where

H (u; v) = p X i=0 C i p i u i v p i : (24)
By di¤erentiating L with respect to t, we get

L 0 (t) = p P i=1 R iC i p i u i 1 v p i @u @t dx + p P i=1 R (p i)C i p i u i v p i 1 @v @t dx: (25)
Using the fact that iC i p = pC i 1 p 1 ; for all i = 1; :::; p;

(26) and interchanging the index, we get

L 0 (t) = p p 1 X i=0 C i p 1 Z u i v p 1 i a i+1 u + b i v + i+1 f + i g dx = I + J;
where

I = p p 1 X i=0 Z C i p 1 u i v p 1 i a i+1 u + b i v dx; (27)
and

J = p p 1 X i=0 C i p 1 Z i+1 + i c(t; x)'(u; v)u i v p 1 i dx: (28)
For the integrals I and since the functions i (x) are C 1 (R), we can apply Green's formula to obtain

I = p p 1 X i=0 C i p 1 Z ar i+1 u i v p 1 i ru + br i u i v p 1 i rv dx;
which can be written as follows

I = I 1 + I 2 ;
where

I 1 = p p 1 X i=0 C i p 1 Z a i+1 r u i v p 1 i :ru + b i r u i v p 1 i :rv dx;
and

I 2 = p p 1 X i=0 C i p 1
Z u i v p 1 i ar i+1 :ru + br i :rv dx:

Using (26) another time and interchanging the indexes, the integral I 1 becomes

I 1 = p(p 1) p 2 X i=0 C i p 2 Z T i (ru; rv) u i v p 2 i dx; (29)
where

T i (ru; rv) = a i+2 jruj 2 + (a + b) i+1 rurv + b i jrvj 2 dx: (30)
Then from the expression of , we have i =:

1 2 + + i = 1 2 i + 1 2 + i ;
where i = i and + i = + i ; and since Supp( ) ( 1; 0), Supp( + ) (0; 1) and Supp( i ) = ( 1; 1) then Supp i = [ 2; 1] and Supp + i = [ 1; 2] :

We have i (x) = i = Z D1\D + 2 (x y) i (y) dy;
where D 1 : 1 < y < 1 and D + 2 : 1 < x y < 0:

This gives i (x) = 8 > > > > < > > > > : i x+1 R 1 (z) dz; for 2 < x < 1; + i x+1 R 0 (z) dz; for 1 < x < 0:
Since we integrate on the interval ( 1; 0), then we have

i (x) = + i x+1 Z 0 (z) dz; for 1 < x < 0: (31)
The same reasoning gives

+ i (x) = + i = Z D1\D + 2 + (x y) i (y) dy;
where

D 1 : 1 < y < 1 and D + 2 : 0 < x y < 1: The calculations give i (x) = 8 > > > > > < > > > > > : i 0 R x 1 (z) dz; for 0 < x < 1; + i 1 R x 1 (z) dz; for 1 < x < 2:
Since we integrate on the interval (0; 1), we take

+ i (x) = i 0 Z x 1 (z) dz; for 0 < x < 1: (32)
Let's summarize, we have

i = ( i = 8 < : (x) + i ; for 1 < x < 0; + (x) i ; for 0 < x < 1; (33)
where

8 > > > > < > > > > : (x) = x+1 R 0 (z) dz, for 1 < x < 0;
and

+ (x) = 0 R x 1 (z) dz, for 0 < x < 1: (34)
Consequently we get

T i (ru; rv) = T i (ru; rv) + T + i (ru; rv) ;
where

T i (ru; rv) = a + i+2 jruj 2 + (a + b) + i+1 rurv + b + i jrvj 2 (x) ; on (0; T max ) ;
and

T + i (ru; rv) = a i+2 jruj 2 + (a + b) i+1 rurv + b i jrvj 2 + (x) dx; on (0; T max ) + :
The above quadratic forms of the two variables ru and rv under the relations ( 15)-( 18) together with the positivity of and + are positive and satisfy the following inequalities

T i (ru; rv) (x) + i : jruj 2 b + i + jrvj 2 a + i+2 ! ; (35)
and

T + i (ru; rv) + (x) i : jruj 2 b i + jrvj 2 a i+2 ! ; (36)
for all i = 0; 1; :::p 2 and all t 2 (0; T max ) where

i = 4ab i i+2 (a + b) 2 i+1 2 8 ;
is positive, for all i = 0; 1; :::p 2: In conclusion the integrals I 1 and I + 1 are non positive on (0; T max ) and then we have In the following our strategy is to subtract the integrals I 2 which are summation until p 1 from the integrals I 1 which are summation until p. To do this, we use the following formula

I 1 =: I 1 + I +
C i p 1 = C i 1 p 2 + C i p 2 ; i = 1; :::; p 2:
Then the integrals I 2 can be written as follows

I 2 = I 21 + I 21 ;
where Then

I 21 = p R
I 2 = p p 2 X i=0 Z C i p 2 [a i+1 :ru + b i :rv] u i v p 2 i dx; (37)
were i = ur i+1 + vr i ; i = 0; 1; :::p 1:

Before the subtraction, we apply Young's inequality to the new form of the integral I 2 given by (37). That is for all > 0 we can …nd a positive constant C such that

j i+1 ruj (p 1) i+2 jruj 2 + C j i+1j 2
(p 1) i+2

;

and j i rvj (p 1) i jrvj 2 + C j ij 2 (p 1) i : Put I 2a = p 2 X i=0 C i p 2 Z u i v p 2 i a i+2 jruj 2 + b i jrvj 2 dx;
and

I 2b = Z p 2 X i=0 C i p 2 i (x; u; v)u i v p 2 i dx; (39)
where As the i 's are positive and uniformly bounded on and from (18), we can choose K su¢ ciently large such that i i+2 (1 )

i (x; u; v) = j i+1j 2 (p 1) i+2 + j ij 2 (p 1) i ; (40
2 i+1 2 > (a + b) 2 4ab
; i = 0; 1; :::p 2;

for some > 0 su¢ ciently small. This gives

I 1 =: p(p 1) p 2 X i=0 C i p 2 Z u i v p 2 i T i (ru; rv) dx < 0;
where

T i (ru; rv) = a i+2 (1 ) jruj 2 + (a + b) i+1 rurv + b i (1 ) jrvj 2 :
Then we can subtract the integral p(p 1)I 2a from I 1 to get I 1 = I 1 + p(p 1)I 2a 0:

Finally we have I C pI 2b :

Using the fact that the functions i (x) are C 1 (R) with compact supports then the coe¢ cients of u and v in the expression of i given by (38) are uniformly bounded and consequently the coe¢ cients i (given by (40)) of the polynomial under the integration sign in formula (39) intervening in the integrals I 2b are uniformly bounded by a constant independent of the time. Consequently we have

I 2b C p Z p 2 X i=0 C i p 2 u i v p 2 i u 2 + v 2 dx:
For the remaining integral J, we have

J = p p 1 X i=0 C i p 1 Z i+1 + i c(t; x)'(u; v)u i v p 1 i dx = J + J + ; where J = p p 1 X i=0 C i p 1 Z + i+1 + + i (x) c(t; x)'(u; v)u i v p 1 i dx and J + = p p 1 X i=0 C i p 1 Z + i+1 + i + (x) c(t; x)'(u; v)u i v p 1 i dx:
Since the sequence + i is increasing with c(t; x) > 0 on , and the sequence i is decreasing with c(t; x) < 0 on , then J and J + are nonpositive on (0; T max ). That is J 0 ; for all t 2 (0; T max ) :

In conclusion, we have Since, the function c(t; x) is uniformly bounded on and the function '(u; v) is of polynomial growth, the reactions are L p ( ) for all p > 1 which gives, from the preliminary remarks the global existence of the solution.

L 0 (

Conclusion

With the exception of a few partial results obtained under very strict constraints the question of the global existence of solutions of systems of coupled reaction-di¤ usion equations with nonlinearities with non constant sign had been unsolved until just lately. The failure of traditional approaches, such as those based on the entropy inequality or those employing the duality argument, etc., is the main reason for the fascinating outcomes in such situations. This question was solved in this study using a relatively straightforward functional yet powerful approach.

Proposition 1

 1 The system (1) admits a unique classical solution (u; v) on (0; T max ): If T max < +1, then lim T max t% (ku(t; :)k 1 + kv(t; :)k 1 ) = +1:

where n : ] 1 ; 1 [ 7 !

 117 [0; 1] are the functions de…ned as follows

1 0;

 1 for all t 2 (0; T max ) ;whereI 1 = p(p 1) + i (ru; rv) :

i

  ) is a second degree polynomial of the variables u and v. Then we haveI 2 p(p 1)I 2a + C pI 2b :In conclusion we subtract I 2a from I 1 as followsI =: I 1 + I 2 I 1 + p(p 1)I 2a + C pI 2b : ; on ; i = 0; 1; :::; p; (41) then using the properties of the convolution, we have i i (x) + i ; on :

  avr 1 ru + bvr 0 rv :v p 2 dx aur i+2 :ru + bur i+1 :rv u i v p 2 i :

	and
	I 22 =	p p	p 2 P i=1 R aur p ru + bur p 1 rv :r u p 2 dx C i 1 p 2 R aur i+1 ru + bur i rv :r u i 1 v p 1 i dx
	=: p	p 2 X	Z	C i p 2
			i=0
				p	p 2 P i=1	C i p 2	R	avr i+1 :ru + bvr i :rv u i v p 2 i dx
		=: p	p 2 X	Z	C i p 2 avr i+1 :ru + bvr i :rv u i v p 2 i dx;
					i=0

  This shows that the Functional L(t) is uniformly bounded on [0; T max [ and ends the proof of the Theorem.Corollary 8 All solutions of problem (1)-(3) with uniformly bounded positive initial data in are global in time.

	which gives
	L(t) L (0) e C	00 p t ; 0 t < T max :	(43)
	Proof. Using (43) and (42), we get
	ku (t; :) + v (t; :)k p Ce	C 00 p

t) C 0 p (u + v) p ; 0 t < T max : But from (41) we can …nd two positive constants C p and C + p such that C p (u + v) p L(t) C + p (u + v) p ; 0 t < T max ; (42) then we have for some positive constant C 00 p the following inequality L 0 (t) C 00 p L (t) ; 0 t < T max ; p t ku 0 + v 0 k p :

This work was supported by the College of Science and Technology, University of Abbes Laghrour Khenchela, project code: C00L03UN400120220001.


Priscilla De F Cardoso

Stéphane Perchat

Laurival A Vilas-Boas

Didier Lereclus

Gislayne T Vilas-Bôas

Diversity of the Rap-Phr quorum-sensing systems in the Bacillus cereus group

Keywords: Rap-Phr, Bacillus cereus group, RNPP, Sporulation, Phosphatase

Bacteria of the Bacillus cereus group colonize several ecological niches and infect different hosts. Bacillus cereus, a ubiquitous species causing food poisoning, Bacillus thuringiensis, an entomopathogen, and Bacillus anthracis, which is highly pathogenic to mammals, are the most important species of this group. These species are closely related genetically, and their specific toxins are encoded by plasmids. The infectious cycle of B. thuringiensis in its insect host is regulated by quorumsensing systems from the RNPP family. Among them, the Rap-Phr systems, which are well-described in Bacillus subtilis, regulate essential processes, such as sporulation. Given the importance of these systems, we performed a global in silico analysis to investigate their prevalence, distribution, diversity and their role in sporulation in B. cereus group species. The rap-phr genes were identified in all selected strains with 30% located on plasmids, predominantly in B. thuringiensis. Despite a high variability in their sequences, there is a remarkable association between closely related strains and their Rap-Phr profile. Based on the key residues involved in RapH phosphatase activity, we predicted that 32% of the Rap proteins could regulate sporulation by preventing the phosphorylation of Spo0F. These Rap are preferentially located on plasmids and mostly related to B. thuringiensis. The predictions were partially validated by in vivo sporulation experiments suggesting that the residues linked to the phosphatase function are necessary but not sufficient to predict this activity. The wide distribution and diversity of Rap-Phr systems could strictly control the commitment to sporulation and then improve the adaptation capacities of the bacteria to environmental changes.

Introduction

Several bacterial processes are regulated by quorum sensing, a cell-cell communication that enables bacteria to regulate their fate with regard to the population density. The Rap proteins and their cognate Phr peptide inhibitors are quorumsensing systems present in the Bacillus cereus group but not extensively studied in these bacteria. The B. cereus group comprises at least seven species (Bacillus cereus sensu stricto, Bacillus thuringiensis, Bacillus anthracis, Bacillus weihenstephanensis, Bacillus mycoides, Bacillus pseudomycoides and Bacillus cytotoxicus) of rod-shaped, sporeforming, Gram-positive bacteria that are found in diverse ecological niches and able to colonize different hosts [START_REF] Liu | Genomic insights into the taxonomic status of the Bacillus cereus group[END_REF]. Due to the complex phylogeny of the group, as phylogenetic clades are polyphyletics and species are paraphyletics, its taxonomy continues to be debated [START_REF] Bazinet | Pan-genome and phylogeny of Bacillus cereus sensu lato[END_REF][START_REF] Guinebretière | Ecological diversification in the Bacillus cereus Group[END_REF][START_REF] Helgason | Bacillus anthracis, Bacillus cereus, and Bacillus thuringiensis-one species on the basis of genetic evidence[END_REF][START_REF] Liu | Genomic insights into the taxonomic status of the Bacillus cereus group[END_REF][START_REF] Raymond | Bacillus thuringiensis and Lysinibacillus sphaericus: Characterization and use in the Field of Biocontrol[END_REF][START_REF] Tourasse | Extended and global phylogenetic view of the Bacillus cereus group population by combination of MLST, AFLP, and MLEE genotyping data[END_REF].

The three main species of the B. cereus group have a significant impact on human activity. B. cereus is a ubiquitous and opportunistic bacterium and includes strains that cause food poisoning with vomiting or diarrhea and severe local infections, such as endophthalmitis or periodontitis (Callegan et al. 2003;[START_REF] Ehling-Schulz | Cereulide synthetase gene cluster from emetic Bacillus cereus: structure and location on a mega virulence plasmid related to Bacillus anthracis toxin plasmid pXO1[END_REF][START_REF] Stenfors Arnesen | From soil to gut: bacillus cereus and its food poisoning toxins[END_REF]. B. thuringiensis is the world's most Communicated by M. Kupiec.

Electronic supplementary material

The online version of this article (https ://doi.org/10.1007/s0029 4-019-00993 -9) contains supplementary material, which is available to authorized users.

1 3 used biopesticide due to its production of insecticidal toxins (designated as Cry proteins) specifically pathogenic to a wide range of insects [START_REF] Schnepf | Bacillus thuringiensis and its pesticidal crystal proteins[END_REF]. B. anthracis is a mammal pathogen, including humans, and is the causative agent of anthrax [START_REF] Liu | Anthrax lethal and edema toxins in anthrax pathogenesis[END_REF]. Although phenotypically different, these species are closely related genetically [START_REF] Rasko | Genomics of the Bacillus cereus group of organisms[END_REF], and the species determinants are encoded by plasmid genes [START_REF] Vilas-Boas | Biology and taxonomy of Bacillus cereus, Bacillus anthracis, and Bacillus thuringiensis[END_REF]). For B. cereus, the enzymatic complex involved in cereulide (emetic toxin) synthesis is encoded by pCER270 [START_REF] Ehling-Schulz | Cereulide synthetase gene cluster from emetic Bacillus cereus: structure and location on a mega virulence plasmid related to Bacillus anthracis toxin plasmid pXO1[END_REF]. Strains are identified as B. thuringiensis if they produce a crystal inclusion during sporulation due to the presence of plasmids carrying genes encoding Cry toxins, generally active against insects or nematodes [START_REF] Deng | Regulation of cry Gene Expression in Bacillus thuringiensis[END_REF][START_REF] Schnepf | Bacillus thuringiensis and its pesticidal crystal proteins[END_REF]. The high toxicity of B. anthracis is due to toxins and its capsule, which are encoded by genes located on the plasmids pXO1 and pXO2, respectively [START_REF] Kolstø | What sets Bacillus anthracis apart from other Bacillus species?[END_REF].

Several microorganisms behaviors, such as biofilm formation, sporulation, motility, genetic exchange (competence and conjugation), and virulence factor production, are regulated by quorum sensing (QS), a cell-cell communication process that allows bacteria and eukaryotic microorganisms to coordinate their biological processes based on the population density [START_REF] Polke | Quorum sensing by farnesol revisited[END_REF][START_REF] Rutherford | Bacterial quorum sensing: its role in virulence and possibilities for its control[END_REF]. In Gram-positive bacteria, this communication is done by signaling oligopeptides that are recognized by cognate regulators, such as the QS systems of the RNPP family (from Rap, NprR, PlcR, and PrgX) [START_REF] Declerck | Structure of PlcR: insights into virulence regulation and evolution of quorum sensing in Gram-positive bacteria[END_REF]). These regulators are formed by tetratricopeptide repeat (TPR) domains that are structural motifs of degenerated residues that mediate protein-protein and protein-peptide interactions [START_REF] D'andrea | TPR proteins: the versatile helix[END_REF]. The activity of these cytoplasmic regulators is activated (NprR and PlcR) or inhibited (Rap) by secreted, matured, and re-imported peptides that function as signaling molecules [START_REF] Perchat | A cell-cell communication system regulates protease production during sporulation in bacteria of the Bacillus cereus group[END_REF]Perego and Hoch 1996;[START_REF] Pottathil | The extracellular Phr peptide-Rap phosphatase signaling circuit of Bacillus subtilis[END_REF][START_REF] Slamti | A cell-cell signaling peptide activates the PlcR virulence regulon in bacteria of the Bacillus cereus group[END_REF]. The genes encoding these signaling peptides are located directly downstream from the coding sequence of their cognate RNPP regulator, and the two genes are transcribed in the same orientation [START_REF] Declerck | Structure of PlcR: insights into virulence regulation and evolution of quorum sensing in Gram-positive bacteria[END_REF]. Except for the Rap proteins, RNPP regulators have an HTH (helix-turn-helix) DNA-binding domain, allowing them to function as transcriptional regulators [START_REF] Declerck | Structure of PlcR: insights into virulence regulation and evolution of quorum sensing in Gram-positive bacteria[END_REF]. During the infectious cycle of B. thuringiensis in insect larvae, three QS systems are successively activated [START_REF] Slamti | Quorum sensing in Bacillus thuringiensis is required for completion of a full infectious cycle in the insect[END_REF]): (i) PlcR-PapR regulates the virulence stage by controlling the expression of virulence genes; (ii) NprR-NprX regulates the necrotrophic stage, allowing bacteria to survive and to sporulate in the insect cadaver; and (iii) Rap-Phr regulates the initiation of the sporulation process.

Sporulation is essential for survival and dispersion of a wide variety of organisms [START_REF] Huang | Sporulation: how to survive on planet Earth (and beyond)[END_REF]. In Bacillus subtilis, this differentiation process is regulated by a complex pathway [START_REF] Sonenshein | Control of sporulation initiation in Bacillus subtilis[END_REF], in which Spo0A is the major regulator of sporulation that must be phosphorylated to be active. External signals, such as starvation, are detected by different sporulation kinases (KinA to KinE), which phosphorylate the Spo0F response regulator [START_REF] Burbulys | Initiation of sporulation in B. subtilis is controlled by a multicomponent phosphorelay[END_REF]. The phosphoryl group is then transferred through the phosphorelay from Spo0F to the phosphotransferase Spo0B, and then to Spo0A (Jiang et al. 2000a). Certain Rap proteins indirectly inhibit the phosphorylation of Spo0A by dephosphorylating Spo0F and thus impair the initiation of sporulation (Perego and Hoch 1996).

Eleven rap genes (from rapA to rapK) were identified on the chromosome of the B. subtilis 168 strain. Functional studies have shown that RapA, RapB, RapE, RapH, RapI, and RapJ can dephosphorylate Spo0F (Jiang et al. 2000b;2011[START_REF] Parashar | Structural basis of response regulator dephosphorylation by rap phosphatases[END_REF], 2013a;;Perego et al. 1996;[START_REF] Smits | Temporal separation of distinct differentiation pathways by a dual specificity Rap-Phr system in Bacillus subtilis[END_REF]). RapC, RapD, RapF, RapH, and RapK regulate competence by inhibiting ComA [START_REF] Auchtung | Modulation of the ComA-dependent quorum response in Bacillus subtilis by multiple rap proteins and Phr peptides[END_REF][START_REF] Bongiorni | Synergistic regulation of competence development in Bacillus subtilis by two Rap-Phr systems[END_REF][START_REF] Core | TPR-mediated interaction of RapC with ComA inhibits response regulator-DNA binding for competence development in Bacillus subtilis[END_REF][START_REF] Ogura | Bacillus subtilis rapD, a direct target of transcription repression by RghR, negatively regulates srfA expression[END_REF][START_REF] Smits | Temporal separation of distinct differentiation pathways by a dual specificity Rap-Phr system in Bacillus subtilis[END_REF]), RapG regulates extracellular protease production by inhibiting DegU [START_REF] Ogura | Binding of response regulator DegU to the aprE promoter is inhibited by RapG, which is counteracted by extracellular PhrG in Bacillus subtilis[END_REF], and RapI also regulates the mobility of the ICEBs1 genetic element [START_REF] Auchtung | Regulation of a Bacillus subtilis mobile genetic element by intercellular signaling and the global DNA damage response[END_REF]. rap-phr genes have also been identified in B. subtilis plasmids. These plasmid systems are involved in the regulation of proteolytic enzyme production [START_REF] Koetje | A plasmid-borne Rap-Phr system of Bacillus subtilis can mediate cell-density controlled production of extracellular proteases[END_REF], sporulation, competence, biofilm formation (Parashar et al. 2013b), and plasmid conjugation [START_REF] Singh | Mobility of the native Bacillus subtilis conjugative plasmid pLS20 is regulated by intercellular signaling[END_REF]. The activities of Rap proteins are inhibited by their cognate Phr peptides. Phr-encoding genes are located downstream from the rap genes and are generally co-transcribed, although many phr genes have a secondary promoter [START_REF] Perego | Pentapeptide regulation of aspartylphosphate phosphatases[END_REF][START_REF] Mcquade | Control of a family of phosphatase regulatory genes (phr) by the alternate sigma factor sigma-H of Bacillus subtilis[END_REF]. The pro-Phr are secreted and processed in the extracellular environment. The mature Phr are then internalized within the bacterial cells by oligopeptide permeases and bind to Rap proteins to inhibit their activity [START_REF] Perego | A peptide export-import control circuit modulating bacterial development regulates protein phosphatases of the phosphorelay[END_REF].

The Rap-Phr systems are also present in bacteria of the B. cereus group. [START_REF] Bongiorni | Rap phosphatase of virulence plasmid pXO1 inhibits Bacillus anthracis sporulation[END_REF] have identified five rap genes in the B. anthracis A2012 strain, among which only two were shown to inhibit sporulation. More recently, [START_REF] Fazion | A plasmid-borne Rap-Phr system regulates sporulation of Bacillus thuringiensis in insect larvae[END_REF] characterized the Rap-Phr system from a small plasmid (pHT8_1) of the B. thuringiensis HD73 strain and demonstrated its involvement in the regulation of sporulation in insect larvae.

In this study, we performed an overview of the Rap-Phr systems in the B. cereus group, including their identification, distribution, and prediction of their sporulation activity. We show that the Rap-Phr systems are widespread in all strains, in both chromosomes and plasmids, and with great sequence variability. A comparison between the B. cereus

Materials and methods

Bacterial genomes

The B. cereus group strains with a complete genome sequence available in the NCBI Genome database (http:// www.ncbi.nlm.nih.gov/genom e/) on April 2015 were selected. The species classification was considered as found in the database in the moment of data collection. The main available features (chromosome size and GC content, MLST sequence type, size and number of plasmids, and the proportion of the total genome they represent) of the genomes from the 49 selected strains are presented in Table 1 and Online Resource Table 1.

Construction of the rap-phr database

Each selected genome sequence was screened for the presence of rap genes. Three strategies were used: (i) using 'rap' and 'response regulator aspartate phosphatase' as keywords; (ii) using each identified sequence for a sequence similarity search by BLASTn against all selected genomes and; (iii) using all chromosomal Rap protein sequences from the B. subtilis 168 strain as a query for a tBLASTn alignment with B. cereus group genomes. All protein sequences were analyzed using InterProScan (http://www.ebi.ac.uk/inter pro/seque nce-searc h) and SMART (http://smart .embl-heide lberg .de) servers for the identification of domains and motifs [START_REF] Jones | InterProScan 5: genome-scale protein function classification[END_REF][START_REF] Letunic | SMART: recent updates, new developments and status in 2015[END_REF]. The detection of TPR domains and the absence of HTH DNA-binding domains were used as the main criteria to validate putative sequences as Rap proteins. Sequences shorter than 333 amino acids or showing uncharacteristic domains for Rap proteins were excluded.

Identification of phr genes was performed considering the gene organization: (i) short open reading frames (encoding 35-120 aa) and location, (ii) overlapping the rap gene (up to 10 bp) at the 3′ terminal end or located immediately downstream (up to 100 bp) from the rap gene, and (iii) transcription from the same DNA strand as the rap gene. When a phr gene was not identified by this strategy, the downstream region of rap was scanned for small open reading frames (ORFs) checking for putative unannotated phr using the VectorNTI software (Invitrogen). The identified systems were numbered according to their location (first the chromosomal genes, then the plasmid ones). The chromosomal genes were numbered according to their order from the replication origin.

A 20 kb region around the rap genes (10 kb upstream and 10 Kb downstream) was analyzed using the ISfinder database [START_REF] Siguier | ISfinder: the reference centre for bacterial insertion sequences[END_REF] to verify the presence of mobile elements within these genomic regions.

Rap protein clustering

Phylogenetic trees were constructed with Rap protein sequences according to the location of their genes in the genome: (i) total (all sequences), (ii) chromosome, and (iii) plasmids. The DAMBE program [START_REF] Xia | DAMBE5: a comprehensive software package for data analysis in molecular biology and evolution[END_REF]) was used to gather sequences with 100% identity. Unique Rap sequences were aligned using the MUSCLE algorithm, and MEGA 6 [START_REF] Tamura | MEGA6: Molecular evolutionary genetics analysis version 6.0[END_REF]) was used to build phylogenetic trees by the Neighbor-Joining Method with the best model corrections for each alignment. RapH from B. subtilis 168 was used as outgroup in the Total tree, which was visualized on iTOL-Interactive Tree Of Life [START_REF] Letunic | Interactive tree of life (iTOL) v4: recent updates and new developments[END_REF]. For Rap clustering delimitation, a value of 0.8 was used as the cutoff using the average distance of the number of amino acid substitutions for chromosomal and plasmid trees. These trees were visualized on MEGA 6. For the Phr peptide, identical sequences recognition was done as described for the Rap proteins.

Multilocus sequencing type (MLST) tree

According to the scheme of [START_REF] Tourasse | The Bacillus cereus group: novel aspects of population structure and genome dynamics[END_REF], sequences of MLST housekeeping genes of all selected genomes were obtained from the 'University of Oslo's Bacillus cereus group MultiLocus and MultiData Typing website (http:// mlsto slo.uio.no)'. Sequences of adk, glpT, glpF, panC, pycA, ccpA, and pta genes were downloaded already concatenated. Alignment and phylogenetic tree development were performed as described for Rap proteins.

Plasmids construction and growth conditions

To assess the effect of Rap proteins on sporulation, seven plasmid rap genes (rap6-BtHD1, rap8-BtHD1, rap10-BtHD1, rap6-Bt407, rap7-Bt407, rap8-Bt407 and rap7-BtHD73) and three chromosomal rap genes (rap1-BcATCC14579, rap2-BcATCC14579 and rap5-BtHD73) were cloned in the plasmid pHT315-P xylA , a multi-copy vector with xylose-inducible promoter [START_REF] Grandvalet | Identification of genes involved in the activation of the Bacillus thuringiensis inhA metalloprotease gene at the onset of sporulation[END_REF]. All genes were amplified by PCR using primers listed in Online Resource Table 2 and ligated to the plasmid pHT315-P xylA using the appropriate restriction sites. For cloning steps, these plasmids were transformed in Escherichia coli K-12 strain TG1 and then in the 1 3

Dam -Dcm -E. coli strain ET12567 (Stratagene, La Jolla, CA, USA) by thermal shock. Finally, each constructed plasmid was transformed by electroporation [START_REF] Lereclus | Transformation and expression of a cloned δ -endotoxin gene in Bacillus thuringiensis[END_REF]) in the acrystalliferous (Cry -) B. thuringiensis var. kurstaki HD73 strain [START_REF] Wilcks | Characterization of plasmid pAW63, a second self-transmissible plasmid in Bacillus thuringiensis subsp. kurstaki HD73[END_REF]. Luria-Bertani (LB) medium was used to cultivate E. coli and B. thuringiensis at 37 °C for DNA preparation. The medium HCT was used to optimize the sporulation of B. thuringiensis [START_REF] Lereclus | Molecular relationships among plasmid of Bacillus thuringiensis: conserved sequences through 11 crystalliferous Strains[END_REF]. Antibiotics were used at the following concentration: ampicillin 100 µg/mL for E. coli and erythromycin 10 µg/mL for B. thuringiensis.

DNA manipulation

Genomic DNA from the three B. thuringiensis strains (HD-1, Bt407, and HD73) and the B. cereus strain (ATCC14579) was extracted using the Puregene Yeast/ Bact. Kit B (Qiagen, France). PCRs were performed in an Applied Biosystems 2720 Thermal cycler (Applied Biosystem, USA) with Phusion High-Fidelity or Taq DNA Polymerase (New England Biolabs, USA) and oligonucleotides (Online Resource Table 2) were synthesized by Eurofins Genomics (Germany). The QIAquick PCR Purification Kit (Qiagen, France) was used to purify the amplified DNA fragments that were subsequently treated with appropriated restriction enzymes (New England Biolabs). Digested DNA fragments were separated on 1% agarose gels and purified from gels using the QIAquick gel extraction kit (Qiagen, France). T4 DNA ligase and restriction enzymes were used following the manufacturer's recommendations (New England Biolabs). E. coli plasmid DNA extractions were performed using the QIAprep Spin Miniprep Kit (Qiagen, France). DNA sequencing was carried out by GATC Biotech (Konstanz, Germany).

Sporulation assay

The sporulation efficiency of B. thuringiensis HD73 strain expressing rap genes was determined in the sporulationspecific medium HCT supplemented with 20 mM of xylose at the beginning of stationary growth phase. After 48 h of growth at 30 °C, serial dilutions were plated before and after heat treatment for 12 min at 80 °C. The sporulation percentage was calculated as 100 × the ratio between heat-resistant spores per milliliter and total viable cells per milliliter. All experiments were repeated at least three times, and the mean values (± standard error of the mean) were calculated.

Statistical analyses

The

Results

Genomic overview of the B. cereus group strains

To study the occurrence, the prevalence and the distribu- andB. cytotoxicus), only one genome for each species was chosen. When the same strain was sequenced twice, only one was selected. Following these criteria, 49 genomes were used in this study (Table 1).

The GC content of these genomes was around 35%, with minor differences among the strains. In the B. cereus strains, the chromosomal size ranged from 5.08 to 5.46 Mb, while in B. thuringiensis the variation was from 5.21 to 6 Mb. All the other species showed chromosomes larger than 5 Mb, except B. cytotoxicus NVH 391-98, which was 4.09 Mb (Table 1). The analysis of the plasmid content of the 49 selected strains revealed a total of 197 plasmids with a size ranging from 2.1 to 502 Kb (Online Resource Table 3). These plasmids can provide a significant increase in genome size (Table 1). In B. cereus, the average increase of the genome size is 5% with a maximum of 12% in the strain B. cereus 03BB108. In B. thuringiensis, the average increase of the genome size was 12%, with a maximum of 24% in the B. thuringiensis MC28 and B. thuringiensis IS5056 strains.

rap-phr genes distribution

In the 49 genomes sequences, 302 rap genes were identified (Online Resource Data 1 and Online Resource Table 4) whose 144 (47.7%) were correctly annotated as 'response regulator aspartate phosphatase' or 'rap'. A phr gene was identified downstream from all rap genes, but 31 of them were not annotated as ORFs (Online Resource Table 4). The rap and phr genes were always located in the same DNA strand with a slight overlapping (usually 1 or 4 nucleotides). The average size was 1099 bp for the rap genes (from 1032 to 1185 bp) and 166 bp for the phr genes (from 120 to 330 bp).

From 2 to 16 rap-phr genes were identified in all strains (Fig. 1a, Online Resource Table 4). Two to eight chromosomal rap-phr systems were found by strain, representing 70% of all the identified rap-phr genes (Fig. 1b). Plasmid rap-phr genes were found in 27 of the 49 strains and on 65 of the 197 plasmids. Some strains harbor a large number of plasmid rap-phr genes, up to 12 plasmid systems for the B. thuringiensis strain IS 5056 (Online Resource Table 3). The size of plasmids harboring rap-phr genes varied from 6.88 to 502 kb. However, the occurrence of these genes is most common (70%) in plasmids larger than 70 kb (Online Resource Table 3). Also, large plasmids (> 200 kb) might contain several rap-phr genes, up to five, such as the pBMB422 plasmid (422.7 kb) of the B. thuringiensis YBT-1520 strain (Online Resources Table 3 and4).

The average number of chromosomal rap-phr genes is similar between B. cereus and B. thuringiensis (4.04 and 4.45, respectively). However, the average number of plasmid rap-phr genes is sixfold higher in B. thuringiensis than in B. cereus (3.6 and 0.6, respectively; P = 0.001). The percentage of plasmids harboring rap-phr genes is higher in B. thuringiensis than in B. cereus (38.6% versus 22.0%, respectively, P = 0.03). Moreover, B. thuringiensis strains show a similar amount of chromosomal and plasmid rap-phr genes, while in the B. cereus strains the number of chromosomal rap-phr genes is almost seven times higher (P < 0.001) than plasmid genes (Table 2).

As the genome of B. cereus group strains are rich in mobile and repeated elements [START_REF] Kolstø | What sets Bacillus anthracis apart from other Bacillus species?[END_REF]), the presence of these elements in the vicinity of rap-phr genes was analyzed. A mobile element was found in the vicinity of 48% of all the rap genes, corresponding to 39% of the chromosomal genes and 68% of the plasmid ones (Online Resource Data 1). The prevalence of mobile elements in the 20-kb region around rap-phr chromosomal genes is similar between B. cereus and B. thuringiensis (Table 2). However, these elements were found in the vicinity of 36% of plasmid rap genes from B. cereus against 72% of B. thuringiensis ones (P < 0.01).

Rap protein clustering

Among the 302 Rap and Phr proteins initially identified, we distinguished 192 different Rap protein sequences and 152 different Phr pro-peptide sequences, corresponding to 63.5% and 50.3% of all sequences, respectively. Rap protein sequences with 100% of identity were found in several strains from different species (Fig. 2). The identical sequences are always identified on different strains and on the same type of replicon (chromosome or plasmid), except for the chromosomal Rap1 from the B. cereus 03BB87 strain, which is identical to the plasmid Rap3 of the pBCX01 plasmid from the B. cereus G9241 strain. To investigate this unique case, a 10-kb region around the rap1 gene from the B. cereus 03BB87 strain was analyzed with BLASTn. By this approach, we detected 99-100% identity to the pXO1 and pXO1-like plasmids. A BLAST alignment between the pXO1 plasmid and the entire B. cereus 03BB87 chromosome highlighted a region corresponding to 3% of the chromosome with 98% of coverage and 99% of identity with the plasmid. Moreover, the B. cereus 03BB87 strain also carries a pXO1-like plasmid, the pBCX01, sharing 97% identity with pXO1 with less than 10% of coverage (data not shown). This might be due to a mistake in the genome assembly or to the integration of DNA regions from the pBCX01 plasmid in the chromosome.

Three phylogenetic trees were constructed using Rap protein sequences, according to the location of their encoding genes in the genome (all sequences, chromosome or plasmid sequences) (Fig. 2 and Online Resource Fig. 1). These trees were built by the Neighbor-Joining method with corrections based on the Jones-Taylor-Thornton matrix model that enables the assessment of the overall divergence among Rap proteins. The phylogenetic tree with all sequences showed a high number of branches with a dispersion of plasmid sequences in several branches. Closely related Rap sequences are encoded by similar kind of plasmids distributed in eight major sets (Fig. 2, sets A-H). Set A comprises rap genes harbored by an identical 8.5 kb plasmid present in different strains and set B by plasmids higher than 70 kb. Set C includes pXO1-like plasmids; set D plasmids larger than 200 kb, while plasmids from set E are smaller than 20 kb. Set F includes plasmids around 70 kb and set G plasmids larger than 200 kb that can also harbor Rap sequences from set B. The set H is the most versatile since it comprises two distinct subsets: one with plasmids larger than 400 kb and another with plasmids ranging from 75 to 235 kb.

The chromosomal tree is divided into 12 Rap groups (Online Resource Fig. 1a). Some groups are composed of several sequences, such as Group C6 which includes 38 Rap sequences. However, some groups have few Rap sequences, such as Group C11 with Rap4 from B. anthracis Ames Ancestor and Rap5 from B. cytotoxicus NHV391-98. Also, Group C10 is formed by three identical sequences from two B. cereus emetic strains (B. cereus AH187 and B. cereus NC7401) and from BcQ1. The plasmid tree is separated into nine Rap groups (Online Resource Fig. 1b). Likewise in the

Phr peptides

In silico determination of the mature Phr sequences is complex due to the high variability of these sequences within the B. cereus group. Indeed, the mature Phr already described from the B. cereus group revealed some differences in size and location within the pro-peptide. The BXA0205Phr from the pXO1 plasmid (Phr5-B. anthracis) is a pentapeptide while the Phr8 from the pHT8_1 plasmid (Phr8-B. thuringiensis HD73) is an heptapeptide but both are located in the C-terminal end. Moreover, the active form of the BA3791Phr from B. anthracis (Phr3-B. anthracis) is located within the C-terminal region of its precursor, but its exact sequence was not determined [START_REF] Bongiorni | Rap phosphatase of virulence plasmid pXO1 inhibits Bacillus anthracis sporulation[END_REF][START_REF] Fazion | A plasmid-borne Rap-Phr system regulates sporulation of Bacillus thuringiensis in insect larvae[END_REF]. However, all these mature peptides present a positively charged residue, the typical feature of Phr active form [START_REF] Pottathil | The extracellular Phr peptide-Rap phosphatase signaling circuit of Bacillus subtilis[END_REF]. The Phr sequences from the B. cereus group present a great variability in their amino acid sequences and in their sizes (Online Resources Data 1). The mature Phr peptides from the B. cereus group described above were sought among the Phr sequences in our database.

While the mature Phr8-B. thuringiensis HD73 (YAHGKDI) was identified only on identical Phr sequences, the Phr5-B. anthracis (GHTGG) was found in several sequences. A great number of longer Phr possess the GDTGG/GDGGG/ GETGG repetition sequences duplication described by [START_REF] Even-Tov | Transient duplication-dependent divergence and horizontal transfer underlie the evolutionary dynamics of bacterial cell-cell signaling[END_REF]. These sequences were defined as the putative autoinducer sequences although they are not generally associated with a positively charged residue. However, the Phr peptides containing these repetitions also bear an ARPDY sequence, which could be the active form.

The relationship between MLST phylogenetic tree and Rap distribution

The 49 selected strains are distributed into six of the seven phylogenetic clusters determined by [START_REF] Guinebretière | Ecological diversification in the Bacillus cereus Group[END_REF]. These clusters, established from the Bacillus cereus group MultiLocus and MultiData Typing website [START_REF] Tourasse | Extended and global phylogenetic view of the Bacillus cereus group population by combination of MLST, AFLP, and MLEE genotyping data[END_REF], are supported by recent results of a pangenomic study of this clade [START_REF] Bazinet | Pan-genome and phylogeny of Bacillus cereus sensu lato[END_REF]. Moreover, the MLST data are still effective in discriminating variation of biology, ecology and host association among this group strains [START_REF] Raymond | In defense of Bacillus thuringiensis, the safest and most successful microbial insecticide available to humanity-a response to EFSA[END_REF]. The cluster III, including B. anthracis strains, emetic strains, and other pathogenic strains (mainly composed by B. cereus strains), and the cluster IV with B. cereus and B. thuringiensis strains from diverse environmental sources (mainly formed by B. thuringiensis strains) are the more extensively represented (Fig. 3). The Rap profile of each strain was analyzed in relation to the MLST tree of the B. cereus group. We observed that any Rap group was not present in all strains and that a Rap group was not exclusively related to an MLST cluster. None of the strains has more than five different Rap chromosomal groups, and a same Rap group can be present more than once in the same strain. As expected, the Cluster IV (mainly composed of B. thuringiensis strains) has a higher number of plasmid Rap. Additionally, phylogenetically related strains show a similar Rap profile both for chromosomal and plasmid groups. For example, the closely related strains B. thuringiensis Bt407 Cry -, B. thuringiensis CT-43, and B. thuringiensis IS5056 show an identical chromosomal profile and similar plasmid profile. These three strains might derive from a same parental strain, for example the B. thuringiensis strain 407 Cry + from the serotype 1 [START_REF] Lereclus | Transformation and expression of a cloned δ -endotoxin gene in Bacillus thuringiensis[END_REF].

Sporulation activity prediction

Considering the importance of sporulation for the survival and dispersion of Bacillus and the role of some Rap-Phr systems in this process, we aimed to predict the activity of Rap proteins from the B. cereus group on sporulation. The RapH residues E45, D46, Q47, L50, F58, L96, D134, E137, and Y175 have been described to be involved in the binding and the dephosphorylation of Spo0F in B. subtilis [START_REF] Parashar | Structural basis of response regulator dephosphorylation by rap phosphatases[END_REF]. First, we used the sequences of Rap proteins interacting with Spo0F to define a consensual sequence of residues potentially involved in the sporulation process (Online Resource Table 5). Next, the Rap protein sequences of each chromosomal and plasmid groups were separately aligned with the RapH sequence, and the presence of the nine key residues was examined (Online Resource Table 5).

Depending on the presence of these residues, the Rap proteins from the B. cereus group were classified as Spo + (predicted phosphatase activity on Spo0F) or Spo-(no predicted phosphatase activity) (Online Resource Table 4). This analysis showed that 97 of the 302 Rap proteins display a Spo + profile (32%) and the predicted Rap Spo + are more frequently found in plasmids (65% of plasmid Rap) than in chromosomes (18% of chromosomal Rap) (Fig. 4a). However, there is no correlation between Rap groups and predicted sporulation function because there are groups with only Spo-or Spo + Rap proteins, as well as mixed groups. Most of the chromosomal groups were exclusively Spo-and the plasmid groups are mainly mixed (Online Resource Table 5). Interestingly, the amount of Rap Spo + is significantly higher (P < 0.01) in B. thuringiensis than in B. cereus (40% and 26%, respectively) (Fig. 4a). Ten B. cereus strains (42%) do not harbor chromosomal Rap Spo + , including four strains that do not have any Rap Spo + at all (Online Resource Table 4). The B. thuringiensis HD-1 strain, widely used as a biopesticide against lepidopteran insects, has nine Rap Spo +. In sharp contrast, five B. thuringiensis strains do not have chromosomal Rap Spo +, and the nematicidal B. thuringiensis YBT1518 strain is the only one B. thuringiensis to have no predicted Rap Spo + at all.

To validate the in silico prediction, ten Rap proteins representative of various plasmid or chromosomal groups with a predicted Spo + (seven Rap) or Spo-(three Rap) activity were selected to study their effect on sporulation (Online Resource Table 5). The corresponding rap genes were expressed under the xylose-inducible promoter P xylA in the B. thuringiensis HD73 strain, and the sporulation efficiency was measured after 48 h at 30 °C in a sporulation-specific medium (HCT). The three Rap Spo-(rap6-BtHD1, rap6-Bt407, and rap7-BtHD73) did not inhibit the sporulation efficiency compared to the control strain, confirming their prediction. Among the Rap Spo + analyzed, the expression of rap8-Bt407, rap10-BtHD1, rap7-Bt407, and rap5-HD73 strongly prevent the sporulation (P < 0.001) while the expression of rap1-BcATCC14579, and rap2-BcATCC14579 slightly inhibited the sporulation efficiency compared to the control strain (P < 0.01) (Fig. 4b, Online Resource Table 6). However, the rap8-BtHD1, predicted Rap Spo + does not display a role in sporulation in our B. thuringiensis HD73 model strain in this growth condition. Hence, the sporulation results allow us to confirm the predicted phenotype for nine of the ten tested Rap.

Discussion

Despite the importance of the Rap-Phr systems in the regulation of various essential pathways, they have been poorly studied in the bacteria of the B. cereus group [START_REF] Bongiorni | Rap phosphatase of virulence plasmid pXO1 inhibits Bacillus anthracis sporulation[END_REF][START_REF] Fazion | A plasmid-borne Rap-Phr system regulates sporulation of Bacillus thuringiensis in insect larvae[END_REF][START_REF] Slamti | Quorum sensing in Bacillus thuringiensis is required for completion of a full infectious cycle in the insect[END_REF]).

Here, we provide a complete and detailed overview of these systems in this group, concerning their prevalence, sequence diversity, relevant association to plasmids and their role in sporulation. We show that the rap genes are widespread in all the studied strains of the B. cereus group and that a putative phr gene is always present immediately downstream from all rap genes. The rap-phr genes are always encoded on the same DNA strand but in different transcription frames, a characteristic of the RNPP family [START_REF] Declerck | Structure of PlcR: insights into virulence regulation and evolution of quorum sensing in Gram-positive bacteria[END_REF]. Genes coding for the Phr peptides were diverse in size and the occurrence of phr genes two times longer than the average could be explained by the duplication of the region coding for the mature signaling peptide, important for the evolutionary diversification of Rap-Phr specificity [START_REF] Even-Tov | Transient duplication-dependent divergence and horizontal transfer underlie the evolutionary dynamics of bacterial cell-cell signaling[END_REF]). The rap-phr genes are located in all chromosomes and numerous plasmids. The total number of chromosome rap-phr genes is similar between B. cereus and B. thuringiensis species. However, chromosome size seems unrelated to the amount of chromosomal rap-phr genes as the B. cytotoxicus NVH 391-98 strain (distantly related to other strains of the B. cereus group) presents the smallest chromosome and the largest number of chromosomal rap-phr genes.

A global analysis of rap genes in the genus Bacillus showed that species from the B. subtilis group contain 11 (± 2) rap genes in opposition to six (± 3) for the B. cereus group species [START_REF] Even-Tov | Transient duplication-dependent divergence and horizontal transfer underlie the evolutionary dynamics of bacterial cell-cell signaling[END_REF]. In agreement with this study, we found a similar number of rap genes in the B. cereus group (6.2 ± 3.2). Another difference between the two Bacillus groups is related to the phr gene occurrence. While 27% of the rap genes from B. subtilis strain 168 do not have an associated phr gene [START_REF] Perego | Forty years in the making: understanding the molecular mechanism of peptide regulation in bacterial development[END_REF], a common pattern for the B. subtilis group [START_REF] Even-Tov | Transient duplication-dependent divergence and horizontal transfer underlie the evolutionary dynamics of bacterial cell-cell signaling[END_REF], a putative phr gene is always located downstream from all identified rap genes in the B. cereus group. Interestingly, Rap proteins from the B. subtilis and B. cereus species constitute two independent clusters, suggesting that the diversification of the Rap sequences occurred after the evolutionary separation of the two bacterial groups [START_REF] Even-Tov | Transient duplication-dependent divergence and horizontal transfer underlie the evolutionary dynamics of bacterial cell-cell signaling[END_REF]. After this separation, Rap-Phr systems from the B. cereus group might have been subjected to genetic variations that also evolved this quorumsensing system to the other RNPP family systems, like PlcR-PapR and NprR-NprX. The role of these QS systems in the production of extracellular proteases or sporulation corresponds to functions performed by some Rap-Phr systems of B. subtilis [START_REF] Auchtung | Modulation of the ComA-dependent quorum response in Bacillus subtilis by multiple rap proteins and Phr peptides[END_REF][START_REF] Ogura | Binding of response regulator DegU to the aprE promoter is inhibited by RapG, which is counteracted by extracellular PhrG in Bacillus subtilis[END_REF]. Interestingly, NprR, which presents a Rap-like structure combined with an HTH DNA-binding domain, was suggested to be the evolutionary intermediate between Rap proteins and the other regulators of the RNPP family (Perchat et al. 2016a). This hypothesis could also explain the difference in the number of rap-phr genes between B. subtilis and B. cereus. Our analysis revealed that 30% of the identified rap-phr genes were plasmid-born. The plasmid rap-phr genes have been described in diverse Bacillus species [START_REF] Koetje | A plasmid-borne Rap-Phr system of Bacillus subtilis can mediate cell-density controlled production of extracellular proteases[END_REF]Parashar et al. 2013b;[START_REF] Singh | Mobility of the native Bacillus subtilis conjugative plasmid pLS20 is regulated by intercellular signaling[END_REF][START_REF] Yang | A plasmidborn Rap-Phr system regulates surfactin production, sporulation and genetic competence in the heterologous host, Bacillus subtilis OKB105[END_REF], including the B. cereus group [START_REF] Bongiorni | Rap phosphatase of virulence plasmid pXO1 inhibits Bacillus anthracis sporulation[END_REF][START_REF] Chao | Complete nucleotide sequence of pBMB67, a 67-kb plasmid from Bacillus thuringiensis strain YBT-1520[END_REF][START_REF] Fazion | A plasmid-borne Rap-Phr system regulates sporulation of Bacillus thuringiensis in insect larvae[END_REF][START_REF] Slamti | Quorum sensing in Bacillus thuringiensis is required for completion of a full infectious cycle in the insect[END_REF][START_REF] Van Der Auwera | Conjugative plasmid pAW63 brings new insights into the genesis of the Bacillus anthracis virulence plasmid pXO2 and of the Bacillus thuringiensis plasmid pBT9727[END_REF]. These plasmid rap-phr genes are carried by a wide range of plasmids but are mainly located in conjugative plasmids greater than 70 kb and are more abundant in B. thuringiensis than in B. cereus. This difference is not only the consequence of a higher number of plasmids in B. thuringiensis, which has only twice as many plasmids as B. cereus and six-times as many rap-phr plasmid genes. Moreover, this difference might be explained by the higher presence of mobile elements close to plasmid rap genes in B. thuringiensis than in B. cereus. We have shown that plasmids larger than 200 kb can host multiple rap-phr genes, and it is also remarkable that rap-phr genes were found on virulence plasmids, such as pXO1 in B. anthracis and pCER270 in B. cereus, as well as in some Cry plasmids in B. thuringiensis. However, we did not find specific rap-phr genes associated to the Cry plasmids. The multiplicity of the rap-phr genes in the B. cereus group creates appropriate conditions for their diversity and evolution, as demonstrated by [START_REF] Even-Tov | Transient duplication-dependent divergence and horizontal transfer underlie the evolutionary dynamics of bacterial cell-cell signaling[END_REF]. Thus, different rap-phr genes can respond to various signals and be regulated differently at transcriptional level.

Identical Rap protein sequences are located on the same type of replicon: (i) in the chromosome as a consequence of common ancestor, or (ii) in plasmids as a consequence of conjugation events. The proportion of identical Phr is higher than that of identical Rap. This characteristic might allow the Phr peptides to act cooperatively on various Rap proteins from different strains. The mature Phr can be located inside or at the C-terminal extremity of the pro-peptide [START_REF] Pottathil | The extracellular Phr peptide-Rap phosphatase signaling circuit of Bacillus subtilis[END_REF], hampering the identification of the active form of some Phr. Due to this difficulty and to the wide variability of pro-Phr sequences, the Phr phylogenetic tree was not estimated and the evolutive correlation between Rap and Phr was not determined. Nevertheless, this coevolution was described for the NprR-NprX and PlcR-PapR systems [START_REF] Perchat | A cell-cell communication system regulates protease production during sporulation in bacteria of the Bacillus cereus group[END_REF][START_REF] Slamti | Specificity and polymorphism of the PlcR-PapR quorum-sensing system in the Bacillus cereus group[END_REF].

The correlation between the Rap-Phr system distribution and the MLST tree based on housekeeping genes revealed that closely related strains harbor a similar Rap-Phr system pattern, suggesting a similar evolutionary history of both genetic characters. This correlation was not observed for the PlcR and NprR regulators from the RNPP family [START_REF] Ko | Population structure of the Bacillus cereus group as determined by sequence analysis of six housekeeping genes and the plcR gene[END_REF][START_REF] Perchat | A cell-cell communication system regulates protease production during sporulation in bacteria of the Bacillus cereus group[END_REF]. Also, the closest strains have a similar profile, even for plasmid genes, suggesting a beneficial association of particular plasmid-chromosome combinations that leads to the maintenance and propagation of these proficient combinations [START_REF] Méric | Lineage-specific plasmid acquisition and the evolution of specialized pathogens in Bacillus thuringiensis and the Bacillus cereus group[END_REF].

One-third of the Rap proteins of the B. cereus group are predicted to have phosphatase activity on Spo0F. However, some strains did not have any predicted Rap Spo + (four B. cereus strains, one B thuringiensis strain, B. mycoides and B. pseudomycoides). This absence of Rap Spo + could be compensated by NprR, which has a Rap-like activity on the sporulation phosphorelay and which is conserved in all the strains of the B. cereus group (Perchat et al. 2016b). In these strains, Rap proteins might have undergone genetic variations leading to the loss of sporulation function and the acquisition of new undetermined functions. Interestingly, Rap proteins predicted to regulate sporulation are mainly encoded by plasmid genes and, therefore, are more abundant in B. thuringiensis. These plasmid Rap-Phr systems could help B. thuringiensis to adapt and survive in its complex ecological niche, the insect. Moreover, many Rap-Phr plasmid systems are located on cryptic plasmids less than 16 kb in size. These plasmid genes could regulate different beneficial functions leading to the maintenance of these plasmids in the bacterial cell. Recently, the plasmid pHT8_1 from the B. thuringiensis HD73 strain has been characterized, and the role of its Rap-Phr system in the regulation of the sporulation process in insect larvae has been demonstrated [START_REF] Fazion | A plasmid-borne Rap-Phr system regulates sporulation of Bacillus thuringiensis in insect larvae[END_REF].

Sporulation assays validated the in silico prediction except for one Spo + Rap. The residues involved in RapH-Spo0F interaction are highly conserved in B. subtilis and B. cereus, and small differences are sufficient to lose this activity. However, these key residues are relatively well conserved in Rap proteins that do not regulate the sporulation pathway, suggesting that the Rap activity on sporulation was the ancestral role of these proteins [START_REF] Even-Tov | Transient duplication-dependent divergence and horizontal transfer underlie the evolutionary dynamics of bacterial cell-cell signaling[END_REF]. Thus, the presence of these residues is a good indicator but is not sufficient to predict the sporulation activity. The role of the Rap Spo-remains the main unsolved question. Given the large amount of plasmid Rap-Phr and of the role of these systems in plasmid conjugation in B. subtilis [START_REF] Singh | Mobility of the native Bacillus subtilis conjugative plasmid pLS20 is regulated by intercellular signaling[END_REF]), it will be interesting to study this phenotype.

Cell-cell communication systems enable a fine regulation of important processes in bacteria. Indeed, some Rap-Phr systems regulate sporulation that allows bacteria to adapt, survive and disseminate. This work highlights the importance of Rap-Phr systems linked to genetic mobile elements in the B. cereus group, especially in B. thuringiensis. This location on mobile elements could increase the spreading of these genes in bacteria of the B. cereus group.

1 3 COFECUB (Comité Français d'Evaluation de la Coopération Universitaire et Scientifique avec le Brésil). Cardoso, P.F. was supported by fellowships from CAPES-finance code 001 and CAPES/COFECUB and Vilas-Boas, L. A. by CAPES/COFECUB. We would like to thank the reviewers who helped to improve this manuscript.

Fig. 1

 1 Fig. 1 Distribution of the identified rap-phr genes in the B. cereus group. a The number of rap-phr genes by strain with the chromosomal systems in blue and the plasmid systems in red. b Percentage of the rap-phr genes by replicon with chromosomes in blue and plasmids in red

Fig. 2

 2 Fig.2Phylogenetic tree of all Rap protein sequences (plasmid and chromosome sequences) inferred by Neighbor-Joining method (conducted in MEGA6) and visualized by iTOL[START_REF] Letunic | Interactive tree of life (iTOL) v4: recent updates and new developments[END_REF]. RapH from B. subtilis 168 was used as outgroup. The evolutionary distances were computed using the JTT matrix-based method and are

Fig. 3

 3 Fig. 3 Distribution of the chromosomal and plasmid Rap-Phr systems groups in relation to the MLST phylogenetic tree (Bacillus cereus group MultiLocus and MultiData Typing website-http:// mlsto slo.uio.no). MLST clusters proposed by Guinebretière et al.

1 3

 1 and B. thuringiensis strains showed that plasmid Rap-Phr systems are more frequently present in B. thuringiensis than in B. cereus. One-third of the Rap proteins were predicted to have a sporulation function and these Rap proteins are preferentially located on plasmids and, therefore, are mainly present in B. thuringiensis.

Table 1

 1 Main features of the 49 selected genomes from the B. cereus group strains Strain (named as in NCBI database) Release date Assembly

	Chromosome	Plasmids	STs (MLST)
	Size (Mb) %GC Nb Size (Mb) % ↑ genome

  (Sequence Types) data was obtained from the University of Oslo's Bacillus cereus group MultiLocus and MultiData Typing website (http:// mlsto slo.uio.no). Plasmid 'Nb' is the sum of different plasmids sequenced for each strain and ' % ↑ genome' is how much all these plasmids increase the genome size of that strain (in relation to chromosome size alone) a This strain was first annotated as B. cereus[START_REF] Johnson | Complete Genome sequences for 35 biothreat assay-relevant Bacillus species[END_REF], and is now classified as B. pseudomycoides (strain BTZ) b Nowadays, B. weihenstephanensis strains are considered as heterotypic synonym of B. mycoides[START_REF] Liu | Genome analysis-based reclassification of Bacillus weihenstephanensis as a later heterotypic synonym of Bacillus mycoides[END_REF] All B. cereus and B. thuringiensis available strains were selected. However, as B. anthracis is a clonal species[START_REF] Helgason | Bacillus anthracis, Bacillus cereus, and Bacillus thuringiensis-one species on the basis of genetic evidence[END_REF][START_REF] Rasko | Genomics of the Bacillus cereus group of organisms[END_REF], only one representative genome sequence (Ames Ancestor strain) was selected, even if around 30 genomes of this species were available. Moreover, our preliminary results revealed identical Rap-Phr profiles in all the B. anthracis isolates, thus confirming the clonal aspect of the B. anthracis strains (data not shown). For the other species(B. mycoides, B. pseudomycoides, B. weihenstephanensis,

	Table 1 (continued) of these species.
	Strain (named as in NCBI database) Release date Assembly	Chromosome	Plasmids	STs (MLST)
				Size (Mb) %GC Nb Size (Mb) % ↑ genome
	B. anthracis 'Ames Ancestor' A2084 2004	GCA_000008445.1 5.23	35.4 2	0.27	5%	1
	B. weihenstephanensis KBAB4 b	2007	GCA_000018825.1 5.26	35.6 4	0.61	12%	118
	B. mycoides ATCC 6462	2015	GCA_000832605.1 5.26	35.5 3	0.38	7%	133
	B. pseudomycoides DSM 12442	2012	GCA_000161455.1 5.78	35.4 -	-	-	132
	B. cytotoxicus NVH 391-98	2007	GCA_000017425.1 4.09	35.9 1	0.01	0%	117
	B. cereus group Average			5.35	35.4 4.1 0.46	9%

tion of Rap-Phr systems in the B. cereus group, those genome sequences with complete assembly level available in NCBI Genome section until April 2015 were selected. Most of these genomes belong to B. cereus, B. thuringiensis or B. anthracis, considering the relevance STs

Table 2

 2 Comparison between B. cereus and B. thuringiensis concerning the Rap-Phr systems Statistical analyses: a unpaired t test; b Mann-Whitney test; c Fisher's exact test; d unpaired t test with Welch correction. ns not significant; *P < 0.05; **P < 0.01; ***P ≤ 0.001

	B. cereus	B. thuringiensis Significance

chromosomal tree, many groups are composed of several sequences, such as Group P2 with 26 Rap proteins. However, Group P4 and Group P5 are composed of unique sequences and Group P6 by two identical sequences.

Acknowledgements

We gratefully acknowledge Jacques Mahillon for the B. thuringiensis var. kurstaki strain HD73 Cry-and Fernanda A. P. Fazion for the plasmid pHT315xyl_rap7061. The project was supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, contract no.: 2065/2014, 23038.001776/2014-29) and

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