R Blossey

A C Maggs

A fluctuation-corrected functional of convex Poisson-Boltzmann theory

Poisson-Boltzmann theory allows to study soft matter and biophysical systems involving point-like charges of low valencies. The inclusion of fluctuation corrections beyond the mean-field approach typically requires the application of loop expansions around a mean-field solution for the electrostatic potential φ(r), or sophisticated variational approaches. Recently, Poisson-Boltzmann theory has been recast, via a Legendre transform, as a mean-field theory involving the dielectric displacement field D(r). In this paper we consider the path integral formulation of this dual theory. Exploiting the transformation between φ and D, we formulate a dual Sine-Gordon field theory in terms of the displacement field and provide a strategy for precise numerical computations of free energies beyond the leading order.

Introduction

The description of electrolytes in soft matter is commonly based on the Poisson-Boltzmann equation, a mean-field theory for the electrostatic potential φ(r) (see, e.g., [1], and references therein). Formally, this theory is derived from the partition function of a Coulomb system by performing a Hubbard-Stratonovich transformation to an (imaginary) fluctuating field φ(r), later identified with the electrostatic potential [2,3]. (For the Hubbard-Stratonovich transformation, see e.g. [4,5,6,7,8,9]). This approach is valid in the weak fluctuation regime, in which the saddle-point of the action yields the standard Poisson-Boltzmann equation, which can be systematically improved by a loop expansion [3]. An alternative approach to go beyond mean-field theory is to invoke a variational approach [10,11]. Going beyond mean field in Poisson-Boltzmann is known to improve many features of the theory, for instance dielectric contrast and image-charge interactions are much better described in theories which allow fluctuations [12].

A difficulty with these approaches is that the action of the functional integral is complex. A Wick rotation gives a real theory, but at the price of rendering the effective free energy concave. Thus, when the electrostatics is coupled to other degrees of freedom, the extremization of the free energy becomes a numerically difficult operation. Extrema result from a combination of minima in the non-electrostatic degrees of freedom and maxima in the electrostatic potential. This complicates many numerical studies of biophysical molecules in which the solvent is described by Poisson-Boltzmann theory, and also has repercussions for the validity limits of theories including fluctuation effects beyond Poisson-Boltzmann theory [13]. This technical problem has been circumvented [14,15] by a reformulation of the Poisson-Boltzmann theory in terms of purely convex functionals, which is achieved by the means of a Legendre transform, with the complex field φ(r) being systematically replaced by the dielectric displacement field D(r). Expressed in terms of the field D(r), the resulting theory yields a convex functional so that standard minimization techniques can be applied, and loop corrections or variational approaches be defined as for the original Poisson-Boltzmann theory. Though it was constructed only from the mean-field, the one-loop correction in the dual theory (very surprisingly) has been shown to yield the same fluctuation spectrum as obtained within the usual formulation of Poisson-Boltzmann theory [16].

This approach, however, working from conventional mean-field theory, does not help in constructing a systematic improvement in the dual formulation of the theory. The aim of the present article is to formulate a systematic method for going beyond the "oneshot" approximation that has been considered until now for constructing dual theories and present a general formulation that is equivalent to the exact, discretized statistical mechanics of the electrolytes that are formulated as integrals over an electrostatic potential. We stress the generality of our approach, even if in this paper we chose to apply the formulation to only the simplest two-component electrolyte.

With this motivation in mind, we revisit the dual approach from a functional integral perspective which relies on the introduction of the dual transform via delta functions. The arising transformation integrals can be performed either in an exact way, or in a systematic fashion by invoking the Poisson summation formula. The latter will first give us access to a saddle-point like limit, but it equally allows us -with no more than simple integration and one-dimensional Fourier transforms -to explicitly calculate the corrections beyond the saddle point which arise from the discrete nature of the charges. In this way we obtain a novel formulation of Poisson-Boltzmann theory in terms of its dual which is exact. Our final result is a discretized free energy function, which, when sampled gives the exact discretized energy of a set of discrete charges interacting with the Coulomb interaction.

We expect that this novel dual formulation will be highly useful for the finite temperature simulation of coupled conformational/electrostatic degrees of freedom, rather than the minimization envisaged with the original mean-field theory. For this purpose we formulate the theory in a formulation adequate for numerical computations on lattices.

The paper is organized as follows. In the second section, we discuss the notion of duality, recall the formulation of the duality transform for the Poisson-Boltzmann functional and the derivation of the dual mean-field functional. In section three we derive a prescription to formulate the functional integral of a dual theory including fluctuations beyond mean field for the case of the symmetric electrolyte; this choice is made for definiteness and provides no restriction to the method. The program is subsequently implemented in section four. We conclude in section five.

Duality in Poisson-Boltzmann theory

We first recall the notion of duality previously employed in the context of the Poisson-Boltzmann theory at the mean-field level [14]. We start with the usual expression [2] for the partition function for a set of charged particles interacting through the Coulomb interaction, described through a fluctuating potential field φ:

Z = D[φ]e -β d 3 rh(φ) with h(φ) = ε (∇φ) 2 2 + g(iφ) -i f φ ,
where g(iφ) is the pressure, while f is the density of fixed charges, which are typically confined to a small part of the system. Introducing the electric field E with the help of a delta function (and being free with non-essential normalization factors)

Z = D[φ]D[E]e -β d 3 rh(φ) δ(E + ∇φ)
yields after Fourier representation of the delta function with multiplier D

Z = D[φ]D[E]D[D]e -β d 3 rh(φ,E,D) with h(φ, E, D) = ε E 2 2 + g(iφ) -iD • (∇φ + E) -i f φ (1) = ε E 2 2 + g(iφ) -iD • E + iφ(divD -f ) .
Careful consideration of boundary conditions shows that φD.dS = 0 so that one requires either φ = 0 or the normal component of the field D n = 0 at the boundaries. Performing the integration over the electric field E, one finds

Z = D[φ]D[D]e -β d 3 rh(φ,D) (2)
with

h(φ, D) = D 2 2ε + g(iφ) + iφ(divD -f ) .
In this expression, divDf ≡ s can be read as a Fourier transform variable such that we arrive at the formal expression

h(φ, D) = D 2 2ε -ln{F(e g(iφ) )}[divD -f ] (3)
with F the Fourier operator. We emphasize that this step requires the evaluation of just a single one-dimensional Fourier transform. In numerical applications this transform can be done once and tabulated, even if the model for g(iφ) is relatively complicated analytically. As emphasized in [17] the function g(φ) is typically the pressure of the uncharged fluid. We then find the main result of this paper

Z = D[D]e -β d 3 r h ef f (D) (4)
in which h ef f (D) is a local functional of D. If the conventional saddle-point approximation is employed, we obtain the standard mean-field theory of soft matter electrostatics. Our aim here is to derive an expression for h ef f (D) valid beyond meanfield theory, on the basis of which an exact sampling of the partition function eq.( 4) can be performed e.g. by Monte-Carlo simulation [18].

In our formulation of the full statistical mechanics of the dual theory it is a crucial feature that the duality transformation φ → D has been carried out solely via the introduction of delta functions. Only this approach guarantees the identity of the fluctuation spectra of the dual theory as discussed in detail [19] in a general field-theory setting. By contrast, theories derived from reparametrizations are called pseudo-dual theories [19] for which this property in general does not hold. This is e.g. the case for the reparametrizations of the Poisson-Boltzmann theory recently developed in [20,21] with the aim of defining convex functionals for the electrostatic potential. These theories fail to produce the correct fluctuation spectra as was shown explicitly in the calculation of the one-loop correction [16]. Even in lattice gauge theories, dual formulations can have similar technical advantages of positivity and convexity and are an active research topic [22].

The above presentation has been at a formal level. In order to be explicit in this program we will now present a detailed calculation of eq.( 3) for the case of a simple model for the pressure,

g(iφ) = -2λ cos(φ eβ) , (5)
corresponding to a perfect gas model of the ions. The full partition function corresponding to (5) (without the complex coupling to the external charge f ) is the well-known Sine-Gordon energy. In three dimensions it has been widely studied due to its links to electrodynamics. Unlike the closely related model of charged hard-spheres there is no phase transition [23]. In its discretized form the theory is regularized by the lattice spacing.

For both the purpose of regularizing and simulating this theory, we will first discretize it. Our dual model then is, by construction, entirely equivalent to this full, discrete Sine-Gordon system, as it includes all the fluctuations beyond mean-field theory that are contained in this starting theory.

Discretization of the dual theory

We now turn to a definition and calculation of the functional integral (2), for the explicit case of the symmetric electrolyte given by (5). The continuum action for the potential that we start with is

h(φ) = ε (∇φ) 2 2 -2λ cos(φ eβ) -i f φ . (6)
We discretize this model at a spacing to a simple cubic lattice. The potential and charges are associated with the sites of the lattice. Derivatives are replaced by lattice differences, d . The discretized energy corresponding to eq. ( 6) is then

h (φ) = ε (d φ) 2 2 -2λ 3 cos(φ eβ) -i 3 f φ .
We change the integration variable from φ to the scaled potential φ/(eβ), which gives the (a-dimensional) statistical weight

βh (φ) = sites ε e 2 β (d φ) 2 2 -2 λ cos(φ) -i¯ φ ,
where we define λ = β 3 λ and ρ = f 3 /e. We note that ε/e 2 β = /(4π B ), with B the Bjerrum length. To define the dual formulation we now consider the discrete electric field, E = -d φ, to be associated the links of the lattice and impose this constraint using a delta-function integral over D , which is also considered as link variable. The entire formulation follows very closely the continuum calculation given above. We need only to define the discrete equivalent of the divergence operator from the adjoint (or matrix transpose) of the difference operator d : div → -d T [24].

The partition function in its discretized form is thus

Z = D[D ]D[φ] e - e 2 β 2 ε D 2 +i(-d T D -¯ )φ-2 λ cos(φ)
.

This expression can be reorganized into

Z = D[D ] e -e 2 β 2 ε D 2 D[φ]e (-isφ+2 λ cos(φ)) = I[D , J[φ]] (7)
with s ≡ -d T D -¯ . Here sums in the exponentials are understood as being over sites for potentials and charges and over links for field components. It is here that we see explicitly that the only step that is needed to complete the program is the integration over the field φ. As the field φ in the expression ( 7) is purely local, we can drop the functional notation and thus have to deal with a simple integral for each site of the discretization z(s) = dφ e (-isφ+2 λ cos(φ)) .

For a general model of electrolyte we can expect that this integral will be difficult to perform analytically, but it can always be treated in a numerical manner, in order to generate the exact effective theory for D . We can make further analytic progress for the case of the two-component electrolyte, which we now present in detail.

Two-component electrolyte

The standard way of proceeding to calculate z(s) would be to invoke complex integration. The argument of the exponential function has zeroes at 2 λ sin(φ) = is, hence at φ = sin -1 (is/2 λ), whereby the inversion of the sine function fixes the saddle to be retained. Here, we will work through the saddle-point evaluation and its corrections in a different manner, in view of our interest to have an approach which is adapted to numerical computations.

The mathematical tool we use in our formalism is the Poisson summation formula which expresses a sum over integer occupation numbers by an equivalent sum over Fourier coefficients. This gives a mathematically rigorous formulation in which the dominant contribution is the mean-field free energy already but gives a framework in which higher-order corrections to the dual theory are expressed as an exact Fourier series, with exponential convergence. Thus we now find an analytic expansion for eq. (8).

In order to implement this program we note that we first rewrite z(s) in the discrete form

z(s) = n δ(s -n) g(n) = n δ(s -n) e -fe(n) (9)
where we then need to calculate the weighting function f e (s). In order to perform this step we expand the exponentiated cosine in eq. ( 8) as a Taylor series of the two complex exponentials using

exp ( λe iφ ) = ∞ n=0 e inφ+nµ n! , ( 10
)
where for convenience we define µ = ln( λ). Substituting eq. ( 10) in eq. ( 8) and using the definition of the delta function we find

z(s) = ∞ n 1 ,n 2 =0 1 n 1 ! 1 n 2 ! e µ(n 1 +n 2 ) δ(s + n 1 -n 2 ) ( 11
)
where n 1 and n 2 represent the occupation numbers of positive and negative ions.

In order to perform the summation, we can invoke the Poisson summation formula which tells us that if g(n, s) is a function defined on integers and g is its continuous Fourier transform

z(s) = n g(n, s) = k g(2πk, s) = k z k (s) ( 12
)
where z k (s) is the contribution to the k-th Fourier mode to the total sum z(s). In order to apply this identity we solve the delta-function constraint and reduce the problem to a single summation, which leads to the expression

z(s) = n g(n, s) = n 1 n! 1 (n + |s|)! e µ(2n+|s|) .
This expression is a simple function of the variable s, which can be tabulated, or interpolated to arbitrary precision for numerical work. Some additional care is, however, needed for our problem: we here require the sum is over positive integers, whereas the Poisson formula applies for a sum over positive and negative integers. This can be achieved by imposing a multiplicative weighting function.

We implement this first for the k = 0 contribution to the full partition sum eq. ( 12). For convenience we go back to a continuous variable n with which we can write this contribution as an integral over the summand of eq. ( 12); it thus is the naive integral over the occupation number variable, forgetting about the discrete nature of the elementary charges. It reads as

z 0 (s) = ∞ -∞ dn w(n) g(n, s)
where the weighting function is given by, e.g.,

w(n) =        0 if n < -1 e -1/n e -1/n +e 1/(1+n) if -1 < n < 0 1 if n > 0 .
This cross-over function is smooth and increases from zero to unity on the interval -1 < n < 0. The choice of w(n) is non-unique; we note that the general theory of Fourier analysis show that the final results does not depend on the particular choice we have made. We also note that the extension of the factorial function to the real numbers, which is needed to represent g(n, s) in its continuous form, is also non-unique. The commonly used extension is Euler's Gamma function, but alternative extensions due to Hadamard are also possible [25].

The integral z 0 (s) is dominated for λ > 1 by a saddle. We will also show that in this limit the contributions z k (s) for |k| ≥ 1 are exponentially smaller and that to high accuracy z(s) can be replaced by z 0 (s). To study the nature of the saddle we work with an improved version of the Stirling formula

ln(n!) ≈ (n + 1/2) ln(n + 1/2) -(n + 1/2) + O(1) = S(n)
which correctly includes the term (ln n)/2 in its expansion. Replacing the single continuous variable n again by two continuous variables n 1 and n 2 we then have the expression

z 0 (s) = n 1 ,n 2 exp (µ(n 1 + n 2 ) -S(n 1 ) -S(n 2 )) δ(s + n 1 -n 2 ). (13)
The saddle-point is best studied by implementing the delta-function constraint with a Lagrange multiplier τ , which leads to the following equations, s = 2 λ sinh(τ )

n 1 + 1/2 = λe -τ (14)
n 2 + 1/2 = λe +τ .

On eliminating τ we find the values of the occupation numbers which extremize the integrand:

n 2 + 1/2 = ( s + (s 2 + 4 λ2 ) 1/2 ) 2 n 1 + 1/2 = (-s + (s 2 + 4 λ2 ) 1/2 ) 2 . (15)
When (s/ λ) is small, both n 1 and n 2 equal the background occupation number λ. The value of the saddle-point eq. ( 13) is

ln(z 0 ) ≈ s ln s 2 λ + 1 + (s/2 λ) 2 -s 2 + 4 λ2 -ln ( λ) (16)
and we define f 0 (s) = -ln z 0 . With s small, z 0 = O(e 2 λ).

We now evaluate the second derivative, ∆(s), of the action at the saddle

∆(s) = 1 n 1 + 1/2 + 1 n 2 + 1/2 = 1 λ2 4 λ2 + s 2 .
For small s, ∆(s) = 2/ λ. At the saddle-point we have the quadratic approximation to the statistical weight,

f (n, s) = f 0 (s) + 1 2 ∆ (n -n s ) 2 , n s = min(n 1 , n 2 ) . (17)
According to the sign of s we use the smaller of n 1 and n 2 as the primary variable mean-field occupation number. Performing the integral over the full occupation number n in eq. ( 17) we find that the effective action for eq. ( 9) is 11), (blue dot-dash) compared to the saddle point approximation eq. ( 16), (red dash) and the quadratic approximation, eq. ( 18), (yellow dotted). Three different values of λ. For already modest values of λ = 3 (bottom) the quadratic approximation approximation is already excellent with a good overlay of the exact and approximate curves. For λ < 1 (top) the mean field and quadratic approximations underestimate the true free energy. The curves are shifted so that f (0) = 0.

f e (s) = f 0 (s) + 1 2 ln(∆(s)) . ( 18
)
With the help of the Poisson summation formula we now calculate the higher Fourier contributions to the sum eq. ( 12) which come from the discrete nature of the elementary charges. For λ > 1 the Fourier components are also calculated in the saddlepoint approximation. We take the quadratic approximation to the energy eq. ( 17) and regroup the positive and negative Fourier coefficients, zk = (z k + z -k ) = e -f 0 (s) 2 cos (2πkn)e -∆(n-ns) 2 /2 dn , which results in the expression

zk = e -f 0 (s) √ ∆ 2 cos(2πkn s )e -2k 2 π 2 /∆ .
If we consider that ∆ = O(2/ λ) then the amplitude of this contribution is zk /z 0 = O(e -π 2 k 2 λ) .

If λ 1 then this contribution is strongly suppressed in comparison with the k = 0 contributions to the partition function.

We now calculate the importance of the crossover function w(n) which constrains the sums and integrals to be over positive occupations. Near n = 0 the function ln(n!) = O(1). The Fourier transform of the weighting function can be estimated by a further saddle-point calculation to be O(1) × e - √ k , [26]. We thus expect that the contribution to the partition sum coming from the constraint of the sum over positive occupation numbers is of O(1).

Finally, by a regrouping of the contributions we conclude that an expansion for the Fourier partition sum eq. ( 12) is given by the formula

z(s) = e -f 0 (s) ∆(s) (1 + 2 cos(2n s )e -2π 2 /∆(s) ) + O(1) ,
where -f 0 (s) ∼ 2 λ and we have kept just the first and largest non-trivial Fourier component for zk . ‡ We conclude that when the discretization is such that λ > 1, we can drop the oscillating terms as well as the end-point corrections to find the simplified expression

f e (s) = f 0 (s) + 1 4 ln s 2 + 4 λ2 λ4 . ( 19
)
This result is to be compared to the exact sum eq. ( 11) in Fig. 1. We see that the the analytic expression is an excellent fit to the full numerical evaluation. Thus for λ > 1, the final effective functional for the displacement field D is given by the

h ef f = links e 2 β ε D 2 2 + f e (-d T D -¯ ) ,
where f e (s) is given by eq. ( 16) and eq. ( 19), and the measure includes the delta-functions of eq. ( 9). This functional is valid beyond mean-field as it includes the fluctuations in the underlying field φ, it is also explicitly an energy function that should be sampled by Monte Carlo or molecular dynamics.

Conclusions

In this Letter we have given an explicit formulation of the duality-transformed Poisson-Boltzmann theory, for the illustrative case of a system of a symmetric electrolyte. The ‡ We have also assumed that the background charge density is not so large that it overwhelms the electrolyte concentration.

partition function is given by a double integral over the potential-field variable φ and the dielectric displacement field D . The potential integral can be evaluated as a systematic series with the help of the Poisson summation formula. The result is a Fourier series for the φ integral. For λ > 1 we evaluated this series analytically, but the Fourier transforms can also be evaluated numerically for small λ. The final result is a theory which has physical content identical to the Sine-Gordon model of a charged lattice gas, which goes beyond mean field theory and includes all fluctuations in an exact manner. The final theory remains clearly analytically intractable. However, it is of now of a form which is simple to simulate, and to couple to external charges. This is in contrast to the original field-theory formulation in terms of the potential, which included a complex coupling to external charges. This opens the possibility of simulation of complex coupled situations where electrostatic fluctuations are important.

Although we have derived our formulation for the case of the symmetric electrolyte our approach is not limited to this choice and should allow the treatment of more complex models of electrolyte, fully including fluctuations beyond mean-field theory in their coupling to external soft-matter systems.

Figure 1 .

 1 Figure 1. Comparison of the evaluation of the sum eq. (11), (blue dot-dash) compared to the saddle point approximation eq. (16), (red dash) and the quadratic approximation, eq. (18), (yellow dotted). Three different values of λ. For already modest values of λ = 3 (bottom) the quadratic approximation approximation is already excellent with a good overlay of the exact and approximate curves. For λ < 1 (top) the mean field and quadratic approximations underestimate the true free energy. The curves are shifted so that f (0) = 0.


Marta Gallardo Ruiz

Jean-François Le Galliard

Thomas Tully
email: thomas.tully@sorbonne-universite.fr

Marta Gallardo

Genetic variation in light vision

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Introduction

Animals use various environmental cues for habitat choices, and different individuals from a single species may show contrasting habitat preferences depending on their sex, stage, size and genetic background for instance [START_REF] Stamps | Habitat selection by dispersers: integrating proximate and ultimate approaches[END_REF][START_REF] Matthysen | Multicausality of dispersal: a review. Dispersal Ecology and Evolution[END_REF]. This intra-specific variation of behavioural responses may influence a wide range of eco-evolutionary processes [START_REF] Sih | Ecological implications of behavioural syndromes[END_REF][START_REF] Ronce | Dispersal syndromes. Dispersal Ecology and Evolution[END_REF][START_REF] Edelaar | Non-random gene flow: an underappreciated force in evolution and ecology[END_REF]. Inter-individual differences in habitat choice behaviours may be genotype-dependent and related to differential performance in specific habitats or niches [START_REF] Sih | Ecological implications of behavioural syndromes[END_REF][START_REF] Edelaar | Non-random gene flow: an underappreciated force in evolution and ecology[END_REF][START_REF] Hawthorne | Genetic linkage of ecological specialization and reproductive isolation in pea aphids[END_REF][START_REF] Cousyn | Rapid, local adaptation of zooplankton behavior to changes in predation pressure in the absence of neutral genetic changes[END_REF][START_REF] De Meester | Evolutionary potential and local genetic differentiation in a phenotypically plastic trait of a cyclical parthenogen, Daphnia magna[END_REF][START_REF] Jaenike | Genetic variation for habitat preference: evidence and explanations[END_REF]. Genetic variation in habitat choice behaviour is common in animals and has been well documented for host plant preferences and cryptic habitat selection in insects (reviewed in [START_REF] Jaenike | Genetic variation for habitat preference: evidence and explanations[END_REF].

One of the cues animals use to select their habitat is light which animals may be positively or negatively attracted to. For example, negative phototaxis acts as a predator-avoidance mechanism in some aquatic organisms [START_REF] De Meester | Evolutionary potential and local genetic differentiation in a phenotypically plastic trait of a cyclical parthenogen, Daphnia magna[END_REF][START_REF] Cousyn | Rapid, local adaptation of zooplankton behavior to changes in predation pressure in the absence of neutral genetic changes[END_REF][START_REF] Michels | Inter-clonal variation in phototactic behaviour and key life-history traits in a metapopulation of the cyclical parthenogen Daphnia ambigua: the effect of fish kairomones[END_REF][START_REF] Borowsky | Responses to light in two eyeless cave dwelling amphipods (Niphargus ictus and Niphargus frasassianus)[END_REF] and may help some soil organisms that are very sensitive to relative humidity to prevent desiccation by looking for deeper and more humid soil layers [START_REF] Salmon | Linking species, traits and habitat characteristics of Collembola at European scale[END_REF][START_REF] Salmon | Species traits and habitats in springtail communities: a regional scale study[END_REF]. Nevertheless, different individuals or populations within the same species may present different phototactic preferences. For example, clonal populations of Daphnia magna (a zooplancton species) exposed to high levels of predatory pressure are more photophobic than clonal populations less exposed to predation [START_REF] Cousyn | Rapid, local adaptation of zooplankton behavior to changes in predation pressure in the absence of neutral genetic changes[END_REF][START_REF] De Meester | Evolutionary potential and local genetic differentiation in a phenotypically plastic trait of a cyclical parthenogen, Daphnia magna[END_REF]. In general, differences in phototaxis may have a heritable, genetic basis (e.g., [START_REF] Markow | Genetic analysis of phototactic behavior in Drosophila simulans[END_REF][START_REF] De Meester | Evolutionary potential and local genetic differentiation in a phenotypically plastic trait of a cyclical parthenogen, Daphnia magna[END_REF][START_REF] Cousyn | Rapid, local adaptation of zooplankton behavior to changes in predation pressure in the absence of neutral genetic changes[END_REF] or could be the result of nongenetic, phenotypic plasticity and personality (is this too anthropomorphic for Daphnia or springtails?) differences among individuals (e.g., [START_REF] Kain | Phototactic personality in fruit flies and its suppression by serotonin and white[END_REF]. Quantifying sources of variation in phototaxis is therefore important to understand the evolution of this wide-spread behavioural trait. Here, we investigate the genetic variation in light sensitivity and light-dependent movement behaviour in the eye-less springtail Folsomia candida Willem 1902 (Collembola, Isotomidae), an hemi-edaphic and cosmopolitan soil organism inhabiting various habitats such as caves, forest litter and man-made habitats [START_REF] Fountain | Folsomia candida (Collembola): a standard soil arthropod[END_REF].

The degeneration or even loss of the visual system is a convergent and frequent evolutionary phenomenon in soil-dwelling and cave animals [START_REF] Christiansen | Morphological adaptations. Encyclopedia of Caves[END_REF]. Nevertheless, even eyeless and eye-reduced species often retain some sensitivity to the ambient light level through extra-ocular photoreceptors (EOP, Taddei-Ferretti and Musio, 2000;[START_REF] Ullrich-Lüter | Unique system of photoreceptors in sea urchin tube feet[END_REF], which are useful for the maintenance of circadian rhythms [START_REF] Friedrich | Biological clocks and visual systems in cave-adapted animals at the dawn of speleogenomics[END_REF] or for orientation and habitat choice [START_REF] Timmermann | Phototactic response and light sensitivity in an epigean and a hypogean population of a barb (Garra barreimiae, Cyprinidae)[END_REF][START_REF] Borowsky | Responses to light in two eyeless cave dwelling amphipods (Niphargus ictus and Niphargus frasassianus)[END_REF]. Indeed, previous works strongly suggest that F. candida is sensitive to light despite being eyeless. In choice-test experiments, F. candida avoids UV light moving to warmer locations exposed to white light, prefers darkness over cool white light [START_REF] Fox | Phototactic responses to ultraviolet and white light in various species of Collembola including the eyeless species, Folsomia candida[END_REF], and displays a doseresponse avoidance of UV-B light relative to darkness [START_REF] Beresford | Lethal and sub-lethal effects of UV-B radiation exposure on the collembolan Folsomia candida (Willem) in the laboratory[END_REF]. Yet, to our knowledge, no study has examined the wavelengths of maximum sensitivity of the ocular or extra-ocular photoreceptors (EOPs)of these animals [START_REF] Barra | Les photorecepteurs des Collemboles, etude ultrastructurale[END_REF][START_REF] Jordana | A new type of arthropod photoreceptor[END_REF][START_REF] Fox | Phototactic responses to ultraviolet and white light in various species of Collembola including the eyeless species, Folsomia candida[END_REF]. In true insects (Pterygota), few species are able to detect wavelengths longer than 600 nm (red light), which suggests a red-blind common ancestor [START_REF] Briscoe | The evolution of color vision in insects[END_REF]. In addition, the behavioural tests mentioned above could not always prevent confounding effects of differences in temperature or humidity associated with the lighting treatment. This is of great importance, as F. candida needs a relative humidity close to saturation [START_REF] Holmstrup | Strategies for Cold and Drought Tolerance in Permeable Soil Invertebrates[END_REF][START_REF] Waagner | Recovery of reproduction after drought in the soil living Folsomia candida (Collembola)[END_REF] and is very sensitive to temperature [START_REF] Boiteau | Locomotor response of Folsomia candida (Collembola: isotomidae) to cooling temperatures[END_REF][START_REF] Boiteau | Role of avoidance behavior in the response of Folsomia candida to above-freezing cooling temperatures[END_REF].

Although sexual reproduction exists in some populations of F. candida [START_REF] Frati | High levels of genetic differentiation between Wolbachia-infected and non-infected populations of Folsomia candida (Collembola, Isotomidae)[END_REF], this species is generally recognized as asexual, and most studies using F. candida as a model species have used parthenogenetic lineages [START_REF] Fountain | Folsomia candida (Collembola): a standard soil arthropod[END_REF]. Earlier studies on several parthenogenetic lineages have uncovered substantial intra-specific genetic and morphological polymor-phism [START_REF] Chenon | Genetic polymorphism in nine clones of a parthenogenetic collembolan used in ecotoxicological testing[END_REF][START_REF] Tully | Two major evolutionary lineages revealed by molecular phylogeny in the parthenogenetic collembola species Folsomia candida[END_REF][START_REF] Tully | Intraspecific phenotypic variation and morphological divergence of strains of Folsomia candida (Willem) (Collembola: isotomidae), the standard test springtail[END_REF]. Intra-specific diversity is organised in two major evolution-ary clades [START_REF] Tully | Two major evolutionary lineages revealed by molecular phylogeny in the parthenogenetic collembola species Folsomia candida[END_REF][START_REF] Tully | Intraspecific phenotypic variation and morphological divergence of strains of Folsomia candida (Willem) (Collembola: isotomidae), the standard test springtail[END_REF], and life history studies have shown that two contrasted biodemographic strategies evolved along the divergence of these two clades [START_REF] Tully | Reproductive flexibility: genetic variation, genetic costs and long-term evolution in a collembolan[END_REF][START_REF] Tully | Facteurs génétiques, maternels et environnementaux de l'expression des traits d'histoire de vie chez le collembole[END_REF][START_REF] Tully | The evolution of postreproductive life span as an insurance against indeterminacy[END_REF][START_REF] Mallard | Within species variation in long-term trajectories of growth, fecundity and mortality in the Collembola Folsomia candida[END_REF]. One clade has a high reproductive potential: when sufficient food is available, these springtails produce on average larger clutches than the ones from the other clade [START_REF] Tully | Reproductive flexibility: genetic variation, genetic costs and long-term evolution in a collembolan[END_REF], but they have shorter mean lifespans than the less fecund clade and also reach a smaller adult body size [START_REF] Tully | Reproductive flexibility: genetic variation, genetic costs and long-term evolution in a collembolan[END_REF][START_REF] Tully | The evolution of postreproductive life span as an insurance against indeterminacy[END_REF][START_REF] Mallard | Within species variation in long-term trajectories of growth, fecundity and mortality in the Collembola Folsomia candida[END_REF]. These two groups of clonal lineages fit well to the typical slow (A) and fast (B) life history syndromes (or r-K life histories, see [START_REF] Reznick | R-and K-selection revisited: the role of population regulation in life-history evolution[END_REF]. But, until now, the ecological conditions in which they have evolved and the time elapsed since the divergence of the two clades remain to be determined. Intra-specific variation in habitat choice behaviour and mobility has so far neither been examined in this species nor in other Collembola. Instead, the few works that relate the habitat preference or distribution and colonization ability of Collembola with their morphological and life history traits are focused on the study of collembolan community composition [START_REF] Salmon | Linking species, traits and habitat characteristics of Collembola at European scale[END_REF][START_REF] Ponge | Spatial and taxonomic correlates of species and species trait assemblages in soil invertebrate communities[END_REF][START_REF] Huebner | Post-fire succession of collembolan communities in a northern hardwood forest[END_REF][START_REF] Salmon | Species traits and habitats in springtail communities: a regional scale study[END_REF][START_REF] Ponge | Decreased biodiversity in soil springtail communities: the importance of dispersal and landuse history in heterogeneous landscapes[END_REF]. Intraspecific variation in phototactic behaviour and life history traits has been well investigated in Daphnia magna. In this species, positive phototactic clones present a fast life history strategy whereas negative and intermediate phototactic clones present a slow life history strategy (e.g., De Meester, 1994).

We tested if springtail clonal variation in light-dependent habitat choice exists using an experimental setup to control the lighting conditions while maintaining constant temperature and moisture. We tracked springtail movements under this setup to quantify their spatial preference for shaded versus illuminated areas as a measure of the habitat choice behaviour. We first tested whether F. candida can use light as an environmental cue for habitat choice under different lighting conditions, including natural shaded and sunny understory spectra and an artificial redlight spectrum. We measured springtail sensitivity to long wavelengths because these wavelengths are dominant under the forest canopy [START_REF] Smith | Sensing the light environment: the functions of the phytochrome family[END_REF] while red and far red are the principal wavelengths that penetrate the soil [START_REF] Bliss | Penetration of light into soil and its role in the control of seed germination[END_REF]. We hypothesised that F. candida should not detect or react to red and far-red light per se if the incapability to detect these wavelengths was a shared condition of most true insects and Collembola, even though most springtails are sensitive to heat generated by red light [START_REF] Briscoe | The evolution of color vision in insects[END_REF]. We further studied the sensitivity to light of twelve clonal lineages of F. candida including eleven lineages belonging to the two evolutionary clades described earlier [START_REF] Tully | Two major evolutionary lineages revealed by molecular phylogeny in the parthenogenetic collembola species Folsomia candida[END_REF][START_REF] Tully | Reproductive flexibility: genetic variation, genetic costs and long-term evolution in a collembolan[END_REF][START_REF] Tully | The evolution of postreproductive life span as an insurance against indeterminacy[END_REF][START_REF] Tully | Intraspecific phenotypic variation and morphological divergence of strains of Folsomia candida (Willem) (Collembola: isotomidae), the standard test springtail[END_REF]. We addressed the following questions: Do light sensitivity and habitat choice behaviour vary between clonal lineages, as has been found in other taxa [START_REF] Jaenike | Genetic variation for habitat preference: evidence and explanations[END_REF][START_REF] De Meester | Evolutionary potential and local genetic differentiation in a phenotypically plastic trait of a cyclical parthenogen, Daphnia magna[END_REF][START_REF] Cousyn | Rapid, local adaptation of zooplankton behavior to changes in predation pressure in the absence of neutral genetic changes[END_REF])? If such clonal variation exists, how is it organised relative to the phylogenetic clades and what are the links between the behavioural responses and the main life history strategies of each clade? We predicted that lineages from the slow life history group would be more photophobic [START_REF] De Meester | Life histories and habitat selection in Daphnia: divergent life histories of D. magna clones differing in phototactic behavior[END_REF][START_REF] De Meester | Life history characteristics of Daphnia magna clones differing in phototactic behavior[END_REF], given that photophobia is likely to be associated with life in more stable habitats, which usually selects for a slow life history [START_REF] Pianka | R-Selection and K-Selection[END_REF][START_REF] Reznick | R-and K-selection revisited: the role of population regulation in life-history evolution[END_REF].

Materials and methods

Maintenance and origin of the studied springtails

We used twelve clonal lineages of the Collembola Folsomia candida labelled AP, BR, BV, DK, GB, GM, HA, ME, PB, TO, US and WI [START_REF] Tully | Two major evolutionary lineages revealed by molecular phylogeny in the parthenogenetic collembola species Folsomia candida[END_REF]. Information about the phylogenetic relation-ships and habitat and geographical origin of all strains except ME can be found in previous studies [START_REF] Tully | Two major evolutionary lineages revealed by molecular phylogeny in the parthenogenetic collembola species Folsomia candida[END_REF][START_REF] Tully | Intraspecific phenotypic variation and morphological divergence of strains of Folsomia candida (Willem) (Collembola: isotomidae), the standard test springtail[END_REF]. Clones AP, BV, BR, GB and HA belong to the "slow clade" A while the "fast clade" B comprises the clones DK, GM, PB, TO, US and WI [START_REF] Tully | Reproductive flexibility: genetic variation, genetic costs and long-term evolution in a collembolan[END_REF]. The new clone ME was collected in November 2013 from some decaying wood beams into an abandoned man-made tunnel in the Mercantour French National Park (South-East of France, 44°7.0260N, 7°16.7270E, 1530 m). The life history strategy of this clone and its phylogenetic relationships with the other clones are currently unknown.

All clonal populations were reared in similar conditions in polyethylene vials (inner diameter 52 mm, height 65 mm) filled with a 30 mm layer of plaster of Paris mixed with Indian ink to increase visual detectability of individuals [START_REF] Tully | Reproductive flexibility: genetic variation, genetic costs and long-term evolution in a collembolan[END_REF]. Populations were kept in incubators at 20°C (+/-0.5°C) in the dark and fed with pellets of a mixture of agar and dried yeast [START_REF] Tully | Reproductive flexibility: genetic variation, genetic costs and long-term evolution in a collembolan[END_REF]. We established synchronised populations of each clone in the same way and at the same time by transferring 10-12 randomly chosen adult females from stock cultures to new culture vials. Females were transferred to new vials every week and old vials were kept at 20 C for laid eggs to hatch since this temperature is optimal for all lineages. Then, vials were checked weekly for the presence of new-borns. If newborns were detected, ad libitum food was provided weekly to ensure optimum body growth. Therefore, the age of the experimental animals was known with a one-week accuracy. However, due to the low fecundity of some clones or accidental flooding in some vials, we had to add medium size (~1.4mm) adults of unknown age taken from the stock cultures in order to keep balanced samples for all clones and treatments (see Table S1 for details on sample size and individual characteristics).

Measurement of natural light conditions

At midday of a sunny day in June 2013, we measured 23 light spectra under the canopy of a temperate forest located in the Foljuif field station, 80km South of Paris, France (48 17013.96"N, 2 40040.34"E), where populations of F. candida have been previously found. We recorded 19 spectra in the shade and four spectra on sunspots in the undergrowth. The climate conditions and plant community structure of this forest have been studied extensively [START_REF] Blandin | Etude d'un ecosysteme forestier mixte, 1: Climat, structure de la vegetation et retombees de litieres [France, Seine-et-Marne[END_REF]. We used a handheld spectrome-ter parameterized for absolute irradiance measurements in the range 200-850 nm (Jaz Series, JAZ-ULM-200, Ocean Optics, USA). We measured the incident light with a cosine-corrected probe (180 field of view) after dark calibration and converted data into units of irradiance spectrum (mW cm 2) using factory calibration of the spectrophotometer. Measurements were taken at ground levels under the canopy of different tree and bush species, as well as under sunspots under the trees and into tree stump holes to sample different light spectra that natural populations of spring-tails are likely to encounter aboveground.

Simulation of environmental conditions

Artificial light and constant climate conditions were simulated in a controlled environment laboratory at the CEREEP Ecotron -Ile De France (Saint-Pierre-les-Nemours, France). We used one 13 m3 controlled environment chamber of the Ecolab, where climate (temperature, relative humidity, rainfall) and lighting conditions can be simulated [START_REF] Verdier | Climate and atmosphere simulator for experiments on ecological systems in changing environments[END_REF]. In the centre of the chamber, we installed vials for the characterisation of photophobia in springtails (see below) on a controlled water table to maintain constant humidity and temperature conditions in the vials during the observations. Controlling for constant humidity and temperature was essential to test exclusively for the effect of light, since slight changes in humidity and temperature could modify the springtails' behaviour and thus affect their apparent or real photophobia [START_REF] Holmstrup | Strategies for Cold and Drought Tolerance in Permeable Soil Invertebrates[END_REF][START_REF] Boiteau | Locomotor response of Folsomia candida (Collembola: isotomidae) to cooling temperatures[END_REF]. Atmospheric temperature in the chamber was maintained at 20°C (+/-0.1°C) and temperature of the water table was set at 10°C. We used a cold water table to keep the temperature inside the vials slightly lower than the air temperature to prevent condensation on the transparent lid that covered the vials (Fig. 1). The temperature measured inside the vials was 19°C and the relative humidity was 93% (DS-1923 iButton loggers) irrespective of the test conditions. Light was provided by means of modular LED (light emitting diodes) arrays (n = 40) allowing to turn on or off cool white LEDs (n=15 per array) and five different LED types with maximum emission wavelengths in the UV (370nm, n=3), green (520nm, n=2), red (660nm, n=5), far red (740nm, n=7), and infrared (840 nm, n = 3). Thus, up to 36,864 combinations of LEDs parameters can be programmed. To automatise the simulation of the most appropriate lighting conditions, we wrote a procedure in R (The R and Development Core Team, 2012) to calculate the sum of squared distance between the reference spectrum and the simulated spectrum, and to select the combination of parameters that minimised this sum. We simulated the "average undergrowth" light spectrum by producing a spectrum which best matched the mean spectra of the 23 light measurements made at various places on the forest floor (see Fig. 2a and Table S2 provided as Supplementary information). We also produced a "maximum undergrowth spectrum", using the four measurements made on sunspots, which should correspond to the strongest undergrowth light conditions that litterdwelling springtails are susceptible to be exposed to in this forest (Fig. 2b and Table S2). We expected a stronger light-avoidance response under this condition compared to under the mean undergrowth light spectrum. In addition, we simulated a "red light" spectrum whose power was close to that of the mean undergrowth light spectrum (Fig. 2a and Table S2). This red light spectrum has two irradiance peaks (red at 660 nm and far red at 740nm) and has no UV light that could promote photophobia in springtails [START_REF] Fox | Phototactic responses to ultraviolet and white light in various species of Collembola including the eyeless species, Folsomia candida[END_REF][START_REF] Beresford | Lethal and sub-lethal effects of UV-B radiation exposure on the collembolan Folsomia candida (Willem) in the laboratory[END_REF]. We further com-pared these three lighting conditions with a control treatment where the room was maintained in the dark. This condition was used to verify whether the experimental set-up in itself could affect the behaviour of the springtails and to compare the mobility of the different clones in the dark.

Experimental protocol

The dark, average undergrowth and red light spectra were tested in June 2014, and the maximum undergrowth spectrum was tested in December 2014. We used 36 individuals of each clonal lineage in each light treatment (n = 1728), and six individuals per clone were tested every day. Springtails were photographed to measure their length (mean = 1.41 mm, range = 0.93-2.19 mm) and kept isolated without food in rearing vials the day before. These individuals were transferred the next morning to new rearing vials wrapped with a thin flexible black plastic sheet, and only exposed to light from above. Each plastic box was filled with plaster of Paris and had a hole in the bottom to ensure that the plaster can be kept constantly wet to saturation (Fig. 1). An opaque black piece of plastic covered half of the vial's opening and descended vertically along the centre of the vial down to 3 mm above the plaster level, in order to maintain in the dark half of the vial while allowing the springtail to move freely between the two halves of the vials. Preliminary experiments showed that in total darkness springtail behaviour is very sensitive to drought and slight humidity differences between the two sides of the vials. The vials were thus covered with transparent lids to prevent any desiccation. When being transferred, animals were separated randomly in two groups of same size and released in the bright and dark sides at the beginning of the experiment (here called 'position') respectively, such that the proportion of animals in each position was exactly 50% for each clone and treatment. One observer (MGR) studied the springtails' movements between the two sides of the vials by recording the position of each animal every hour during six hours (from 9 am to 5 pm). For the control dark treatment, we observed the springtails' positions as quickly as possible using a soft light head torch to minimise disturbance. At the end of the experiment, we inspected all vials to check that no individual had escaped or deceased to ensure valid observations when the springtails did not move.

Statistical analyses

We used the software R (The R and Development Core Team, 2012) for the statistical analysis and to produce the graphs. The raw data and the R-scripts of the analysis, and figures are provided as Supplementary information. We calculated the movement (or transition) probabilities (the probability that an individual has moved from one side to the other side of the vial after one hour), using pairs of consecutive observations of the same individual to form a binomial variable which equals zero when the individual stayed in the same half of the vial and equals one when the animal had moved to the other. This binomial variable was analysed with nested generalised linear mixed models (GLMMs) for repeated measurements approach with a binomial error distribution and logit link function, using treatment (light spectrum type), position (dark or light) and clonal group as the fixed factors, and individual as the random factor. Observations were thus clustered in individuals and individuals were nested within clones. In the full model, we analysed all main effects as well as two and three-way interactions. We implemented the full model with the glmer procedure in package lme4 for R 3.1.2 [START_REF] Bates | lme4: Linear mixed-effects models using S4 classes[END_REF]The R and Development Core Team, 2012) and tested for the significance of fixed effects with Chi-square tests. Since the threeway interaction Spectrum*Position*Clone was significant (see Table S3 provided as Supplementary information), we analysed each treatment independently to get a better understanding of the behavioural differences among clones under each treatment.

For subsequent analyses we further included the body length of each individual in the analysis to test for a size effect on individual movement probability. Since mean body length can vary substantially among clones [START_REF] Mallard | Within species variation in long-term trajectories of growth, fecundity and mortality in the Collembola Folsomia candida[END_REF], we calculated for each treatment the corrected body lengths as the sum of the overall mean body length and the body length residuals computed here as the difference between individual lengths and mean length of their clone. This "corrected body length" was then used to study the effect of body length while controlling for differences among clones. The full models included position, clone and corrected length as fixed factors, the two-way interactions and individual identity as the random factor. We used a backward stepwise selection to get final models that only included significant factors and interactions. We also report full models in Table S4 following Forstmeier and Schielzeth's recommendation [START_REF] Forstmeier | Cryptic multiple hypotheses testing in linear models: overestimated effect sizes and the winner's curse[END_REF]. For the full and final models, we calculated marginal and conditional R2, which quantify the amount of variance explained by the models [START_REF] Nakagawa | A general and simple method for obtaining R2 from generalized linear mixedeffects models[END_REF]. We additionally performed GLMMs with clade, position and their interaction as fixed factors, and clone and individual within clone as nested random factors to test for phylogenetic variation. The clone ME, whose evolutionary clade has not yet been characterized, was excluded from these analyses.

Results

In the homogenous dark environment, the mobility of spring-tails was not influenced by their position in the vial (Table 1; Fig. 3) and was independent of their body length. The individual mobility (average transition probability) differed significantly among clones (Table 1, Fig. 4a), and this variability was mostly due to difference between the two evolutionary clades (x21 = 15.86, p < 0.001). In complete darkness, the clones with a fast life history (clade B) had on average a high-er propensity to switch between sides (0.44 per hour) than clones with a slow life-history strategy (0.34, Fig. 4a). Contrary to our expectations, F. candida proved to be slightly sensitive to red light (Table 2), independently of their length: springtails preferentially moved from red light to dark (0.46 per hour) than from dark to red light (0.42, see Fig. 3). Clones also differed in their mean mobility, but these differences were not due to clade (x21 = 0.07, p = 0.79, Fig. 4b). Springtails showed photophobia when exposed to average undergrowth light: they moved from the light to the dark side significantly more often than in the reverse direction (0.43 versus 0.37, see Fig. 3 and Table 3). The different clones (x211 = 18.09, p = 0.08) and the two clades (x21 = 0.40, p = 0.40) had similar movement probabilities (Fig. 4c andTable 3). In this light condition, movement probability varied with body length depend-ing on position: while in the dark compartment the length did not influence mobility, but when exposed to light, smaller than average springtails were more prone to move than longer ones (Fig. 5a). When exposed to maximum undergrowth light, springtails moved away on average more often (0.45 per hour) than when they were in the dark side of the vials (0.36, see Fig. 3 and Table 4). Differences among clones (x211 = 15.67, p = 0.15) and clades were not significant (x21 = 1.04, p = 0.31, Fig. 4d), and there was a non-significant trend (Position* Corrected Length: x211 = 1.22, p = 0.27) for longer than average individuals to move less often than smaller individuals, as observed in the average undergrowth light treatment (Table 4). However, there was a significant interaction between effects of clone and of corrected length: for most clones, size matters little and smaller individuals tended to move more, while for clone GB the mobility was positively influenced by relative length (x211 = 21.49, p = 0.03) (Fig. 5b) Clone GB was characterised by the slowest life history strategy.

Discussion

Limits of the experimental protocol

Our experimental protocol allowed us to study the behaviour of isolated individuals in darkness and under three light conditions while maintaining constant temperature and humidity. As the springtails' mobility is very sensitive to these environmental factors, this represents a significant improvement to test for photophobia. Differences in temperature between the illuminated and dark sides of the boxes fell within the measurement error of our thermometer devices (<0.5°C). Thus, even though we cannot strictly exclude that the bright side may be slightly warmer than the dark side, the potential temperature difference was so small that we could not detect it. Thus we interpret our result as a light sensitivity but one has to remember that temperature gradients could also play a role. For instance, temperature receptors such as micro setae on the antennas or legs may help springtails to perceive small temperature gradients, in particular when doing 'turning' movements (loops), a frequent behaviour when in an unknown environment [START_REF] Bengtsson | Irregular walks and loops combines in small-scale movement of a soil insect: implications for dispersal biology[END_REF].

In addition, our experimental design is not suited to know how many times springtails moved between sides during the one hour census interval. Our estimations of transition probabilities are based on the hypothesis that each springtail does no or only one move in an hour, which may be wrong given that some individuals may have moved multiple times between two censuses. Future studies should track more continuously individuals through space to obtain better estimates of dispersal distances and capacities.

Light sensitivity and its ecological significance

We found that, despite being eyeless, F. candida is slightly but significantly sensitive to light. Photophobia was detected in the presence of the three light sources, including two light spectra chosen to mimic natural conditions encountered by springtails. Hence, this study confirms earlier general findings of negative phototaxis in this species [START_REF] Fox | Phototactic responses to ultraviolet and white light in various species of Collembola including the eyeless species, Folsomia candida[END_REF][START_REF] Beresford | Lethal and sub-lethal effects of UV-B radiation exposure on the collembolan Folsomia candida (Willem) in the laboratory[END_REF][START_REF] Boiteau | Geotaxis in the springtail Folsomia candida[END_REF][START_REF] Salmon | Responses to light in a soildwelling springtail[END_REF]. But our study goes one step further by showing that the intensity of photophobia response varied depending on the light spectrum: Light avoidance by springtails increased from the red light spectrum at an intensity equivalent to that of the average light spectrum under the canopy, to average undergrowth spectrum, and to maximum undergrowth spectrum. F. candida is thus sensitive both to light intensity and to its quality. This suggests that despite being blind, these springtails can use light as an environmental cue for orientation, movement and habitat choice under natural conditions. Light avoidance behaviour was not very strong, because individuals also moved from the dark to the illuminated side of the vials and switched their position frequently. This may be due to the fact that the intensity of the light spectrum tested was low compared for example to direct sunlight, but could also be related to the small spatial scale of the setup and the short time scale of our tests.

Multiple examples of eye-reduced and eyeless species that retain brightness and colour sensitivity exist (e.g. [START_REF] Ramirez | Understanding the dermal light sense in the context of integrative photoreceptor cell biology[END_REF][START_REF] Friedrich | Biological clocks and visual systems in cave-adapted animals at the dawn of speleogenomics[END_REF]. Extra-ocular photoreceptors (EOP) or nonvisual photoreceptors can be responsible of light sensitivity when eyes are absent. EOP exist in the form of dermal photoreceptors in invertebrates [START_REF] Tosini | Dermal photoreceptors regulate basking behavior in the lizard Podarcis muralis[END_REF][START_REF] Binder | The role of dermal photoreceptors during the sea lamprey (Petromyzon marinus) spawning migration[END_REF][START_REF] Xiang | Light-avoidance-mediating photoreceptors tile the Drosophila larval body wall[END_REF][START_REF] Ullrich-Lüter | Unique system of photoreceptors in sea urchin tube feet[END_REF], the sixth caudal ganglion in decapods [START_REF] Wilkens | Photosensitivity in the sixth abdominal ganglion of decapod crustaceans: a comparative study[END_REF][START_REF] Larimer | A functional caudal photoreceptor in blind cavernicolous crayfish[END_REF], genitalia [START_REF] Arikawa | Multiple extraocular photoreceptive areas on genitalia of butterfly Papilio xuthus[END_REF] and neurons of the optic lobes in butterflies [START_REF] Lampel | Expression of UV-, blue-, long-wavelength-sensitive opsins and melatonin in extraretinal photoreceptors of the optic lobes of hawk moths[END_REF], and photosensitive cells in the hydra [START_REF] Taddei-Ferretti | Photobehaviour of Hydra (Cnidaria, Hydrozoa) and correlated mechanisms: a case of extraocular photosensitivity[END_REF]. F. candida lacks external eye facets and any other external eye structures as revealed by scanning electron microscopy [START_REF] Fox | Phototactic responses to ultraviolet and white light in various species of Collembola including the eyeless species, Folsomia candida[END_REF]. Whether F. candida is only facet-less but has internal photoreceptors where eyes should be or whether it uses other types of extraocular photoreception such as a dispersed photoreception dermal sense [START_REF] Ramirez | Understanding the dermal light sense in the context of integrative photoreceptor cell biology[END_REF] or neural photoreception [START_REF] Lampel | Expression of UV-, blue-, long-wavelength-sensitive opsins and melatonin in extraretinal photoreceptors of the optic lobes of hawk moths[END_REF][START_REF] Wilkens | Photosensitivity in the sixth abdominal ganglion of decapod crustaceans: a comparative study[END_REF], remains an open question.

Photophobia may help springtails to avoid open spaces when they forage in the litter close to the surface, and together with positive geotaxis [START_REF] Boiteau | Geotaxis in the springtail Folsomia candida[END_REF], it may serve for orientation in vertical movements. Being able to avoid open spaces and the surface of the litter may provide several benefits. First of all, all clonal lineages tested here were non-pigmented and thus little protected against the deleterious effects of UV radiations. Long-term (two weeks) exposure to UV-B light increases mortality and causes DNA damage in this species [START_REF] Beresford | Lethal and sub-lethal effects of UV-B radiation exposure on the collembolan Folsomia candida (Willem) in the laboratory[END_REF]. Behavioural tactics of habitat selection for dark sheltered zones or deeper soil layers can help springtails to escape from potentially dangerous UV-radiation. This may also explain why unpigmented or slightly pigmented Collembola are preferentially distributed in edaphic or hemiedaphic rather than epigeal habitats and in woodlands rather than in grasslands [START_REF] Salmon | Linking species, traits and habitat characteristics of Collembola at European scale[END_REF][START_REF] Salmon | Species traits and habitats in springtail communities: a regional scale study[END_REF]. However, the photophobia of F. candida cannot be exclusively related to UV-radiation avoidance, since F. candida was also sensitive to red light. Longwaves are dominant wave-lengths under the forest canopy [START_REF] Smith | Sensing the light environment: the functions of the phytochrome family[END_REF], and thus sensitivity to these wavelengths could be related to a more general aboveground avoidance. Moreover, as ambient light level is often correlated with other climatic factors (notably temperature and humidity), photophobia is useful to target deeper, more humid and more stable microhabitats preferred by most Collembola species, especially those that are the most vulnerable to desiccation [START_REF] Salmon | Linking species, traits and habitat characteristics of Collembola at European scale[END_REF][START_REF] Salmon | Species traits and habitats in springtail communities: a regional scale study[END_REF]). Another explanation is that photophobia may indirectly help springtails to avoid predators. Being devoid of compound eyes F. candida probably cannot form images and thus cannot see approaching predators [START_REF] Hauzy | Spatial heterogeneity and functional response: an experiment in microcosms with varying obstacle densities[END_REF], as it is the case of the sister eyeless species Folsomia fimentaria [START_REF] Baatrup | Predation of the mite Hypoaspis aculeifer on the springtail Folsomia fimetaria and the influence of sex size, starvation, and poisoning[END_REF]. This species is thus very vulnerable in illuminated environments to visually active-chasing predators such as hunting ground-beetles, some of which are ferocious predators of springtails [START_REF] Ernsting | Effects of food deprivation and type of prey on predation by Notiophilus biguttatus F. (Carabidae) on springtails (Collembola)[END_REF].

Intra-specific variability in movement and light sensitivity

In general, smaller individuals were more mobile than larger ones and we found that smaller springtails exposed to light were also more likely to seek refuge in the dark side than the larger ones in the average undergrowth light treatment. The risk of staying in an illuminated environment may not be the same for large and small individuals. For example, it has been shown that the carabid Notiophilus biguttatus a common visual hunter and voracious predator of springtails preys more efficiently on small rather large Collembola [START_REF] Ernsting | Components of predatory behaviour underlying density-dependent prey-size selection by Notiophilus biguttatus F. (Carabidae, coleoptera)[END_REF]. This interpretation is tenuous given that the effect of body size on movement was small, variable across clones and not repeatable across light treatments. These variable effects of body size are difficult to explain. The intensity of photophobia may also depend on other individual state parameters such as the nutritional status or the phase in the moulting and oviposition cycle. There could also be a trade-off between the preference for safer dark deep zones and the need for foraging and dispersal at the illuminated surface [START_REF] Dromph | Effect of starvation on phototaxis and geotaxis of collembolans[END_REF].

We searched for clonal variation in both the mean movement propensity and in the light sensitivity of movement. We first found that the various lineages moved differently in the dark: the average transition probabilities varied from 0.2 to almost 0.5 depending on the lineages. Quite remarkably, this genetic diversity was structured between the two previously described phylogenetic clades: clones from the clade characterized by a fast lifehistory strategy had on average higher transition probabilities than those with slow life-history strategy, suggesting that they move faster and can potentially disperse over longer distances in natural conditions. Even though we did not measure dispersal per se (defined as the movement of individuals or propagules that leads to gene flow, see [START_REF] Clobert | Dispersal Ecology and Evolution 1 st[END_REF], mobility or exploratory activity has generally been positively correlated to dispersiveness and can thus be used as a rough proxy for dispersal in natural populations [START_REF] Ronce | Dispersal syndromes. Dispersal Ecology and Evolution[END_REF]. The higher movement propensity of the fast life-history clade agrees with the evolutionary scenario of the 'colonizer syndrome' where fast life-history species or populations are more dispersed than their slow life-history counterparts [START_REF] Ronce | Dispersal syndromes. Dispersal Ecology and Evolution[END_REF]. The combination of increased dispersal, high fecundity but short lifespan and low competitive ability allows fast individuals to colonise new habitats more efficiently. Individuals with reduced movement propensity, low fecundity and high competitive ability perform better at the core range and in stable habitats, where populations have reached their carrying capacity and where infraspecific competition is strong [START_REF] Burton | Trade-offs and the evolution of life-histories during range expansion[END_REF]. Future studies should assess whether differences in short-distance movement probabilities translate into clonal differences in long-distance movement behaviour and therefore dispersal distances.

Studies on the intraspecific variation of phototaxis are scarce but differences in phototaxis among genetically distinct lineages have been reported in several species. It has been shown that several Daphnia clonal populations differ in their photophobia in response to contrasting predatory pressures [START_REF] Cousyn | Rapid, local adaptation of zooplankton behavior to changes in predation pressure in the absence of neutral genetic changes[END_REF][START_REF] De Meester | Evolutionary potential and local genetic differentiation in a phenotypically plastic trait of a cyclical parthenogen, Daphnia magna[END_REF]. Artificially selected Drosphila strains also differ in their phototactic preferences both in quality and degree as a result of different genotypes and the presence of specific mutations [START_REF] Kain | Phototactic personality in fruit flies and its suppression by serotonin and white[END_REF]. But contrary to these findings, we did not find any significant differences in our model organism in light sensitivity among clones and clades for the two undergrowth spectra, including when springtails were exposed to the brightest light spectrum. This could indicate that photophobia is a shared, evolutionary rigid behavioural trait in this species. It also implies that other studies that failed to find intraspecific variation in phototaxis may be missing in the published bibliography.

In addition, when exposed to light, the genetic differences in movement behaviour observed in the dark vanish. Our observations also suggest that moving from one side of the vial to the other side in homogeneous dark conditions or in heterogeneous light conditions is associated with different underlying behaviours: exploration for the former and escape for the latter. This is also supported by the fact that body length did not affect movement in the dark while small individuals were found to be slightly more mobile than larger ones in the heterogeneous environments. Table S1

Tables

Figure 1 :

 1 Figure 1: Fig. 1. a) Photographs of the experimental boxes. Boxes were wrapped with a thin flexible black plastic sheet (1), and only exposed to light from above. Boxes were covered with a transparent plastic lid (2) to prevent desiccation and disturbance due to airflow. The arrow shows the gap between the plaster of Paris floor (3) and the opaque black piece of plastic (4) that descended vertically along the

Figure 2 :

 2 Figure 2: Light conditions used during the experiment. a) Natural and simulated average under-canopy spectra and experimental red light spectrum. The three spectra have equivalent total power. b) Natural and simulated sunspots under the canopy spectra. The two spectra have equivalent total power.

Figure 3 :

 3 Figure 3: Models that best describe transition probabilities in the dark treatment after backward selection (N = 431). Dropped variables are presented in order in which they were removed. Marginal and conditional R2 of the final model are 0.111 respectively.

Table 1 :

 1 Models that best describe transition probabilities in the dark treatment after backward selection (N = 431). Dropped variables are presented in order in which they were removed. Marginal and conditional R2 of the final model are 0.111 respectively. Table 2: Models that best describe transition probabilities in the red light treatment after backward selection (N = 432). Marginal and conditional R2 of the final model are 0.015 and 0.068 respectively.

Figure 4 :

 4 Figure 4: Mean movement probability between the two sides of the test boxes (95% CI) for each clone. Black symbols denote clones with a slow life-history strategy; white symbols denote clones with a fast life-history strategy; and the grey symbol denotes the clone ME whose life-history strategy remains uncharacterised. Mean values were averaged over the two sides of the test boxes.

Table 3 :

 3 Models that best describe transition probabilities in the average undergrowth spectrum after backward selection (N = 432). Marginal and conditional R2 of the final model are 0.011 and 0.074 respectively. Table 4: Models that best describe transition probabilities in the maximum undergrowth spectrum after backward selection (N = 432). Marginal and conditional R2 of the final model are 0.039 and 0.089 respectively.

Figure 5 :

 5 Figure 5: a) Predicted mean movement probability between the two sides as a function of individuals' length, for each position in the average undergrowth spectrum. White boxplots and dotted line denote that the individual was on the bright side prior to the observation, while grey boxplots and solid line denote that the individual was on the dark side. The dashed-line represent the predicted response of the individuals that were on the bright side prior to the observation, while the solid line represent the predicted response of the individual that were on the dark side. b) Predicted movement probability between the two sides as a function of individuals' length for each clone in the maximum undergrowth spectrum.

.

  Sample size, age and body length per clone studied in the experimental setup.

	Total	Number	Mea	Range	Mean	Range of
	num-	of indi-	n	of age	body	length
	ber of	viduals	age	(days)	length	(mm)
	individ	of un-	(days		(mm)
	uals	known	)
		age (or
		length)
	AP 144	42	46.2 23-96 1.34 0.95-1.99
	BR 144	42	32.3 17-76 1.41 0.99-2.02
	BV 144	24	68.4 23-117 1.50 1.02-1.98
	DK 144	26	62.7 23-110 1.29 0.97-1.76
	GB 144	61	48.6 20-104 1.58 1.11-2.19
	GM 144	24	65.1 23-116 1.38 0.99-1.80
	HA 144	24	66.1 23-117 1.51 1.08-2.05
	ME 144	24	69.7 12-117 1.41 0.97-1.99
	PB 144	24 (1)	67.5 23-116 1.35 0.97-1.77
	TO 144	24	68.8 23-116 1.39 1.04-1.91
	US 144	24	66.5 23-110 1.36 0.93-2.16
	WI 144	24	60.8 23-109 1.40 1.06-1.89

Table S2 .

 S2 Summary of the total irradiance of natural and simulated spectra.

	Spectrum	Total	irradi-
		ance
		(µW.cm-2)
	Natural average undergrowth	31.5
	Simulated average undergrowth	37.3
	Red and far-red	23.8
	Natural maximum undergrowth	70.5
	Simulated maximum undergrowth	52.0

Table S3 .

 S3 The influence of clone identity, position (dark or light), spectrum type (four treatments) and their two and three-way interactions (full model) on transition probability of springtails between each side of the experimental boxes (N=1728 individuals). * p<0.05, ** p<0.01, *** p<0.001.

		Chi	D. f. p-value
		square
	Clone	31.02	11 0.001	***
	Position	26.98	1	<0.001 ***
	Spectrum	9.09	3	0.028	*
	Clone*Position	8.03	11 0.710
	Clone*Spectrum	47.87	33 0.045	*
	Position*Spectrum	8.33	3	0.039	*
	Clone*Position*Spectrum 47.51	33 0.049	*

Table S4 .

 S4 Full models describing the influence of clone, position, corrected length and their two-way interactions on transition probabilities of springtails between each side of the experimental boxes. Full models were fitted separately for each spectrum type corresponding to dark, red light, average undergrowth spectrum and maximum undergrowth spectrum treatments. Marginal and conditional R2 are provided. * p<0.05, ** p<0.01, *** p<0.001.

		Chi squared	D.f.	p-value		R2m	R2c
	Dark treatment (N=431)
	Clone	23.73	11	0.0139 *	*	0.0037	0.1141
	Position	0.74	1	0.3880
	Residual Length	0.03	1	0.8547
	Clone* Position	16.86	11	0.1120
	Clone* Corrected Length	14.10	11	0.2273
	Position* Corrected Length	0.66	1	0.4165
	Red light treatment (N=432)
	Clone	20.21	11	0.0425 *	*	0.0247	0.0731
	Position	4.77	1	0.0290 *	*
	Corrected Length	0.04	1	0.8472
	Clone* Position	9.13	11	0.6095
	Clone* Corrected Length	8.94	11	0.6276
	Position* Corrected Length	0.0046	1	0.9462
	Average undergrowth treatment (N=432)
	Clone	17.8053	11	0.0862		0.0384	0.0827
	Position	9.4785	1	0.0021 **	**
	Corrected Length	3.8588	1	0.0495 *	*
	Clone* Position	17.5853	11	0.0917
	Clone* Corrected Length	12.5436	11	0.3242
	Position* Corrected Length	7.1203	1	0.0076	**
	Maximum undergrowth treatment (N=432)
	Clone	15.4355	11	0.1634		0.04574	0.0961
	Position	23.0414	1	<0.001	***
	Corrected Length	0.0164	1	0.8981
	Clone* Position	11.7662	11	0.3815
	Clone* Corrected Length	21.7120	11	0.0267	*
	Position* Corrected Length	1.1535	1	0.2828

Acknowledgments

We want to thank to the staff members of the Ecotron IleDe-France, especially Simon Chollet, Mathieu Llavata and Florent Massol. This research was supported by the Centre National de la Recherche Scientifique and a grant from the Ecole doctorale Diversité du Vivant at the Université Pierre and Marie Curie to M.G. R. (contract 859/2013). This work was made possible thanks to the CNRS Research Infrastructure Ecotrons supported by the Conseil régional d'Ile-de-France (DIM R2DS I-05-098/R and 2011-11017735) and the Agence Nationale de la Recherche grant of the "Investissements d'avenir" program (ANR-11-INBS-0001 AnaEE France).


Tensile Damage in Concrete: Analysis of Experimental Technique

  Dorothée Boudon-Cussac, François Hild, Gilles Pijaudier-Cabot To cite this version: Dorothée Boudon-Cussac, François Hild, Gilles Pijaudier-Cabot. Tensile Damage in Concrete: Analysis of Experimental Technique. Journal of Engineering Mechanics -ASCE, 1999, 125 (8), pp.906-913. 10.1061/(ASCE)0733-9399(1999)125:8(906). hal-02342789


Estelle Couallier

Shuli Liu

Erika Clavijo Rivera

Liliana Villafana Lopez

Patrick Bourseau

Matthieu Frappart

Liliana Villafana

Membrane fitration for the recovery of lipids from microalgae extracts

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

MEMBRANE FILTRATION FOR THE RECOVERY OF LIPIDS FROM MICROALGAE EXTRACTS

Estelle COUALLIER*, Shuli LIU, Erika CLAVIJO RIVERA, Liliana VILLAFANA LOPES, Patrick BOURSEAU, MaFhieu FRAPPART CNRS, GEPEA UMR-CNRS 6144, bd de l'Université, CRTT-BP 406, 44602 Saint-Nazaire CEDEX -France; hJps://www.gepea.fr/ * estelle.couallier@univ-nantes.fr In this work we focus on the recovery of lipids from aqueous extracts of microalgae. Aber the culture of Parachlorella Kessleri, in starving condiVons to enhance their lipid producVon, the micro-organisms were ground and the resulVng mixture were centrifuged to extract aqueous fracVons containing a large quanVty of dispersed lipids.

Conclusions and PerspecJves

Clari&ication

  fouling with model and real supernatants Despite the concentraVon difference (MM: model mixture, RM: model mixture) Higher flux and lower fouling with real supernatants Impact of the shear rate on the membrane performances (MM: model mixture, RM: model mixture) Higher flux of dynamic filtraVon compared to cross flow filtraVon using the model mixture The membrane filtraVon allows the concentraVon of lipids from the aqueous extracts. The dynamic filtraVon offers interesVng flux. The model mixtures allows the predicVon of the fouling with concentrated extracts. The retenVon of the other compounds will depend on the cell disrupVon condiVons: this will be deepened in a future study (PhD Shuli LIU). The membrane modificaVon through hydrophilisaVon or fouling will be characterised through inovaVve technics in a futrure work (post-doctoral study, collaboraVon with ISCR Rennes and INRA Nantes.


R Blossey
email: ralf.blossey@univ-lille.fr

H Schiessel

Chromatin remodelers as active Brownian dimers

Keywords: nucleosome, chromatin remodeler, helicase, active Brownian dimer

Chromatin remodelers as active Brownian dimers

Introduction.

Active systems currently are one of the most intensive fields of research within the statistical physics community. Built on a large body of work dealing with individual motors (see, e.g. [1]), the field has turned towards studies of the collective behaviour of 'active' constituents, see, e.g. [2]. Here we are concerned with chromatin remodeling motors which actively displace and remove nucleosomes from the chromatin fiber, which have so far received only little attention in the statistical physics or biophysics literature [1]. The number of modeling attempts of individual remodeler dynamics has been rather limited so far; see, e.g., the references [3,4,5,6]; but also the collective behaviour of chromatin remodelers is beginning to attract attention [7,[START_REF] Padinhateeri | Proc. Natl. Acad. Sci[END_REF][START_REF] Florescu | [END_REF]10].

This lack of attention may in part be explained by the structural complexity and size of remodelers which has so far allowed to resolve only few and in particular often partial structures; this also impeded studies of remodeler dynamics, except in artificial constructs (see below). Very recently, the more widespread use of cryoelectron microscopy and FRET-imaging techniques have led to numerous new results, in particular on the (small) chromatin remodeler Chd1 [11,12,13,14,15].

Chromatin remodelers are built around evolutionarily conserved two-domain ATPase units which belong to the helicase-related superfamily II (SF2), and can be grouped into a small number of families which differ from each other by their accessory subunits [16]. These molecular motors play crucial rules in numerous chromatin-based processes such as the activation or the repression of transcription and DNA repair [17,18,[START_REF] Blossey | Chromatin: Structure, Dynamics[END_REF]; their biological relevance is underscored by the recent understanding that remodeler dysfunction is one source of regulatory diseases such as cancer [START_REF] Kumar | [END_REF]. A better understanding of how the motors act on the nucleosomes is thus of key relevance.

In this paper we show that, at a coarse-grained level, the DNA displacement around the nucleosome of a stripped-down version of chromatin remodeling enzymes can be very well captured in terms of a different type of model, an active Brownian dimer model. Such models were developed several years ago in the context of the simpler linear motors [21].

Mapping of a chromatin remodeler to an active Brownian dimer. Our mapping relies on recent structural biology results which we now briefly review. Liu et al. [22,23] have described the structure of a truncated version of a basic remodeler which they call ScSnf2 in complex with a nucleosome core-particle with a 167-bp DNA fragment containing the '601' positioning sequence [24]. Cryo-electron microscopy of this fully functional complex yielded a resolution of about 4 Å. From the observed populations of the complex, the remodeler was found to bind at the nucleosome preferably in different locations on DNA, i.e. either on superhelical location SHL 2, on SHL 6 and simultaneously on both SHL 2 and SHL 6 [22]. We take this basic information as key ingredients to be reflected in our model, which therefore should be able to describe a basic chromatin remodeler without any further additional recognition domains.

Although precise dynamical information cannot directly be inferred from the cryo-EM data, the following facts can be considered as established, also in conjunction with earlier work: (i) the remodeler structure consists of two lobes (also called 'cores', in fact presumably properly folded protein domains), which form multiple contact surfaces with both DNA turns; (ii) upon remodeling, the remodeler injects twist defects into the DNA turns which are expelled at the end contact of the remodeler domains with the DNA; (iii) the remodeler thus performs a spatially restricted rotation at the nucleosome turns: its full rotation around the DNA is impeded by the presence of the histone core. This limited rotary motion of the remodeler is a remainder of its helicase-like nature: a helicase would fully rotate around a single DNA double-strand during its linear motion along the DNA [25]. This rotation is represented in some models of helicases, but not by the Brownian dimer model. In our case, this neglect is justified because of the specific arrangement of the remodeler at the nucleosome. Figure 1 (left) displays a sketch of the remodeler profile at the DNA turns around the nucleosome. Turning to modeling this behaviour, condition (iii) allows to neglect the rotary motion of the remodeler; conditions (i) and (ii) then allow to restrict the dynamics to the footprints of the two lobes on DNA, which we denote as in [21,26] as x 1 and x 2 . In the course of the remodeler action, footprint x 1 is first shifted towards location x 2 via twist defect injection; the displacement of footprint x 2 follows in due course leading to twist defect ejection. The DNA length x ≡ x 1 -x 2 -x 0 where x 0 is the equilibrium extension of the DNA around the nucleosome thus relaxes after one remodeler step which is typically 1-2 bp large [27,28,29]; the variable x is therefore small, irrespective of the location of x 1 and x 2 along the internucleosomal DNA. This mapping of the displacement dynamics is illustrated in Figure 1 (right). The motion of the footprints thus is indeed akin to an inchworm motion of DNA around the histone octamer [23].

Core1& 2°& 1°& Core2& x 1 x 2 x 1 x 2 x 1 x 2 t 0 t 1 t 2
Modeling the inchworm. This inchworm motion can be rationalized in terms of molecular motor models, for which there are two philosophies. The most common one relies on Brownian ratchet models, where the key ingredient is the ratchet potential, a potential with a sawtooth-profile, which controls motor motion in conjunction with the ATP-consumption cycle. For chromatin remodelers such models have been developed for the case of a remodeler from the ISWI family which is capable to position nucleosomes on DNA; this could be demonstrated experimentally in single-molecule positioning assays [4,5]. For the present case in which the motion is that of a motor with two footprints, a ratchet model has been discussed in the literature [33]. The other modeling philosophy relies on active Brownian dimer models [21,26], which is the approach we follow here.

In the overdamped case which we assume in the following, it is given by coupled Langevin equations for the two footprint coordinates x i ,

γ i (x) ẋi (t) = -∂ i U (x) + g i (x)ξ i (t) , (1)
where ξ i (t) is a Gaussian noise with zero mean ξ i (t) = 0 and variance ξ i (t)ξ j (t) = δ ij δ(t -t ). Three key factors appear in this equation. The first is a spatially-dependent friction term γ i (x) ≥ 0 which qualitatively models the remodeler-nucleosome interaction.

The second term is the potential U (x) which we ascribe to the interaction of the remodeler domains between the two footprints. Both depend on the relative coordinate x introduced before. Finally, the factor g i (x) describes the noise correlations. It is given by

g i (x) ≡ 2γ i (x)k B T + A i (2)
where the factor A i describes the contribution of non-equilibrium noise due to ATPconsumption. This active Brownian dimer model is obviously highly coarse grained, since the nonequilibrium ATP-dependent driving force is not associated with particular configurations of the remodeling-based ATP cycle (we comment on this further in the discussion). Crucial in these kinds of models is the asymmetry in the spatially-dependent coupling of the motor to the twist-stretched DNA: neglecting the spatial dependence destroys the propagation mode in the model.

The dynamics of the two footprints can be decoupled into a center of mass motion and the relative coordinate. Due to the dependence of γ i (x) and U (x) on only the relative coordinate x the equation of the center of mass coordinate x cm is a function of the relative coordinate and the noise, while the equation for the relative coordinate is independent from the center of mass coordinate. The dynamics of the latter can be cast into a Fokker-Planck equation for the probability distribution of the relative coordinate P (x, t) which reads as [26]

∂ t P (x, t) = -∂ x [a(x)P (x, t)] + 1 2 ∂ x [b(x)∂ x [b(x)P (x, t)]] (3)
which has a stationary solution given by

p(x) = N b(x) exp x dy 2a(y) b 2 (y) , ( 4
)
where N is the normalization factor. In the following, based on the model assumptions, we choose for γ i (x) the following expressions. Shifting the x-dependence into one of the footprint coordinates, we define γ 1 = γ 0 and γ 2 = γ 0 + γx, similar to the choice in [21].

The zeroth-order term γ 0 can be understood as the translational friction term deriving from thermal nucleosome displacement [START_REF] Kulic | Chromatin Dynamics: Nucleosomes go Mobile through Twist Defects[END_REF][START_REF] Schiessel | Biophysics for Beginners[END_REF]. For the present case, the detailed form of the x-dependence in γ 2 is irrelevant, since x is always small in our case. We now exploit this case, which was not done in [26], where other choices were studied by numerical means. The smallness of x allows us to obtain analytical results for the model. Finally, it suffices to approximate the domain coupling U (x) by a harmonic potential with an effective elastic constant κ, U (x) = 1 2 κx 2 . In our case the coefficients in the Fokker-Planck equation are given by the expressions

a(x) = - 1 γ 0 + 1 γ 0 + γx κx - k B T 2
γ (γ 0 + γx) 2 , (5)
where the T -dependent term has its its origin in the noise term of the Langevin equation, eq.( 2) which has both thermal and 'active' contributions, and

b 2 (x) = 2k B T 1 γ 0 + 1 γ 0 + γx + A (γ 0 + γx) 2 . (6)
where we have set A 1 = 0 and A 2 = A. In the limit γ γ 0 , the coefficients simplify to

a(x) ≈ - 2 γ 0 κx - k B T 2 γ γ 2 0 ≈ - 2 γ 0 κx , (7)
where we neglect the last, temperature-dependent term, as it is small compared to the first term. Further, we have

b 2 (x) ≈ 4k B T γ 0 + A γ 2 0 . (8)
In this case, the probability distribution reduces to a simple Gaussian given by

p(x) = 1 √ 2πσ 2 exp - (x -µ) 2 2σ 2 . ( 9
)
with zero mean value, µ = 0, and the variance

σ 2 = A + 4k B T γ 0 4γ 0 κ . (10)
Note that for A = 0, p(x) is the equilibrium Boltzmann distribution. From this distribution we can then compute the average inchworm velocity using the expression derived in Ref. [26]. The authors have given the exact expression

ẋcm = - 1 2 ∆γ i (x)∂ x U (x) + (11)
k B T 2 ∂ x ∆γ i (x) + i (-1) i A i 4 ∂ x γ i (x) γ i (x) 3
with the averages • to be taken with respect to p(x) and the definition

∆γ i (x) = γ -1 1 (x) -γ -1 2 (x) . (12)
In our case the expression reduces to

ẋcm = - γ 2γ 2 0 κx 2 + k B T γ 2γ 2 0 + A 4 γ γ 3 0 , (13)
whereby the two last terms follow from the averages to be performed on the approximate equation

∆γ i (x) = γ -1 1 (x) -γ -1 2 (x) = γ -1 0 -(γ 0 + γx) -1 ≈ γ γ 0 x . (14)
The remaining average over x 2 produces the final result ẋcm = -A 8

γ γ 3 0 - k B T 2
γ γ 2 0 + k B T 2
γ γ 2 0 + A 4 γ γ 3 0 = A 8 γ γ 3 0 . ( 15
)
Thus, performing the average the T -dependent terms -which are themselves small within our approximation -cancel each other out exactly, as they should on general grounds: in the absence of active driving, the average inchworm velocity must be zero [26]. The final result for the inchworm velocity depends only on both the active noise A and the stretch-dependent driving via γ and γ 0 , but neither on the harmonic potential κ nor temperature. These dependencies enter back into the final result when going to higher orders in our approximation. Keeping the next order in the displacement x leads to a skewed distribution for the displacement, p(x), as found numerically in [26]. A finite mean of the distribution, e.g. with a positive value µ > 0, goes hand in hand with its skewness towards negative values of x in order to allow for the cancellation of temperature-dependent terms in the limit A → 0.

One can compare the putative inchworm motion with experiments on artificial motor constructs, as done for the remodeler RSC by Sirinakis et al. [START_REF] Sirinakis | [END_REF]. In that paper the maximum velocity of the remodeler is obtained from the location of the maximum of the probability distribution of observed velocities as a function of ATP consumption [AT P ], which shows a weaker logarithmic dependence on driving, v max ∼ ln[AT P ]. In the experiments of ref. [START_REF] Sirinakis | [END_REF], the motor was employed on a linearized DNA sequence under external load; the propagation of the DNA was then actually by the formation of loops. Clearly, this is a different regime of helicase action, since for the remodeler acting on the nucleosome, the external load is given by the DNA itself.

Discussion. In this paper we have formulated a coarse-grained minimal model of a chromatin remodeler which combines three elements: i) structural insights into the build-up of a 'minimal' chromatin remodeling module; ii) a mapping to the class of active Brownian dimer models introduced several years ago for simple helicases; iii) an exact solution of the model which furthermore exploits the fact that the twist-stretch coupling is always small in the present case. The model allows to rationalize the notion of inchworm motion in the context of remodelers and makes a simple prediction of the expected inchworm velocity as a function of active driving and the twist-stretch coupling. Higher-order corrections to our analytic limit can be identified with the strength of the remodeler lobe coupling and temperature.

Obviously, the model we present is extremely reduced. It can, however, easily be extended in both more microscopic and macroscopic directions: in order to include detail of nucleosome structure and coupling it to the proper ATP-cycle, modeling approaches such as cited before in terms of ratchet models [33], or very recent novel active models [34] can be applied. In this way, more information about the force-generation leading to the inchworm motion would be built in. On the other hand, it would also be of high interest to take the coarse-grained Brownian dimer for the confrontation with experiments: measuring the inchworm velocity as a function of the active driving force would allow to estimate the friction parameters in eq. (15). A further outlook is that the Brownian dimer model can be taken as a starting point for the derivation of chromatin hydrodynamics models beyond phenomenology [10].

Figure 1 .

 1 Figure 1. Two representations of the action of the remodeler on nucleosomal DNA. Left: side-view of the engagement of the two remodeler lobes with the two turns of DNA, following [22, 23]. Right: Mapping of the remodeler-DNA contacts on the two remodeler footprints denoted by coordinates along the DNA centered at x i , i = 1, 2. Each black bar represents the length of DNA along the nucleosome, with the red bars indicating the footprints (contact surfaces) of the remodeler along the DNA. Top bar: initial configuration at time t 0 . Middle bar: after a first step step t 1 , the left footprint has moved to the right while the other is still unchanged. Lower bar: at time t 2 the right footprint has also moved and the displacement step has been completed.

Acknowledgement. We thank Dr. Giovanni Brandani and Dr. Andrew Flaus for helpful discussions on chromatin remodelers. RB also is grateful for discussions on molecular motor models with Prof. Christian Maes.