Rémy Muller
email: remy.muller@ircam.fr

Thomas Hélie

TRAJECTORY ANTI-ALIASING ON GUARANTEED-PASSIVE SIMULATION OF NONLINEAR PHYSICAL SYSTEMS

This article is concerned with the accurate simulation of passive nonlinear dynamical systems with a particular attention paid on aliasing reduction in the pass-band. The approach is based on the combination of Port-Hamiltonian Systems, continuous-time statespace trajectories reconstruction and exact continuous-time antialiasing filter realization. The proposed framework is applied on a nonlinear LC oscillator circuit to study the effectiveness of the method.

INTRODUCTION

The need for accurate and passive-guaranteed simulation of nonlinear multi-physical systems is ubiquitous in the modelling of electronic circuits or mechanical systems.

Geometric numerical integration [START_REF] Hairer | Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations[END_REF] is a very active research field that provides a theoretical framework for structure and invariant preserving integration of dynamical systems. Port-Hamiltonian Systems (PHS) [START_REF] Van Der Schaft | Port-hamiltonian systems theory: An introductory overview[END_REF] [3] that focus on the energy storage functions and power continuous component interconnections belong to this field and offer a well adapted framework to preserve the system energy (resp. passivity). In the context of nonlinear physical audio systems, it has been applied successfully to the modelling of the wah-wah pedal [START_REF] Falaize | Simulation of an analog circuit of a wah pedal: a port-Hamiltonian approach[END_REF], Fender Rhodes [START_REF] Falaize | Passive simulation of the nonlinear port-Hamiltonian modeling of a Rhodes Piano[END_REF], brass instruments [START_REF] Lopes | Energy Balanced Model of a Jet Interacting With a Brass Player's Lip[END_REF] and the loudspeaker nonlinearities [START_REF] Falaize | Passive simulation of electrodynamic loudspeakers for guitar amplifiers: a port-Hamiltonian approach[END_REF]. Automatic generation of the system equations from a graph of components has been investigated in [START_REF] Falaize | Passive guaranteed simulation of analog audio circuits: A port-hamiltonian approach[END_REF] However the presence of aliasing errors in the numerical simulation is annoying for three reasons. First it causes audible inharmonic audio artefacts. Second it deteriorates the accuracy of the numerical scheme leading to poor convergence rate. Third it requires the use of significant oversampling. This problem is even more pronounced in the case of systems such as sustained instruments that rely on nonlinearities to achieve auto-oscillation.

Aliasing errors in the context of finite elements simulation and some alternatives have been discussed in [START_REF] Boyd | Chebyshev and Fourier Spectral Methods[END_REF] (ch 11). Antialiased waveform generation without oversampling has been proposed in [START_REF] Stilson | Efficiently-variable Non-oversampled Algorithms in Virtual-analog Music Synthesis: A Root-locus Perspective[END_REF]. Static nonlinearity anti-aliasing has also been proposed in [START_REF] V. Zavalishin | Reducing the aliasing of nonlinear waveshaping using continuous-time convolution[END_REF] [START_REF] Bilbao | Antiderivative antialiasing for memoryless nonlinearities[END_REF] by combining exact anti-derivatives and finitedifferences.

Continuous-time input reconstruction has been used in [START_REF] Sarkka | Accurate discretization of analog audio filters with application to parametric equalizer design[END_REF] to simulate the frequency response of LTI systems with higher accuracy. It is also central in collocation-based Runge-Kutta methods that rely on non-uniform polynomial interpolation of the vector field. Splines and in particular uniform B-splines [START_REF] Unser | Think analog, act digital[END_REF] [15] [START_REF] Unser | B-Spline signal processing: Part II-Efficient design and applications[END_REF], [START_REF] Unser | Cardinal exponential splines: Part II-Think analog, act digital[END_REF] also offer a particularly interesting framework to represent and manipulate piecewise continuous-time signals through their digital representations using the standard tools of linear algebra and digital signal processing.

In this article, we try to combine the geometric and the signal processing viewpoints: we choose a physically informed piecewise smooth polynomial reconstruction model based on a discrete sequence of points generated by a passive-guaranteed simulation method.

The paper is organized as follows. We first recall some results about Port-Hamiltonian systems in Section 3, then we consider passive numerical methods in section 4, we talk about piecewisecontinuous trajectory reconstruction in section 5 and continuoustime filtering of piecewise polynomials in section 6. Finally we apply our method to a non linear LC oscillator circuit in section 7.

PROBLEM STATEMENT

Objective

The objective is to simulate nonlinear passive physical audio systems in such a way that:

(i) The nonlinear dynamics is accurately reproduced, (ii) The power balance decomposed into its conservative, dissipative and source parts is satisfied, (iii) The observation operator is designed to reduce the aliasing induced by the nonlinearities.

Approach

To address this problem, the following strategy is adopted. First, trajectories are approximated in the continuous-time domain by smooth parametric piecewise-defined functions, such that the three following properties are fulfilled:

(P1) Regularity: functions and junctions are C k with k ∈ N, (P2) Accuracy: the approximation has accuracy order p, (P3) Passivity: the power balance is globally satisfied for each frame. Second, the anti-aliased output is built a posteriori in three steps:

1. Observe the output from the approximated dynamics in the continuous-time domain, 2. Apply a continuous-time anti-aliasing filter in order to respect the Shannon-Nyquist sampling theorem, 3. Sample the filtered trajectories to convert them back to discretetime.

Methodology

In this article, we restrict ourselves to piece-wise continuous globally C1 polynomial trajectories of the form

x(t) = ∞ n=-∞ xn t -tn h rect ]0,1] t -tn h , t ∈ R (1) with x ∈ R N , xn(τ ), τ ∈ [0, 1]
being a local polynomial model of order r, tn = hn, n ∈ Z and h being the time step parameter. The continuity hypothesis (P1) is expressed mathematically by.

x( )

n+1 (τ ) = x( ) n (τ ) ∀n ∈ Z, ≤ k (2)
For property (P2) the local approximation error between the exact solution and its approximation is defined by

e(h) = x(t0 + h) -x(t0 + h) (3)
provided that x(t0) = x(t0) and it is required that for some p.

e(h) = O(h p+1 ) (4)
Finally to express property (P3) we require the power-balance

E (t) = -P d + Pe (5)
where P d and Pe are respectively the dissipated and external power and E (t) is the instantaneous energy variation of the system.

PORT-HAMILTONIAN SYSTEMS

In this article, nonlinear passive physical audio systems are described under their Port-Hamiltonian formulation. The theory of Port-Hamiltonian Systems (PHS) [START_REF] Van Der Schaft | Port-hamiltonian systems theory: An introductory overview[END_REF] [3] extends the theory of Hamiltonian mechanics to non-autonomous and dissipative open systems. It provides a general framework where the dynamic statespace equations derives directly from an energy storage function and power-conserving interconnection of its subsystems.

Explicit differential form

Consider a system with input u(t) ∈ U = R P , with state x(t) ∈ X = R N and output y(t) ∈ Y = R P with the structured statespace equations [START_REF] Van Der Schaft | Port-hamiltonian systems theory: An introductory overview[END_REF] x = (J(x

) -R(x)) ∇H(x) + G(x)u = f (x, u) y = G(x) T u (6)
where H gives the stored energy of the system

E(t) = (H • x)(t) (7)
with H ∈ C 1 (X, R + ), ∇ being the gradient operator, J = -J T a skew-symmetric matrix and R = R T 0 a positive-semidefinite matrix. The energy variation of this system satisfies the powerbalance given by the derivative chain rule

E (t) = ∇H(x) T x (8)
which can be decomposed as

E (t) = Pc -P d + Pe (9)
with.

Pc = ∇H(x) T J(x)∇H(x) = 0 (10)

P d = ∇H(x) T R(x)∇H(x) ≥ 0 (11) Pe = ∇H(x) T G(x)u (12)
The Pc term is null because J is skew-symmetric: it represents conservative power exchange between storage components in the system. The P d term is positive because R ≥ 0: it represents the dissipated power. Finally the term Pe represents the power brought to the system by the external ports. Equation (9) express the system's passivity property: with external inputs switched off (u = 0) the energy can either be constant (conservative case P d = 0) or decaying (dissipative case P d > 0).

Component-based approach and semi-explicit DAE form

More generally, PHS can be expressed in Differential Algebraic Equation form. When we consider physical systems containing N energy-storage components, M dissipative components and P external interaction ports described by Pc the stored energy level en and its variation law defined by e n = ∇Hn(xn)x n for the state variable xn.

P d the dissipated power qm(w) ≥ 0 with the component's flux and effort variables being in algebraic relation of a single variable w.

Pe the external power upyp brought to the system through this port with up being the controllable input of the system and yp being the observable output.

For a storage component, en = Hn(xn) gives the physical energy storage law. If x n is a flux (resp. effort) variable then ∇Hn(xn) is the dual effort (resp. flux) variable.

Similarly, for a dissipative component, the power is qm = Rm(wm) so that if wm is a flux (resp. effort) variable then z(wm) = Rm(wm) wm

is the effort (resp. flux) and gives the dissipation law. We then consider a passive system obtained by interconnection of these components given by

  x w -y   b = S(x, w)   ∇H(x) z(w) u   a (13)
with S = -S T being skew-symmetric, H(x) = N i=1 Hn(xn) and z(w) = [z1(w1), . . . , zm(wm)] T .

The S matrix represents the power exchange between components: since S = -S T we have a • b = a T Sa = 0 which again leads to the power balance 1 .

∇H(x) • x Pc=E (t) + z(w) • w P d -u • y Pe = 0 (14)
The explicit form (6) can be found by solving the second row of [START_REF] Sarkka | Accurate discretization of analog audio filters with application to parametric equalizer design[END_REF]. The S matrix represents a Dirac structure [START_REF] Van Der Schaft | Port-hamiltonian systems theory: An introductory overview[END_REF] that expresses the power-balance and can be constructed from a component connection graph [8] [18].

PASSIVE NUMERICAL INTEGRATION

Whereas most numerical schemes concentrate their efforts on the the temporal derivative or the numerical integration quadrature, discrete gradient methods preserve the energy (resp. passivity) given by the power-balance ( 9), ( 14) in discrete-time by providing a discrete equivalent of the chain rule derivation property E (t) = ∇H(x) T x . A discrete gradient [START_REF] Mansfield | On the construction of discrete gradients[END_REF] ∇H is required to satisfy the following conditions.

H(x + δx) -H(x) = ∇H(x, δx) T δx (15) ∇H(x, 0) = ∇H(x) (16)
In this article, we will focus on the average vector field [START_REF] Celledoni | energy resp. dissipation in numerical PDEs using the 'average vector field' method[END_REF].

Average Vector Field

In the general case, the AVF method is defined by.

δxn δt = 1 0 f (xn + τ δxn)dτ, xn+1 = xn + δxn (17)
When the matrices J(x), R(x), G(x) are approximated by constant matrices J, R, Ḡ, we obtain the separable structure-preserving approximation of ( 17)

δxn δt = (J -R)∇H(xn, δxn) + Gūn (18)
with the discrete gradient being defined by

∇H(x, δx) = 1 0 ∇H(x + τ δx)dτ (19)
and it satisfies the discrete power balance

δE = ∇H T δx δt = ∇H T (J -R)∇H + ∇H T Gu = 0 -P d + Pe
Then, by the fundamental theorem of calculus, for mono-variant components, i.e. separable Hamiltonians of the form H(x) = N i=1 Hi(xi), we have for each coordinate:

∇Hi(xi, δxi) = H i (x i +δx i )-H i (x i ) δx i δxi = 0 ∇H(xi) δxi = 0 (20)
which satisfies the discrete gradient conditions ( 15)-( 16). For nonseparable Hamiltonians, a discrete-gradient can also be uniquely defined, see [START_REF] Lopes | Explicit second-order accurate method for the passive guaranteed simulation of port-Hamiltonian systems[END_REF] for more details.

To summarize, this method relies on two complimentary approximations: the differential operator dx dt → δx δt and the vector field f → f to achieve energy (resp. passivity) conservation. The discrete PHS equivalent of ( 6) is given by the numerical scheme.

   δxn δt = J -R ∇H(xn, δxn) + Gun yn = G T ∇H(xn, δxn) xn+1 = xn + δxn (21)

Accuracy order

As shown in [START_REF] Celledoni | Energy-preserving runge-kutta methods[END_REF], the AVF has accuracy order p = 2, it is a Bseries method, is affine-covariant and self-adjoint. When approximated as in Eq [START_REF] Mansfield | On the construction of discrete gradients[END_REF] by evaluating matrices J, R, G for x * = xn the accuracy is only of order 1. Order 2 is achieved when either J, R, G are independent of x or when evaluated at the mid-point x * = xn + δxn 2 in the conservative case. It is also possible to restore the accuracy order p = 2 in the general case using a Runge-Kutta refinement [START_REF] Lopes | Explicit second-order accurate method for the passive guaranteed simulation of port-Hamiltonian systems[END_REF].

Implicit resolution

The discrete system is implicit on δxn and admits a unique solution when H is convex. In the general case, an iterative solver is required (typically a fixed-point or Newton iteration), but when the Hamiltonian is quadratic we can avoid the need for an iterative resolution. Furthermore, when the Hamiltonian is convex the method can also be made non-iterative by quadratization of the Hamiltonian [START_REF] Lopes | Explicit second-order accurate method for the passive guaranteed simulation of port-Hamiltonian systems[END_REF].

Proof. When the Hamiltonian is quadratic of the form H(x) = 1 2 x T Qx, the discrete gradient reduces to the mid-point rule

∇H(x, δx) = 1 0 Q(xn + δxnτ )dτ = Q xn + 1 2 δxn
the implicit dependency on δx can thus be solved by matrix inversion

δxn = δt I - δt 2 A -1 Axn + Gun (22)
with A = (J -R)Q Given the sequence of points {xn} obtained by a passiveguaranteed method, we would like to reconstruct piece-wise C kcontinuous polynomial trajectories informed by the system dynamics.

PIECEWISE-CONTINUOUS TRAJECTORIES

The idea is to exploit the dynamic equation at each junction point xn where the approximation is known to be O(h p+1 ).

Indeed, if we had the samples of the exact trajectory, by the Weierstrass approximation theorem, arbitrarily close polynomial approximations converging uniformly to the exact solution could be obtained by computing its derivatives to any desired order.

Since we only have an approximation of order p = 2, we restrict ourselves to a regularity k = 1. This gives four constraints

x(0) = xn, x(1) = xn+1, x (0) = f (xn), x (1) = f (xn+1)
that can be satisfied by a cubic polynomial (r = 3). We choose to represent it using the Bézier form,

x(τ ) = 3 i=0 XiB 3 i (τ ), B n i (t) = n i (1 -t) n-i t i (23)
with {Xi} being its control polygon and B n i (t) being the Bernstein polynomial basis functions, because they have important geometric and finite differences interpretations [START_REF] Farouki | The bernstein polynomial basis: A centennial retrospective[END_REF].

This choice immediately leads to the following equations,

X0 = xn X1 = xn + 1 3 f (xn) (24)
X3 = xn+1 X2 = xn+1 - 1 3 f (xn+1) (25)
where the internal control points X1, X2 are computed from the end points xn, xn+1 by first order forward / backward prediction using the derivative rule.

x (t) = n-1 i=0 DiB n-1 i (t), Di = n(Xi+1 -Xi) (26)
An example trajectory is shown in Figure 1.

ANTI-ALIASED OBSERVATION

Given an observed signal ũ(t) = y(t) belonging to the class of piecewise polynomials, in order to reject the non-band-limited part of the spectrum, we would like to apply an antialiasing filter operator given by its continuous-time ARMA transfer function H(s), then sample its output ỹ(t) to get back to the digital domain.

Since our anti-aliasing filter will be LTI, we will make use of exact exponential integration and decompose its output on a custom basis of exponential polynomial functions.

Without loss of generality we only consider single-input singleoutput filters (SISO) since we can always filter each observed output independently.

State-space ARMA filtering of polynomial input

We want to filter the trajectory by an ARMA filter given by its Laplace transfer function

H(s) = Y (s) U (s) = b0s N + b1s N -1 + . . . + bN s N + a1s N -1 + . . . + aN (27)
This filter can be realized in state-space form as

x = Ax + Bũ (28) ỹ = Cx + Dũ (29)
Common choices are the observable and controllable state-space forms. Furthermore when the denominator can be factored with distinct roots, it is possible to rewrite the transfer function using partial fraction expansion as.

H(s) = c0 + c1 s -λ1 + . . . + cN s -λN (30)
which leads to the canonical diagonal form

A =    λ1 . . . λN    B =    1 . . . 1    (31) C = c1 . . . cN D = c0 (32)

Exact exponential integration

The exact state trajectory is given by the integral

x(t) = xh (t) + xe(t) = e At x0 + t 0 e A(t-τ ) Bũ(τ )dτ (33)
as the sum of the homogeneous solution to the initial conditions xh and the forced state-response with zero initial conditions xe given by the convolution of the input with the kernel e At . Furthermore when A is diagonal we have

e At =    e λ 1 t . . . e λ N t    (34)
which greatly simplifies the computation of the exponential map.

In that case (33) can be evaluated component-wise as

xi (t) = e λ i t xi 0 + t 0 e λ i (t-τ ) ũ(τ )dτ i ∈ {1 . . . N } (35)
where we used the notation x i to detonate the i-th coordinate of the vector x

Polynomial input

With ũ(t) being a polynomial of degree K in monomial 2 form and coefficients ũk

ũ(t) = K k=0 ũk t k k! (36)
we can expand the forced response xe in (35) as a weighted sum

t 0 e λ i (t-τ ) K k=0 ũk t k (τ ) k! dτ = K k=0 ũk ϕ k+1 (λi, t) (37)
with the basis functions {ϕ k } being defined by the convolution

ϕ k (λ, t) = t 0 e λ(t-τ ) τ k-1 (k -1)! dτ k ≥ 1 (38)
One of the main advantages of using a polynomial input (rather than a more general model) lies in the fact that these basis functions can be integrated exactly, avoiding the need of a quadrature approximation formula. See Appendix 12 for a detailed derivation and a recursive formula, and Figure 2 for their temporal shapes.

Using those we can decompose the local state trajectories as.

xi (t) = xi 0 ϕ0(λi, t) + K k=0 ũk ϕ k+1 (λi, t) (39)
We note that the initial condition is equivalent to an impulsive input xi 0 δ(t). This filtering scheme can thus be generalized to non polynomial impulsive inputs.

Numerical update scheme

Since we only wish to sample the trajectory on a fixed grid tn ∈ Z, we just need to evaluate the local state trajectory x(t) and the output y(t) at t = 1 to finally get the following numerical scheme

xi n+1 = xi n ϕ0(λi) + K k=0 ũk,n ϕ k+1 (λi, 1) (40)
ỹn+1 = N i=1 ci xi n+1 + c0 ũn(1) (41)
where the coefficients ϕ k (λi, 1) can be pre-computed and the components xi n+1 evaluated in parallel.

Filter examples

Low-pass filter of order 1

We consider a first order low-pass filter with transfer function H(s) = a s+a . The temporal response to a piecewise polynomial input {t 2 , 1 -t, 0, 1} is shown in Figure 3 for a ∈ {1, 3, 6, 10}.

Butterworth Filter of order 3

To further illustrate the non-band-limited representation capacity of piece-wise polynomials, and the effectiveness of the filtering scheme, we have shown in Figure 4 the response of a third-order Butterworth filter with cutoff ωc = π to a triangular input signal. Its Laplace transfer function for a normalized pulsation ωc = 1 is given by H

(s) = 1 (s 2 +s+1)(s+1) with poles λ1 = -1-i √ 3 2 , λ2 = -1+i √ 3 2 , λ3 = -1 and coefficients c0 = 0, c1 = -3+i √ 3 6 , c2 = -3-i √ 3 6 , c3 = 1.
2 We use the monomial form here instead of Bernstein polynomials because this is the one that leads to the most straightforward and meaningful derivation.

APPLICATION: NONLINEAR LC OSCILLATOR

In order to illustrate the proposed method, we consider the simplest having non linear dynamics. For that purpose, we use a parallel autonomous LC circuit with a linear inductor and a saturating capacitor with the Hamiltonian energy storage function given by

H(q, φ) = ln(cosh(q)) C0 + φ 2L (42)
where the state q is the charge of the capacitor and φ the flux in the inductor. Its circuit's schematic is shown in figure 5 and its energy storage law are displayed in 6

L + - VL IL C(q) + - VC IC Figure 5: A nonlinear LC oscillator circuit
By partial differentiation of the Hamiltonian function H by respectively q and φ we get the capacitor's voltage and the inductor's current, while applying the temporal derivative on q, φ gives the capacitor's current and inductor's voltage. This gives the Branch Component Equations. Applying Kirchhoff Current and Voltage Laws gives the constraints IC = -IL, VC = VL. We can summarize the previous equations with the conservative autonomous Hamiltonian system.

VC = ∂qH = tanh(q) C0 IC = q (43) IL = ∂ φ H = φ L VL = φ (44)
x = J∇H(x) (45)
with.

x = q φ , J = 0 -1 1 0 , ∇H = ∂qH ∂ φ H (46)
Its state space and temporal trajectories are shown in Figure 7. We can see that the numerical scheme preserves the energy since the discrete points lie exactly on the orbit of the reference trajectory. The reconstructed state-space trajectory also shows a good match with the reference for most of the interpolated segments, except around transition regions at the bottom and top.

The spectrum of the flux φ is shown in Figure 8. One can see that the reference spectrum contains harmonics above twice the representable bandwidth where they pass below -90 dB.

The ZOH and FOH spectrums contains spectral images of the non bandlimited spectrum that decay respectively at -6dB/oct and -12dB/oct. Their aliased components in the audio bandwidth start around -80 dB at the Nyquist frequency and decay slowly toward approximately -100 dB at low frequencies.

Contrary, our method, informed by the dynamic, exhibits both reduced aliasing in the audio bandwidth and sharpened spectrum The 10x oversampled reference is compared to the AVF method's discrete output with zero-order hold (ZOH), first-order hold (FOH), the proposed method (proposed cubic) and its 12th order Butterworth filtered spectrum (proposed + AA). The Nyquist frequency is materialized in blue and the multiples of the sampling rate in red.

around the Nyquist frequency. It also has a higher spectral images decay rate thanks to its C 1 regularity. Its aliased components start at -85 dB at the Nyquist frequency and decay much faster to reach -100 dB at about 14 kHz where they reach a kind of aliasing noise floor caused by higher harmonics fold-back. Finally, as expected, the 12th-order Butterworth half-band lowpass filter removes components above the Nyquist frequency thanks to the piecewise continuous cubic input.

DISCUSSION

First, we highlight the fact that the vector field approximation in [START_REF] Unser | Cardinal exponential splines: Part II-Think analog, act digital[END_REF] acts as a first-order antialiasing filter: it is a projection of the vector field on a rectangular kernel. It prevents high-order spectral images from disturbing the low frequency dynamic during the numerical simulation and it is consistent with the underlying piecewise linear approximation model.

Second, the numerical scheme is energy-preserving. From a signal processing perspective, the lowpass filtering effect on the vector field is compensated by the finite difference approximation of the derivative. This is a direct generalization of the mid-point / bilinear methods to nonlinear differential equations.

Third, using the fact that the trajectory approximation has accuracy order p = 2 at the junctions, we can re-exploit the differential equation to reconstruct an informed C 1 -continuous cubic trajectory. It exhibits reduced aliasing in the passband and better high-frequency resolution.

We observe that on the studied example, our method manages to reduce aliased components that are folded once into the audio band. However components caused by multiple folding of the spectrum cannot be removed anymore. This is related to the Papoulis generalized sampling expansion [START_REF] Papoulis | Generalized sampling expansion[END_REF] who states that a band-limited function can be perfectly reconstructed from its values and derivatives sampled at half the Nyquist rate.

Some difficulties arise when trying to generalize the above ideas to higher order trajectories and filtering kernels. First, the line-integral ( 17) is no longer computable in closed form when the trajectory model is non-affine. Second, higher order kernels have longer temporal support which can lead to non-causal integrals.

CONCLUSION AND PERSPECTIVES

Our main contribution is an approach based on smooth piecewise defined trajectories coupled with a guaranteed-passive simulation. The method proceeds in three steps: 1) an energy-preserving passive numerical scheme is applied, 2) C k -continuous trajectories are reconstructed, 3) Exact continuous time lowpass filtering and sampling is performed. We have proposed a first instance of this method using the class of piecewise polynomials with regularity k = 1 and accuracy order p = 2 that exhibits reduced aliasing.

Further work will concern increasing the regularity k and accuracy order p, merging the numerical scheme and the interpolation steps by considering energy-preserving methods with a builtin regular continuous model and considering other classes of models such as rational and exponential functions.

In this regard, exponential integrators [START_REF] Hochbruck | Exponential integrators[END_REF] that integrate the linear part of the dynamic exactly (as we have done in section 6) and rely on approximations for the nonlinear part are of great interest.

Finally we would like to further investigate the link between multi-stages / multi-derivatives general linear methods, their accuracy orders, numerical dispersion and internal bandwidth, and to analyze their behavior and representation capabilities within the framework of Reproducing Kernels Hilbert Spaces and generalized sampling theory [START_REF] Nehab | A fresh look at generalized sampling[END_REF] [27] [START_REF] Unser | Sampling-50 years after shannon[END_REF].

APPENDIX: ϕ-FUNCTIONS

The ϕ-functions, that appear when doing exact integration of an LTI system with polynomial input given in monomial form, are defined by the convolution integral

ϕ k (λ, t) = t 0 e λ(t-τ ) τ k-1 (k -1)! dτ k ≥ 1 (47)
and by definition ϕ0(λ, t) := e λt (48)

For λ = 0 it is immediate that

ϕ k (λ = 0, t) = t k k! (49)

Recurrence relation

We first prove that they satisfy the recurrence formula The ϕ-functions represent thus the tail of the truncated taylor series expansion of e λt up to a scaling factor. This is clear when rewriting (56) as

e λt = k-1 n=0 (λt) n n! + λ k ϕ k (λ, t) (57)

Figure 1 :

 1 Figure1: Example of a cubic trajectory with conservative endpoints. The affine trajectory used to compute the average vector field is shown (in green), the associated cubic interpolated approximation (in blue), its control polygon (in red), and the exact manifold (in dashed black).

5 Figure 2 :

 52 Figure 2: Normalized ϕ-functions for k ∈ {0 . . . 4}. The real parts of the impulse (blue), step (red), ramp (green), quadratic (magenta) and cubic (black) responses are shown for a complex pole λ = i2π (left plot) and a real pole λ = -5 (right plot) over the unit interval t ∈ [0, 1].

Figure 3 :

 3 Figure 3: Exact continuous-time responses of a first order lowpass filter to a polynomial input (in blue).

Figure 4 :

 4 Figure 4: Exact continuous-time response of the order 3 Butterworth filter with cutoff pulsation ωc = π to a triangle input at the Nyquist frequency.

Figure 6 :

 6 Figure 6: Respective energy storage functions (left plot) and their gradients (right plot), of the nonlinear capacitor (in red) and linear inductor (in blue), for C = 1, L = 1.

Figure 7 :

 7 Figure7: Comparison of simulated orbits with discrete points (in blue) computed using the AVF method, reconstructed cubic trajectory (in green) and reference trajectory computed at 10x sampling rate (in red).

Figure 8 :

 8 Figure8: Continuous-time spectrum of the nonlinear LC circuit flux φ for a fundamental frequency of 500 Hz and a sampling frequency of 44.1 kHz. The 10x oversampled reference is compared to the AVF method's discrete output with zero-order hold (ZOH), first-order hold (FOH), the proposed method (proposed cubic) and its 12th order Butterworth filtered spectrum (proposed + AA). The Nyquist frequency is materialized in blue and the multiples of the sampling rate in red.

ϕ

  k+1 (λ, t) = ϕ k (λ, t) -ϕ k (0, Using integration by partsb a u(τ )v (τ )dτ = [uv] b a -b a u (τ )v(τ )dτ with [a, b] = [0, t], u(τ ) = e λ(t-τ ) , v (τ ) = τ k-1 (k-1)! and its primitive v(τ ) = τ k k! gives ϕ k (λ, t) = e λ(t-τ ) τ k λϕ k+1 (λ, t)which after using (49) and identification givesϕ k+1 (λ, t) = ϕ k (λ, t) -ϕ k (0, t) λ 12.2.Explicit formUsing (50) recursively for λ = 0, the first basis functions are given by ϕ0(λ, t) = e λt (51)ϕ1(λ, t) = e λt -1 λ (52) ϕ2(λ, t) = e λt -(1 + λt) λ 2 (53) ϕ3(λ, t) = e λt -(1 + λt + (λt) 2 2! ) λ 3 (54) ϕ4(λ, t) = e λt -(1 + λt + (λt) 2 2! + (λt) 3It is immediate to verify that (56) is satisfied for k = 0. Then assuming that (56) is true for some k ∈ N and using the recurrence (50) we proveϕ k+1 (λ, t) = ϕ k (λ, t) -ϕ k (0, t) λ = 1 λ k+1 e λtis also true for k + 1. By induction (56) is thus satisfied for all k ∈ N.

The minus sign in -y in Eq. (13) is used to restore the receiver convention used for internal components.

Proceedings of the 20 th International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September[START_REF] Falaize | Passive simulation of the nonlinear port-Hamiltonian modeling of a Rhodes Piano[END_REF][START_REF] Lopes | Energy Balanced Model of a Jet Interacting With a Brass Player's Lip[END_REF][START_REF] Falaize | Passive simulation of electrodynamic loudspeakers for guitar amplifiers: a port-Hamiltonian approach[END_REF][START_REF] Falaize | Passive guaranteed simulation of analog audio circuits: A port-hamiltonian approach[END_REF][START_REF] Boyd | Chebyshev and Fourier Spectral Methods[END_REF] 2017

ACKNOWLEDGMENTS

This work has been done at laboratory STMS, Paris, within the context of the French National Research Agency sponsored project INFIDHEM. Further information is available on the project web page.

The contribution of this author has been done at laboratory STMS, Paris, within the context of the French National Research Agency sponsored project INFIDHEM. Further information is available at http://www.lagep.cpe.fr/wwwlagep7/anr-dfg-infidhem-fev-2017jan-2020/


Cl Ement Trunet

Ed Eric Carlin

Louis Coroller
email: louis.coroller@univ-brest.fr

Investigating germination and outgrowth of bacterial spores at several scales

Keywords:

Background: Spore-forming bacteria are a major cause of food spoilage and food poisoning. Spores that resist physical and chemical treatments used in the food industry may germinate and multiply. Spore germination, outgrowth and growth constitute a complex and highly heterogeneous process. Scope and approach: Various techniques and methods can be used to observe the germination, outgrowth and early multiplication process of spore-forming bacteria and/or to quantify the impact of environmental conditions on its progress over time within a spore population. These techniques can be classified by different criteria: (i) the scale of analysis, from populations or cells to molecules, and (ii) the number of analyzed objects (cells) and (iii) the potential of the method to describe and/or quantify the impact of lethal or sub-lethal treatments or environmental conditions. Such treatments are applied to a spore population or a single spore and take into account parameters at the cellular level (growth capacity, morphological properties) to molecular level (proteomics, transcriptomics, spore molecular composition). Key findings and conclusion: A better understanding and quantification of the germination, outgrowth and growth process require the implementation of several complementary methods. Methods providing information at single and population levels, as well as at molecular and cellular levels, are essential to assess and control the fate of spore-forming bacteria development in food systems.

Introduction

Spore-forming bacteria are largely studied in fundamental research on Gram-positive bacteria and for the numerous industrial applications of the spores themselves, of their enzymes and of their metabolites. Spores of Bacilli or Clostridia can remain dormant for years and have a much greater resistance to very diverse physical and chemical agents than their vegetative cells. After germination and multiplication, they can be responsible for food poisoning and food spoilage [START_REF] Mallozzi | Spore-forming Bacilli and Clostridia in human disease[END_REF][START_REF] Setlow | Germination of spores of Bacillus species: What we know and do not know[END_REF][START_REF] Wells-Bennik | Bacterial spores in food: Survival, emergence, and outgrowth[END_REF]. Recovery after chemical or physical stress is a complex phenomenon comprising the germination of spores, the restoration of metabolic activity in suboptimal or favorable conditions and the emergence of the first vegetative cell able to multiply [START_REF] Setlow | Summer meeting 2013 e when the sleepers wake: The germination of spores of Bacillus species[END_REF]. Germination, outgrowth and growth are characterized by a high variability of the behavior of individual cells. This variability is, moreover, highly influenced by environmental conditions. Monitoring the germination and growth of bacterial spores meets two major objectives: (i) to obtain a better understanding of a major physiological process of cell differentiation in the microbial world and (ii) to implement a quantitative evaluation of the impacts of physical and chemical treatments and/ or of environmental conditions on the ability of spores to germinate and grow, and hence become a troublemaker for public health, food production and distribution, and other industrial activities.

This review provides a description of the different techniques and methods for monitoring spore germination and growth after exposure to sublethal treatments and/or suboptimal environments. The discussion focuses on the information provided by different investigation methods with regards to the scales of examination (from cell to molecules, from populations to single cells, from millions to dozens of single events) and their potential for describing and/or quantifying the impact of treatment or environmental conditions.

Overview of spore germination and characterization of spore properties

Germination encompasses the deep morphological and physiological modifications of dormant spores to become vegetative cells. The physiological stages observed during germination and growth can be defined by criteria linked to the changes in spore properties. At first, refractivity loss is due to the rehydration of the spore core. Then, the size of the cell increases as the spore converts to a rod-shaped vegetative cell during the germination, outgrowth and growth process. Simultaneously the respiratory activity is restored, the membranes become fluid and the resistance to wet heat is lost. At a molecular level the CaDPA which was accumulated during sporulation, maintaining low water content in the spore core, is released and RNA expression and protein and other (macro) molecule synthesis starts again [START_REF] Setlow | Germination of spores of Bacillus species: What we know and do not know[END_REF]. These physiological and molecular events may be used to differentiate dormant and germinated spores, or outgrowing and vegetative cells during the germination, outgrowth and growth process (Fig. 1).

The term "germinant" designates nutrient and non-nutrient factors triggering spore germination [START_REF] Setlow | Spore germination[END_REF]. Nutrient germinants are low molecular weight compounds including, but not restricted to, amino acids, sugars and purine nucleosides possibly acting in combination and whose effects can be reinforced by cogerminants such as cations. Non-nutrient germinants include the dipicolinic acid-calcium complex (CaDPA, specific to spore-forming bacteria) or high hydrostatic pressure [START_REF] Setlow | Summer meeting 2013 e when the sleepers wake: The germination of spores of Bacillus species[END_REF]. Germination leads to the rehydration of the spore and the release of the CaDPA concentrated in the spore core. Then, the cortex is hydrolyzed by cortex lytic enzymes (CLEs) such as SleB and CwlJ, the two major CLEs in Bacillus sp. Cortex hydrolysis allows the core to expand and complete rehydration to approximately 80% of spore weight [START_REF] Setlow | Germination of spores of Bacillus species: What we know and do not know[END_REF]. Spores lose their high resistance to moist heat and chemicals early after core hydration. The spore becomes permeable as the inner-membrane surrounding the core expands. In the dormant stage, the low inner membrane permeability prevents small molecules from penetrating the core [START_REF] Sunde | The physical state of water in bacterial spores[END_REF]. Finally, the hydration of the spore and the release of CaDPA lead to the loss of spore refractivity (Fig. 1). Metabolic activity and macromolecule synthesis are restored during outgrowth. The spores swell up and escape from the outermost layers, the proteinaceous coat and exosporium [START_REF] Abee | Germination and outgrowth of spores of Bacillus cereus group members: Diversity and role of germinant receptors[END_REF]. This step leads to the formation of a new vegetative cell able to multiply. The transformation of a germinated spore into a vegetative cell is characterized by morphological changes: the cell size increases, the new cell comes out of the coat and exosporium, and the restoration of a metabolic activity marks a deep physiological modification [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF]. The shifts from dormant spores to germinated spores, then from germinated spores to vegetative cells are highly heterogeneous [START_REF] Setlow | Dynamics of the assembly of a complex macromolecular structure -the coat of spores of the bacterium Bacillus subtilis: B. subtilis spore coat[END_REF]. This variability can be the result of differences in the expression of genes governing sporulation and/or environmental conditions among the population of cells during sporulation as the germination and outgrowth behavior depends on the molecular reserves produced during spore formation. This variability results in different spore dehydration rates, different amounts of enzymes involved in germination and a different number of germinant receptors which are put in place during sporulation [START_REF] Chastanet | Broadly heterogeneous activation of the master regulator for sporulation in Bacillus subtilis[END_REF][START_REF] Setlow | Germination of spores of Bacillus species: What we know and do not know[END_REF]. This variability in turn affects heat resistance, germination and growth ability (Hornstra, de Vries, Wells-Bennik, de Vos, [START_REF] Hornstra | Characterization of germination receptors of Bacillus cereus ATCC 14579[END_REF][START_REF] Setlow | Summer meeting 2013 e when the sleepers wake: The germination of spores of Bacillus species[END_REF].

The study of physical and/or physiological spore transformations requires different levels of analysis, and thus different observation and quantitative methods. Monitoring germination and outgrowth consists in following characteristics which are representative of the spore transformation, over time and within a given spore population. The previously described spore transformations are quantifiable criteria for distinguishing the successive physiological stages of spores during germination and growth. These changes can be observed in populations as well as in individual spores using different techniques and methods, providing information on cellular behavior as well as on molecular characteristics (Fig. 2AeB). Flow cytometry and microscopy methods examine sub-populations and single cells at cellular or molecular levels (Fig. 2CeD). Molecular methods apply to populations or single cells giving information at genomic, transcriptomic, proteomic or other molecular levels (Fig. 2E-F-G).

3. Culture dependent methods and indirect measurements (Fig. 2AeB) Cultural methods require media (broth or solid) containing the necessary elements to provide spore germination, outgrowth and cell multiplication (Fig. 2A). Cells forming a colony on agar plates have definitely experienced germination, outgrowth and multiplication. The detection threshold is 10 to 10 2 spores CFU mL À1 and can be continued with the dilution procedure. The sensitivity is high as the presence of a few spores can be detected among populations of a million or even billion cells that can be eliminated by a moderate heat treatment or by exposure to an ethanol solution. Culture dependent methods have been applied to estimate the impact of sporulation conditions, the intensity of heat treatment and the recovery environment on the ability of spores to form a colony on agar plates, particularly after a heat process. More generally they can be used to quantitatively assess growth, survival or recovery in laboratory media or in real foods after physical or chemical treatments of the spore-forming bacteria of concern. These data are used to develop mathematical models predicting bacterial behavior to optimize, control and improve food quality and safety [START_REF] Baril | Modeling heat resistance of Bacillus weihenstephanensis and Bacillus licheniformis spores as function of sporulation temperature and pH[END_REF][START_REF] Leguerinel | Relationship between the apparent heat resistance of Bacillus cereus spores and the pH and NaCl concentration of the recovery medium[END_REF][START_REF] Legu Erinel | Modelling the influence of the incubation temperature upon the estimated heat resistance of heated Bacillus spores[END_REF][START_REF] Mafart | Modeling combined effects of temperature and pH on heat resistance of spores by a linear-Bigelow equation[END_REF][START_REF] Mtimet | Modeling the behavior of Geobacillus stearothermophilus ATCC 12980 throughout its life cycle as vegetative cells or spores using growth boundaries[END_REF][START_REF] Trunet | Modeling the recovery of heat-treated Bacillus licheniformis Ad978 and Bacillus weihenstephanensis KBAB4 spores at suboptimal temperature and ph using growth limits[END_REF].

Germination is characterized by the rehydration of spores, revealed by the transformation of spores from phase-bright to dark-phase (Fig. 2B). This corresponds to a decrease in the A 600 of spore suspensions, necessarily at relatively high concentrations (i.e. 10 7 -10 8 spore/ml) to exceed the detection limit of spectrophotometric measurements. For example the impact of suboptimal sporulation and incubation conditions on the germination of B. weihenstephanensis or of B. cereus has been evaluated by monitoring changes in A 600 of spore suspensions [START_REF] Garcia | Comparative analysis of Bacillus weihenstephanensis KBAB4 spores obtained at different temperatures[END_REF][START_REF] Van Melis | Germination inhibition of Bacillus cereus spores: Impact of the lipophilic character of inhibiting compounds[END_REF]. A 62% decrease in initial A 600 for Bacillus weihenstephanensis or a 45% decrease for B. cereus corresponded to a 99.9% germination rate. Unsurprisingly, spectrometric methods do not involve direct observation of spores and therefore do not allow any assessment of the variability in individual cell behaviors. Moreover, these techniques can only be applied to transparent liquids.

Furthermore, by studying changes in absorbance of a spore suspension, the shift from a dormant spore to a vegetative cell able to multiply can also be examined. Vegetative cell multiplication leads to an increase in A 600 with the increase in turbidity. Monitoring the evolution of A 600 allows to estimate the lag time corresponding to the time to first cell division, encompassing the germination time and the time needed for growth. With this Using modern microplate readers that can be set to a wide range of incubation temperatures, culture conditions are miniaturized in microwells and A 600 readings are automated and finally computed.

A large number of data are potentially generated and the influence of a large number of variable factors on inactivation, germination and spore incubation can be tested. This method is not very indicative for the quantification of the number of germinating spores that give rise to multiplying vegetative cells. An increase in A 600 can be due to anything from one to millions of spores, the only difference being in the time to a measurable change in A 600 . Moreover the heterogeneity within the cell population is not observable. As A 600 reading is also poorly sensitive and does not discriminate cells from other materials, monitoring germination by spectrometry uses highly concentrated spore suspensions (about 10 7 spore/ml) and therefore requires the preparation of high amounts of clean debris-free spore suspensions. However single spores can be isolated in microplate wells by limit dilution techniques or sorted by flow cytometry [START_REF] Smelt | Modelling the effect of sub(lethal) heat treatment of Bacillus subtilis spores on germination rate and outgrowth to exponentially growing vegetative cells[END_REF][START_REF] Stringer | Lag time variability in individual spores of Clostridium botulinum[END_REF]) (Fig. 2B). By monitoring A 600 in cultures formed from these isolated cells over time, the times to spore germination and first cell division from single spores can be estimated. A shorter lag time and a lower variability of the lag time of heat-activated Bacillus subtilis and Clostridium botulinum spores were revealed [START_REF] Smelt | Modelling the effect of sub(lethal) heat treatment of Bacillus subtilis spores on germination rate and outgrowth to exponentially growing vegetative cells[END_REF][START_REF] Stringer | Heterogeneity of times required for germination and outgrowth from single spores of nonproteolytic Clostridium botulinum[END_REF]. The variability in lag times depends on sporulation conditions, on the intensity of the inactivation treatment by heat and on the incubation conditions [START_REF] Garcia | Comparative analysis of Bacillus weihenstephanensis KBAB4 spores obtained at different temperatures[END_REF][START_REF] Smelt | Modelling the effect of sub(lethal) heat treatment of Bacillus subtilis spores on germination rate and outgrowth to exponentially growing vegetative cells[END_REF][START_REF] Stringer | Lag time variability in individual spores of Clostridium botulinum[END_REF]. Germination can also be monitored by studying the changes in the concentration of CaDPA in the incubation medium (Fig. 2B) that corresponds to its release concomitant with the hydration to the spore core. Several methods are available for DPA assay. One for instance uses the photoluminescence of DPA in the presence of terbium. The complex Tb-DPA is excited at 270 nm and the emitted fluorescence wavelength is 545 nm. The concentration in the released DPA is maximal when the germination of the spore population is completed. The relative DPA release at time t is calculated as the ratio between RFU at t and maximum RFU at full germination [START_REF] Yi | Studies of the commitment step in the germination of spores of Bacillus species[END_REF][START_REF] Zhang | Effects of cortex peptidoglycan structure and cortex hydrolysis on the kinetics of Ca 2þ -dipicolinic acid release during Bacillus subtilis spore germination[END_REF]. By monitoring the DPA release, early events such as "commitment" and germination can be studied or the impact of heat treatment on germination time quantified [START_REF] Kort | Assessment of heat resistance of bacterial spores from food product isolates by fluorescence monitoring of dipicolinic acid release[END_REF][START_REF] Paidhungat | Mechanisms of induction of germination of Bacillus subtilis spores by high pressure[END_REF].

The bioluminescence of spores of engineered strains containing the lux gene (Fig. 2B) has also been exploited. Particularly, the lux gene was under the control of specific promoters, such as the sspB promoter. The sspB gene codes for a small acid-soluble protein (SASP) found in the spore core. As the spore germinates, the core is hydrated and the spores emit bioluminescence almost instantaneously as the intact luciferase, packaged during sporulation, leads to light emission. Relative Light Units are measured over time to evaluate the percentage of germinated spores within the population [START_REF] Ciarciaglini | Germination-induced bioluminescence, a route to determine the inhibitory effect of a combination preservation treatment on bacterial spores[END_REF][START_REF] Hill | Bioluminescence and spores as biological indicators of inimical processes[END_REF]. This method was notably used to quantify the impact of mild acidity, lactic acid and pasteurization on the germination of B. subtilis spores or to study the first steps in the germination process of Bacillus subtilis with different genetic backgrounds [START_REF] Ciarciaglini | Germination-induced bioluminescence, a route to determine the inhibitory effect of a combination preservation treatment on bacterial spores[END_REF][START_REF] Setlow | Properties of spores of Bacillus subtilis blocked at an intermediate stage in spore germination[END_REF]. Difficulty in strain transformation, Fig. 1. Successive stages of the germination and outgrowth of bacterial spores, observed under phase-contrast microscopy and methods applied for monitoring. Commitment to germination can be followed by the release of CaDPA [START_REF] Zhang | Monitoring of commitment, blocking, and continuation of nutrient germination of individual Bacillus subtilis spores[END_REF]. Spores remain phase-bright. Germination can be monitored by phase contrast microscopy on individual spores or on limited spore populations [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF][START_REF] Tychinsky | Dynamic phase microscopy, a new method to detect viable and killed spores and to estimate the heterogeneity of spore populations[END_REF]; by Raman spectroscopy [START_REF] Chen | Real-time detection of kinetic germination and heterogeneity of single Bacillus spores by laser tweezers Raman spectroscopy[END_REF] to follow CaDPA release into culture medium [START_REF] Zhang | Effects of cortex peptidoglycan structure and cortex hydrolysis on the kinetics of Ca 2þ -dipicolinic acid release during Bacillus subtilis spore germination[END_REF]. Spores turn from phase-bright to phase dark. Outgrowth can be monitored using phase contrast microscopy associated with a time lapse image system [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF]; by electron microscopy [START_REF] Saif Zaman | Imaging and analysis of Bacillus anthracis spore germination[END_REF] differences of phenotypes compared to the wild strain and the potential instability of the transformation could limit the application of this method. The methods described in this section 3 "Culture dependent methods and indirect measurements" do not allow direct observation of spores or biochemical analyses related to secreted or released cellular compounds. Moreover, the different cellular states cannot be distinguished and the differential quantification of dormant spores, germinated spores and vegetative cells is not possible. The next section will deal with methods of direct spore observation, such as microscopy and flow cytometry.

4. Microscopy and flow cytometry methods (Fig. 2CeD) Microscopy and flow cytometry discriminate between different physiological stages of germination and outgrowth by spore optical properties (refractivity), morphology (spheroid vs rod), and/or physiological activity (enzymatic or respiratory activity, membrane permeability). Both methods are based on the direct observation of cells, which may be marked with detectable probes. Phase contrast microscopy differentiates dormant spores (phase-bright spores) from germinated spores (phase-dark) by their higher refractive index and from vegetative cells by their smaller size [START_REF] Chen | Real-time detection of kinetic germination and heterogeneity of single Bacillus spores by laser tweezers Raman spectroscopy[END_REF][START_REF] Hornstra | On the origin of heterogeneity in (preservation) resistance of Bacillus spores: Input for a "systems" analysis approach of bacterial spore outgrowth[END_REF][START_REF] Tychinsky | Dynamic phase microscopy, a new method to detect viable and killed spores and to estimate the heterogeneity of spore populations[END_REF]. Various physical media may be used for microscopic observation of spore germination and outgrowth. The spores can be deposited on a microscope slide, and then dried. The medium of germination (such as an L-alanine or inosine solution) and/or growth is either the suspension medium of the spores or is added after drying the slide [START_REF] Kong | Monitoring rates and heterogeneity of high-pressure germination of Bacillus spores by phase-contrast microscopy of individual spores[END_REF]. Spores may also be placed on a semi-solid medium (such as a 1% agarose gel) previously prepared on a microscope slide [START_REF] Pandey | Quantifying the effect of sorbic acid, heat and combination of both on germination and outgrowth of Bacillus subtilis spores at single cell resolution[END_REF]Pandey & Ter Beek et al., 2015;[START_REF] Stringer | Heterogeneity of times required for germination and outgrowth from single spores of nonproteolytic Clostridium botulinum[END_REF][START_REF] Webb | Does proximity to neighbours affect germination of spores of non-proteolytic Clostridium botulinum?[END_REF]. [START_REF] Pandey | Quantifying the effect of sorbic acid, heat and combination of both on germination and outgrowth of Bacillus subtilis spores at single cell resolution[END_REF], Pandey and Ter Beek et al. (2015) also added a closed air-containing chamber made of an adhesive Gen-eFrame ® (primarily designed for in situ hybridization and amplification) to the slide. Spores have also been deposited on strips of Fig. 2. A synthetic view of the potential of different techniques and methods used to monitor germination, outgrowth and growth of bacterial spores at different scales of investigation, with some selected references. Along the X-axis, methods are discriminated by the number of examined cells. Along the Y-axis they are discriminated by the scale of investigation: from cell (cellular level) to specific cell components (molecular level). A) Cultural methods, on solid media, allow the enumeration of spores able to germinate, outgrow and give a vegetative cell able to form a colony. The detection threshold for these methods depends on the applied dilution procedure and can be between 10 and 10 2 CFU mL À1 (1. Legu erinel et al., 2006) B) A 600 measurements monitor the spore refractivity loss (A 600 decreases when spores lose their refractivity). The initial spore concentration must be high, ±10 7 spore mL À1 , to observe a significant A 600 decrease (2. [START_REF] Garcia | Comparative analysis of Bacillus weihenstephanensis KBAB4 spores obtained at different temperatures[END_REF]. The increase in A 600 also allows the growth restoration of a spore population to be monitored. The spore concentration must be>10 6 spore mL À1 to exceed the sensitivity of the spectrophotometer (3. [START_REF] Daelman | Development of a time-to-detect growth model for heattreated Bacillus cereus spores[END_REF]. The full recovery process of single spores can be obtained by combining microplate A 600 readings with wells inoculated with highly diluted spore suspensions (4. [START_REF] Smelt | Modelling the effect of sub(lethal) heat treatment of Bacillus subtilis spores on germination rate and outgrowth to exponentially growing vegetative cells[END_REF]. Spore germination can be monitored by titration in the culture medium of CaDPA released by spores (5. [START_REF] Zhang | Monitoring of commitment, blocking, and continuation of nutrient germination of individual Bacillus subtilis spores[END_REF]. Bioluminescence methods consist in observing bioluminescence of lux-containing engineered spores (6. [START_REF] Ciarciaglini | Germination-induced bioluminescence, a route to determine the inhibitory effect of a combination preservation treatment on bacterial spores[END_REF]. C) Phase contrast microscopy can be used to discriminate dormant spores (phase-bright) from germinated spores and vegetative cells (phasedark) by phase contrast microscopy. Micro-colonies formed after germination, outgrowth and growth restoration of spores fixed on semi-solid media or on strips (Anopore ® ) can be observed by phase contrast microscopy of spores on (7. [START_REF] Webb | Does proximity to neighbours affect germination of spores of non-proteolytic Clostridium botulinum?[END_REF]. This technique can be associated with live imaging automatic data treatment (SporeTracker for instance) (8. [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF]. Epifluorescence microscopy gives supplementary information about membrane permeability or metabolic activity thanks to fluorescent probes (9. [START_REF] Bruno | An Acridine orange spore germination fluorescence microscopy versus spectral paradox[END_REF]. This technique associated with live imaging reveals the metabolic activity or the membrane permeabilization during germination (10). Generally, a concentration >10 6 spore mL À1 is necessary to observe spores by microscopic methods. Atomic force microscopy (AFM) highlights the morphological evolution of spores during germination (11. Pinz on-Arango et al., 2009). D) Flow cytometry takes into account morphological criteria (size, refringency) associated with physiological criteria (membrane permeability, metabolic activity …). A high number of spores (>100 000) can be individually analyzed by flow cytometry) (12. [START_REF] Van Melis | Characterization of germination and outgrowth of sorbic acid-stressed Bacillus cereus ATCC 14579 spores: Phenotype and transcriptome analysis[END_REF][START_REF] Abee | Germination and outgrowth of spores of Bacillus cereus group members: Diversity and role of germinant receptors[END_REF]. A cell sorter would allow further characterization of single spores or of their progeny (13). E) Omics associated with millifluidic techniques will allow the observation of gene expression during the germination of single spores (16). F) Raman spectroscopy can be implemented to monitor single spore germination by following CaDPA release or to detect protein denaturation (14. [START_REF] Zhang | Characterization of single heat-activated Bacillus spores using laser tweezers Raman spectroscopy[END_REF]. G) Omics techniques analyze gene expression or protein composition in spore populations (15. [START_REF] Abhyankar | In pursuit of protein targets: Proteomic characterization of bacterial spore outer layers[END_REF].

Comment citer ce document : Anopore ® permeable membranes [START_REF] Den Besten | Short-and long-term biomarkers for bacterial robustness: A framework for quantifying correlations between cellular indicators and adaptive behavior[END_REF]. Time lapse image systems promote the acquisition of images at relatively high frequency, possibly of spores followed individually when placed on a medium (Fig. 2C). Image analysis is required to monitor the diverse cellular states. Phase-bright and phase-dark spores can be discriminated by their pixel intensity on spore images [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF][START_REF] Pandey | Quantifying the effect of sorbic acid, heat and combination of both on germination and outgrowth of Bacillus subtilis spores at single cell resolution[END_REF]Pandey & Ter Beek et al., 2015;[START_REF] Stringer | Contrasting effects of heat treatment and incubation temperature on germination and outgrowth of individual spores of nonproteolytic Clostridium botulinum bacteria[END_REF][START_REF] Ter Beek | Models of the behaviour of (thermally stressed) microbial spores in foods: Tools to study mechanisms of damage and repair[END_REF][START_REF] Webb | Does proximity to neighbours affect germination of spores of non-proteolytic Clostridium botulinum?[END_REF]. The different spores or cells can be enumerated and the size can be measured using dedicated software. For instance the SporeTracker software quantifies the decreasing pixel intensity during spore germination and the increasing area as the outgrowing cell size increases. SporeTracker can even detect the new cell popping out of the external spore layers, coat and exosporium; this event is called the "burst" [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF].

The time required to complete each step of the transformation of each cell within the population (germination, then outgrowth and first division) can be measured with these microscopical or flow cytometry techniques. Interestingly, the first spores to germinate are not the quickest to form a dividing cell, suggesting a rather time-independent pattern in the successive completion of germination and outgrowth events [START_REF] Smelt | Behaviour of individual spores of non proteolytic Clostridium botulinum as an element in quantitative risk assessment[END_REF]. The impact of sublethal heat treatment, environmental conditions (incubation temperature or pH), or preservatives (tea antimicrobial compounds, sorbic acid for instance) on the times to completion of the spore transformations can be estimated and statistically analyzed [START_REF] Pandey | Quantifying the effect of sorbic acid, heat and combination of both on germination and outgrowth of Bacillus subtilis spores at single cell resolution[END_REF]Pandey & Ter Beek et al., 2015). The influence of spore density and relative proximity on the germination of Clostridium botulinum spores has been demonstrated using these observation techniques. Practical experiments are performed on populations of several dozen to several thousands of cells, which gives some statistical and biological breadth to the observed phenomena [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF][START_REF] Stringer | Contrasting effects of heat treatment and incubation temperature on germination and outgrowth of individual spores of nonproteolytic Clostridium botulinum bacteria[END_REF].

Dynamic phase microscopy is performed with a laser phase microscope equipped with a He-Ne laser emitting at 633 nm. Dynamic phase microscopy estimates phase thickness, i.e. the difference in the optical path (nm) according to the nature of the different structures. Bacterial spore phase thickness can be defined as the spatial distribution of optical path difference between interfering beams, which reflects the state of an object. This technique has shown some potential to study the activation of spores by heat, a sublethal treatment promoting the germination of many species of spore-forming bacteria. Dormant spores (refractive) and germinated spores of B. licheniformis had significantly different phase thickness values [START_REF] Tichinsky | Dynamic phase microscopy: Measurements of translational displacements at sub-nanometer scale[END_REF][START_REF] Tychinsky | Dynamic phase microscopy, a new method to detect viable and killed spores and to estimate the heterogeneity of spore populations[END_REF]. Moreover, heat inactivated spores had a lower phase thickness value than dormant spores. The phase thickness values of spores which had been heat-activated but placed in an environment unsuitable for germination revert to values of untreated spores, confirming the reversibility of the activation treatment shown by other techniques [START_REF] Setlow | Summer meeting 2013 e when the sleepers wake: The germination of spores of Bacillus species[END_REF]. Epifluorescence microscopy can also provide opportunities to study spore germination, outgrowth and growth by detecting and interpreting the fluorescence signal emitted by fluorescent probes marking spore structural elements (Fig. 2C). Dormant spores are impermeable to many molecules. The outer layers (coat, outer membrane and cortex) are permeable to small molecules (from 2 kDa to 8 kDa) [START_REF] Driks | Bacillus subtilis spore coat[END_REF][START_REF] Henriques | Structure, assembly, and function of the spore surface layers[END_REF]. The first hurdle in staining a spore is its impermeability to larger molecules such as fluorescent dyes. Syto9 passes through intact membranes, intercalates DNA and emits a 540 nm-wavelength light when excited at 488 nm. Dormant spores hardly stain with Syto9, with only a green halo at the spore surface possibly due to mother cell DNA on the spore surface [START_REF] Cronin | The use of flow cytometry to study the germination of Bacillus cereus endospores[END_REF]. As the spore germinates and the core hydrates, Syto9 reaches spore DNA and germinated spores show a green fluorescence [START_REF] Cronin | The use of flow cytometry to study the germination of Bacillus cereus endospores[END_REF][START_REF] Reineke | Mechanisms of endospore inactivation under high pressure[END_REF]. In addition to DNA probes, such as Syto9 or propidium iodide (PI), fluorescent markers such as CFDA may reveal perturbations of the membrane integrity and be applied to the study of spore metabolic activity recovery during outgrowth [START_REF] Cronin | The use of flow cytometry to study the germination of Bacillus cereus endospores[END_REF]. Nucleoid morphology can be analyzed by epifluorescence microscopy to monitor spore germination. The staining of nucleoids is not possible in dormant spores due to the impermeability of the structure, but germinated spores, as forespores, contain ringshaped nucleoids. The nucleoids of germinated spores are associated with a/b small acid-soluble proteins (SASPs), which seem to have a role in germination and outgrowth: B. subtilis a À -b -SASP spores require a significantly longer time to recover a vegetative cell state [START_REF] Ragkousi | Analysis of nucleoid morphology during germination and outgrowth of spores of Bacillus species[END_REF].

Transmission Electronic Microscopy (TEM) is based on the variation of electron transmission through biological material and is particularly well suited to the observation of multilayered structures, which bacterial spores are, and more specifically to the evolution of spore structures during germination and outgrowth [START_REF] Dey | Germination and amplification of anthrax spores by soil-dwelling amoebas[END_REF][START_REF] Saif Zaman | Imaging and analysis of Bacillus anthracis spore germination[END_REF]. The phenomenon called "bottle cap" that consists in the opening of a precise region of the spore exosporium leading to the escape of the new vegetative cell has thus been observed in Clostridium sporogenes spores [START_REF] Brunt | Apertures in the Clostridium sporogenes spore coat and exosporium align to facilitate emergence of the vegetative cell[END_REF]. The "bottle cap" phenomenon may not be so different of the "burst" observed with B. subtilis spores by phase contrast microscopy [START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF]. The aperture allowing the cell to escape from external spore layers has also been observed by scanning electron microscopy (SEM) [START_REF] Bassi | A combination of a SEM technique and X-ray microanalysis for studying the spore germination process of Clostridium tyrobutyricum[END_REF][START_REF] Brunt | Apertures in the Clostridium sporogenes spore coat and exosporium align to facilitate emergence of the vegetative cell[END_REF][START_REF] Steichen | Non-uniform assembly of the Bacillus anthracis exosporium and a bottle cap model for spore germination and outgrowth: Non-uniform assembly of the B. anthracis exosporium[END_REF]. This method, in contrast to TEM, allows the observation of entire spores (versus a cross-section with TEM). These two methods are complementary for the analysis of spore structure evolution during germination and outgrowth. Atomic Force Microscopy generates images from the signal variations emitted by a mechanical probe displaced at a nanoscale on the surface of the "observed" object. Only changes in external layers of the spores can therefore be detected. Nevertheless, this method has successfully revealed alterations in coat architecture and topology during germination or after heat treatment (Pinz on-Arango, Scholl, Nagarajan, [START_REF] Pinz On-Arango | Atomic force microscopy study of germination and killing of Bacillus atrophaeus spores[END_REF][START_REF] Plomp | In vitro highresolution structural dynamics of single germinating bacterial spores[END_REF][START_REF] Tabit | The effects of wet heat treatment on the structural and chemical components of Bacillus sporothermodurans spores[END_REF]. TEM, SEM and AFM are destructive, with some possible artifacts due to harsh sample preparation conditions. Electron cryotomography (ECT) is also destructive, but objects are preserved in a near-native state thanks to ultra-quick freezing. Multi-directional EM images are computerized to give reconstructed 3-D images [START_REF] Tocheva | Electron cryotomography[END_REF]. Peptidoglycan rearrangements and cortex thickening and thinning have thus been monitored throughout sporulation, spore maturation, and outgrowth of B. subtilis [START_REF] Tocheva | Investigating germination and outgrowth of bacterial spores at several scales[END_REF]. However, as with TEM, SEM and AFM, the number of cells that can reasonably be observed and analyzed is inversely correlated to the structural details that can be revealed. These images are priceless for basic studies, but of rather poor utility for population studies with statistical significance or for the acquisition of large quantities of data.

With flow cytometry, spores or cells are aligned along a liquid flow in a capillary and detected with a laser beam. The light signals are analyzed by computing. The forward scattered (FSC) light signal provides information on cell size and the side-scattered light signal on cell inner structure or complexity. In addition appropriate fluorescent probes, commonly the same as those used in fluorescent microscopy, may provide information on cell permeability, viability or metabolic activity. , 2011). Spheroidal germinated spores and rod-shaped vegetative cells differ in size. Several characteristics can be obtained for each single cell in one pass through the detector, with size and fluorescence simultaneously providing information on the morphological or physiological states of the evolving spore populations. Other characteristics can be obtained with additional specific detectors [START_REF] Mathys | Flow cytometric assessment of Bacillus spore response to high pressure and heat[END_REF][START_REF] Van Melis | Characterization of germination and outgrowth of sorbic acid-stressed Bacillus cereus ATCC 14579 spores: Phenotype and transcriptome analysis[END_REF]. In contrast to phase contrast microscopy, the monitoring of a single cell over time is rather limited, but the number of cells that can be observed and characterized can be substantially higher (several tens of thousands of events is not uncommon). Flow cytometry has been used for instance to estimate the impact of high pressure (HP), heat treatments, or sorbic acid on Bacillus cereus or Bacillus licheniformis spore survival, germination and outgrowth [START_REF] Cronin | The use of flow cytometry to study the germination of Bacillus cereus endospores[END_REF], 2008;[START_REF] Mathys | Flow cytometric assessment of Bacillus spore response to high pressure and heat[END_REF][START_REF] Reineke | Mechanisms of endospore inactivation under high pressure[END_REF][START_REF] Van Melis | Characterization of germination and outgrowth of sorbic acid-stressed Bacillus cereus ATCC 14579 spores: Phenotype and transcriptome analysis[END_REF].

5. Molecular methods (Fig. 2E-F-G)

Molecular methods used for monitoring spore germination, outgrowth and growth can be presented as destructive or nondestructive methods performed on spore populations, spore population extracts or on specific components. Some methods can be implemented at single cell level.

As in any other biological process, the spore germination, outgrowth and growth process is increasingly analyzed with 'omic' technologies. Genomics, transcriptomics and proteomics establish a link between the spore behavior under different stresses and the fundamental basis of this behavior [START_REF] Abhyankar | The influence of sporulation conditions on the spore coat protein composition of Bacillus subtilis spores[END_REF]. The socalled 'omics' data are increasingly used as new tools for hazard identification and characterization, as well as for exposure assessment. These 'omics' data may provide a better understanding of the diversity of physiological states and thus lead to the identification of biomarkers, such as specific genes or specific proteins for example, that may be used as quantitative data to build predictive models applied to Microbial Risk Assessment [START_REF] Brul | Omics" technologies in quantitative microbial risk assessment[END_REF][START_REF] Mcmeekin | Ecophysiology of foodborne pathogens: Essential knowledge to improve food safety[END_REF][START_REF] Membr E | Latest developments in foodborne pathogen risk assessment[END_REF]. Transcriptomic analysis requires gene expression and mRNA synthesis, which are not essential in the earliest germination events [START_REF] Korza | Changes in Bacillus spore small molecules, rRNA, germination and outgrowth after extended sub-lethal exposure to various temperatures: Evidence that protein synthesis is not essential for spore germination[END_REF]. However, transcriptomic analyses are well suited to studying what happens in lag or early growth from bacterial spores. RNA extraction from spores requires some attention because of the particular structure of spores [START_REF] Moeller | A method for extracting RNA from dormant and germinating Bacillus subtilis strain 168 endospores[END_REF] (Fig. 2G). RNA is extracted from germinated spores or outgrowing cells over time and analyzed using different methods such as microarrays [START_REF] Keijser | Analysis of temporal gene expression during Bacillus subtilis spore germination and outgrowth[END_REF], RNA-seq [START_REF] Nagler | Identification of differentially expressed genes during Bacillus subtilis spore outgrowth in high-salinity environments using RNA sequencing[END_REF] or RT qPCR [START_REF] Bassi | Transcriptome analysis of Bacillus thuringiensis spore life, germination and cell outgrowth in a vegetable-based food model[END_REF]. Gene expression can be analyzed by specific pipelines such as T-Rex or JBrowse [START_REF] Skinner | JBrowse: A nextgeneration genome browser[END_REF][START_REF] De Jong | T-REx: Transcriptome analysis webserver for RNA-seq expression data[END_REF]. Thus, the evolution of cellular activity can be followed during germination and outgrowth and the impact of the incubation environment on this activity can be observed. For example, spore transcripts decrease during germination and early outgrowth until they are undetectable as the RNA stored in the spores is used as a source for de novo RNA and an initial source of nucleotides [START_REF] Keijser | Analysis of temporal gene expression during Bacillus subtilis spore germination and outgrowth[END_REF]. The impact of salinity on Bacillus subtilis has been investigated by a transcriptomic approach revealing a significant impact on stress gene expression during outgrowth [START_REF] Nagler | Identification of differentially expressed genes during Bacillus subtilis spore outgrowth in high-salinity environments using RNA sequencing[END_REF]. Proteins have a major role in structure and protection, such as coat proteins, and in differentiation from germinant receptors or cortex lytic enzymes [START_REF] Setlow | Germination of spores of Bacillus species: What we know and do not know[END_REF]. Proteomic methods can be applied in order to study germination and outgrowth (Fig. 2G). The extracted proteins are digested and analyzed by liquid chromatography associated with mass spectrometry [START_REF] Abhyankar | Gelfree proteomic identification of the Bacillus subtilis insoluble spore coat protein fraction[END_REF][START_REF] Abhyankar | Spore proteomics: The past, present and the future[END_REF][START_REF] Brul | Challenges and advances in systems biology analysis of Bacillus spore physiology; molecular differences between an extreme heat resistant spore forming Bacillus subtilis food isolate and a laboratory strain[END_REF]. Several proteins are essential for early germination steps: the germinant receptors, DPA channel proteins, cortex lytic enzymes (CLEs) and lipoproteins that are potentially implicated in the transduction of the germinant binding [START_REF] Setlow | Spore germination[END_REF][START_REF] Setlow | Germination of spores of Bacillus species: What we know and do not know[END_REF]. Analyzing dormant spores, germinated spores and outgrowing cells reveal an evolution of their proteome during germination and outgrowth process. For example, YpeB is 14-fold less detected in germinated spores than in dormant spores. YpeB is required for the incorporation of the CLE SleB in the inner spore membrane during sporulation, and then for its degradation during germination [START_REF] Chen | Levels of germination proteins in Bacillus subtilis dormant, superdormant, and germinating spores[END_REF]. The changes in spore proteome over time can be monitored during the germination and outgrowth process. A set of proteins appeared to be significantly reduced during the germination step [START_REF] Jagtap | Early events of Bacillus anthracis germination identified by time-course quantitative proteomics[END_REF]. The 'omics' methods are destructive and still applicable only to large spore populations (>10 7 spores mL À1 ). According to recent works, single cell 'omics' analyses are crucial to be able to take into account the heterogeneity of a spore population [START_REF] Abhyankar | Omics" for microbial food stability: Proteomics for the development of predictive models for bacterial spore stress survival and outgrowth[END_REF][START_REF] Blainey | The future is now: Single-cell genomics of bacteria and archaea[END_REF].

Remarkable details of spore germination at individual cell levels have been obtained with Raman spectroscopy (Fig. 2F). Raman spectroscopy is based on the excitation of molecules by a laser. Each spore or cell in this case is considered as a molecular mix. The emitted radiation is collected by a detector and the variation of emitted energy provides information on the energetic levels of the molecules composing the spore or the cell. This method, in combination with confocal microscopy, Raman spectrometry and laser tweezers (exploiting the dielectric properties of cells for separation) have been used to monitor the germination of single spores. CaDPA Raman spectra bands can be identified in several species of spore-forming bacteria such as Geobacillus stearothermophilus, B. subtilis or B. thuringiensis (Chen et al., 2006 , ; Gelder et al., 2007; Zhou, Dong, Setlow, & Li, 2013). The disappearance of the CaDPA spectral bands indicates the CaDPA release from the spore core and has been used to follow the germination of B. subtilis or G. stearothermophilus spores, for instance [START_REF] Chen | Real-time detection of kinetic germination and heterogeneity of single Bacillus spores by laser tweezers Raman spectroscopy[END_REF][START_REF] Vepachedu | Analysis of the germination of spores of Bacillus subtilis with temperature sensitive spo mutations in the spoVA operon[END_REF][START_REF] Wang | Kinetics of germination of wet-heattreated individual spores of Bacillus species, monitored by Raman spectroscopy and differential interference contrast microscopy[END_REF], 2015;[START_REF] Zhang | Characterization of single heat-activated Bacillus spores using laser tweezers Raman spectroscopy[END_REF][START_REF] Zhou | Kinetics of germination of individual spores of Geobacillus stearothermophilus as measured by Raman spectroscopy and differential interference contrast microscopy[END_REF]. Moreover, the comparison of Raman spectra of dormant spores, germinated spores and heat-treated spores has shown that wet heat treatment causes spore protein denaturation, and hence inhibits germination [START_REF] Zhang | Characterization of single heat-activated Bacillus spores using laser tweezers Raman spectroscopy[END_REF]. Spore germination can also be followed through the release of DPA with X-Ray analysis. Electrons from an external source strike the spore atoms, leading to energy emission as X-Ray photons with X-Ray characteristics of the element [START_REF] Bassi | A combination of a SEM technique and X-ray microanalysis for studying the spore germination process of Clostridium tyrobutyricum[END_REF]. CaDPA can also be detected by mass spectrometry or Fourier Transform Infrared Spectroscopy (FTIR) and can even be a biomarker of the presence of bacterial spores [START_REF] Goodacre | Detection of the dipicolinic acid biomarker in Bacillus spores using Curie-point pyrolysis mass spectrometry and Fourier transform infrared Comment citer ce document[END_REF].

Millifluidic or microfluidic technologies could be employed in order to monitor spore germination, outgrowth and growth. These methods are emerging in microbiology and a device has been specifically developed to study Bacillus subtilis spore germination and outgrowth. This device is based on the changes in electrical conductivity of a polyaniline nanofiber network. Electrical changes are induced by the spores passing through a measuring interface and the signal depends on the state of the cell (dormant spores, germinated spores or vegetative cells) [START_REF] Zabrocka | A microfluidic device for real-time monitoring of Bacillus subtilis bacterial spores during germination based on non-specific physicochemical interactions on the nanoscale level[END_REF]. Many techniques and methods can be implemented to monitor the germination, outgrowth and growth process. They operate at different scales, from cellular to molecular, and on different number of studied cells, from a whole population to single spores. It is worth noting that all these methods are complementary: some provide better knowledge of the biological process while others quantitatively determine the impact of different factors, such as heat treatment or incubation pH, on the spores' ability to complete germination, outgrowth and growth.

The ideal method, which does not yet exist, would give quantitative molecular information from the single cell level to the cell population level. By combining different methods, such as 'omics' allied to microfluidics, results can be obtained at both cellular and molecular levels. Such methods are emerging, like mass cytometry, derived from mass spectrometry and applied to analyses of individual cells. At the time of writing, this method is available mainly for mammalian cells, as other techniques like flow cytometry for example were. It is expected to be extended and adapted to microorganisms as mass cytometry has been used to discriminate different strains of E. coli [START_REF] Leipold | Development of mass cytometry methods for bacterial discrimination[END_REF]. We can expect an extension of this method to single bacterial spores, targeting specific spore molecules such as CaDPA. Moreover, advances in microfluidic methods, like the Droplet PCR technique used to study genomes at single cell level (Guti errez-Aguirre, Ra cki, Dreo, & Ravnikar, 2015; Ottesen, Hong, Quake, & Leadbetter, 2006), offer novel approaches and new opportunities to develop methods allowing gene expression analysis or the proteome study of single spores. The behavior of spores in a food matrix can be different from the behavior observed in laboratory media. Thus, the impact of the food matrix has to be taken into account to improve the control of spore-forming bacteria in foods [START_REF] Warda | Influence of food matrix on outgrowth heterogeneity of heat damaged Bacillus cereus spores[END_REF]. Clearly, not all the methods we have described in this review can be applied to study spore behavior in a food matrix. Generally, food-based media are developed to mimic the food matrices and culture-dependent methods or, more recently, microscopy and transcriptomic methods have been used to study spore germination and outgrowth [START_REF] Bassi | Transcriptome analysis of Bacillus thuringiensis spore life, germination and cell outgrowth in a vegetable-based food model[END_REF][START_REF] Warda | Influence of food matrix on outgrowth heterogeneity of heat damaged Bacillus cereus spores[END_REF]. The study of germination, outgrowth and growth, as well as the impact of environmental factors on this process, by methods at different scales provides better comprehension of these phenomena leading to better control of the behavior of spore-forming bacteria.

  Fig.1. Successive stages of the germination and outgrowth of bacterial spores, observed under phase-contrast microscopy and methods applied for monitoring. Commitment to germination can be followed by the release of CaDPA[START_REF] Zhang | Monitoring of commitment, blocking, and continuation of nutrient germination of individual Bacillus subtilis spores[END_REF]. Spores remain phase-bright. Germination can be monitored by phase contrast microscopy on individual spores or on limited spore populations[START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF][START_REF] Tychinsky | Dynamic phase microscopy, a new method to detect viable and killed spores and to estimate the heterogeneity of spore populations[END_REF]; by Raman spectroscopy[START_REF] Chen | Real-time detection of kinetic germination and heterogeneity of single Bacillus spores by laser tweezers Raman spectroscopy[END_REF] to follow CaDPA release into culture medium[START_REF] Zhang | Effects of cortex peptidoglycan structure and cortex hydrolysis on the kinetics of Ca 2þ -dipicolinic acid release during Bacillus subtilis spore germination[END_REF]. Spores turn from phase-bright to phase dark. Outgrowth can be monitored using phase contrast microscopy associated with a time lapse image system[START_REF] Pandey | Live cell imaging of germination and outgrowth of individual Bacillus subtilis spores; the effect of heat stress quantitatively analyzed with SporeTracker[END_REF]; by electron microscopy[START_REF] Saif Zaman | Imaging and analysis of Bacillus anthracis spore germination[END_REF] or by epifluorescence microscopy (van Melis, den Besten, Nierop Groot, & Abee, 2014). The spore core hydrates and the cell emerging from the spores expands. Growth can be monitored by changes in turbidity (A 600 ) or by cultural methods[START_REF] Roubos-Van Den Hil | Soya bean tempe extracts show antibacterial activity against Bacillus cereus cells and spores[END_REF]; by epifluorescence microscopy[START_REF] Van Melis | Quantification of the impact of single and multiple mild stresses on outgrowth heterogeneity of Bacillus cereus spores[END_REF]; by transcriptomic methods[START_REF] Segev | Molecular kinetics of reviving bacterial spores[END_REF] or by flow cytometry[START_REF] Van Melis | Characterization of germination and outgrowth of sorbic acid-stressed Bacillus cereus ATCC 14579 spores: Phenotype and transcriptome analysis[END_REF]. Rod-shaped cells divide and multiply.

  Flow cytometry can be used to monitor germination, outgrowth and growth. Germinated spores Comment citer ce document : Trunet, C., Carlin, F., Coroller, L. (Auteur de correspondance) (2017). Investigating germination and outgrowth of bacterial spores at several scales. Trends in Food Science and Technology, 64, 60-68. DOI : 10.1016/j.tifs.2017.03.008 Manuscrit d'auteur / Author manuscript Manuscrit d'auteur / Author manuscript Manuscrit d'auteur / Author manuscript Version définitive du manuscrit publiée dans / Final version of the manuscript published in : Trends in Food Science & Technology (2017), Vol. 64, p. 60-68, DOI: 10.1016/j.tifs.2017.03.008 Journal homepage : http://www.journals.elsevier.com/locate/trends-in-food-science-and-technology differ from dormant spores due to their higher probe fluorescence (of Syto9 for instance) generated by the permeation of spore layers and core hydration during germination (van Melis, Nierop Groot, Tempelaars, Moezelaar, & Abee

  Comment citer ce document : Trunet, C., Carlin, F., Coroller, L. (Auteur de correspondance) (2017). Investigating germination and outgrowth of bacterial spores at several scales. Trends in Food Science and Technology, 64, 60-68. DOI : 10.1016/j.tifs.2017.03.008 Manuscrit d'auteur / Author manuscript Manuscrit d'auteur / Author manuscript Manuscrit d'auteur / Author manuscript Version définitive du manuscrit publiée dans / Final version of the manuscript published in : Trends in Food Science & Technology (2017), Vol. 64, p. 60-68, DOI: 10.1016/j.tifs.2017.03.008 Journal homepage : http://www.journals.elsevier.com/locate/trends-in-food-science-and-technology 6. Conclusion

Trends in Food Science & Technology (2017), Vol. 64, p. 60-68, DOI: 10.1016/j.tifs.2017.03.008 Journal homepage : http://www.journals.elsevier.com/locate/trends-in-food-science-and-technology method

  Version définitive du manuscrit publiée dans / Final version of the manuscript published in :

	Manuscrit d'auteur / Author manuscript
	Manuscrit d'auteur / Author manuscript
	Manuscrit d'auteur / Author manuscript	Comment citer ce document :
		Trunet, C., Carlin, F., Coroller, L. (Auteur de correspondance) (2017). Investigating
		germination and outgrowth of bacterial spores at several scales. Trends in Food Science and Technology,
		64, 60-68. DOI : 10.1016/j.tifs.2017.03.008

, a prolonged germination and lag phase and a lower germination rate were observed with spores produced in suboptimal sporulation conditions, then heat-treated or incubated in suboptimal conditions

[START_REF] Collado | Kinetics of deactivation of Bacillus cereus spores[END_REF][START_REF] Daelman | Development of a time-to-detect growth model for heattreated Bacillus cereus spores[END_REF][START_REF] Gaillard | Quantifying the combined effects of the heating time, the temperature and the recovery medium pH on the regrowth lag time of Bacillus cereus spores after a heat treatment[END_REF]

; Roubos-van den Hil, Dalmas, Nout, & Abee, 2010).

in Food Science & Technology (2017), Vol. 64, p. 60-68, DOI: 10.1016/j.tifs.2017.03.008 Journal homepage : http://www.journals.elsevier.com/locate/trends-in-food-science-and-technology

Trunet, C., Carlin, F., Coroller, L. (Auteur de correspondance) (2017). Investigating germination and outgrowth of bacterial spores at several scales. Trends in Food Science and Technology, 64, 60-68. DOI : 10.1016/j.tifs.2017.03.008 Manuscrit d'auteur / Author manuscript Manuscrit d'auteur / Author manuscript Manuscrit d'auteur / Author manuscript Version définitive du manuscrit publiée dans / Final version of the manuscript published in : Trends

C. Trunet et al. / Trends in Food Science & Technology 64 (2017) 60e68

Acknowledgements

This work is a partial fulfilment of Cl ement Trunet's PhD thesis and was supported by the Conseil R egional de Bretagne under the Spore'Up contract, by ADRIA D eveloppement (Quimper, France), and by Bretagne Biotechnologie Alimentaire (Rennes, France) and the French National Association of Technical Research (ANRT, Paris) (0801/2012). The authors wish to thank the reviewers of their manuscript for their absolutely relevant comments and useful recommendations to improve both substance and form, and No emie Desriac for her critical reading of the manuscript.