domain: math phys

text:
R G Novikov
email: novikov@cmap.polytechnique.fr

Explicit formulas and global uniqueness for phaseless inverse scattering in multidimensions

We consider phaseless inverse scattering for the Schrödinger equation with compactly supported potential in dimension d ≥ 2. We give explicit formulas for solving this problem from appropriate data at high energies. As a corollary, we give also a global uniqueness result for this problem with appropriate data on a fixed energy neighborhood.

Introduction

We consider the Schrödinger equation

-∆ψ + v(x)ψ = Eψ, x ∈ R d , d ≥ 2, E > 0, (1.1) where v ∈ L ∞ (R d ), supp v ⊂ D,
D is an open bounded domain in R d .

(1.2)

For equation (1.1) we consider the classical scattering solutions for some a priori unknown f . In addition, the function f = f (k, l), k, l ∈ R d , k 2 = l 2 = E, arising in (1.3) is the classical scattering amplitude for equation (1.1).

ψ + = ψ + (x, k), x ∈ R d , k ∈ R d , k 2 = E,
In order to find ψ + and f from v one can use, in particular, the Lippmann-Schwinger integral equation

ψ + (x, k) = e ikx + ∫ D G + (x -y, k)v(y)ψ + (y, k)dy, G + (x, k) = -(2π) -d ∫ R d e iξx dξ ξ 2 -k 2 -i0 ,
(1.4) and the formula

f (k, l) = (2π) -d ∫ D
e -ily v(y)ψ + (y, k)dy, (1.5) where x, k, l ∈ R d , k 2 = l 2 = E > 0; see e.g. [BS], [F2].

The scattering amplitude f for equation (1.1) at fixed E is defined on

M E = {k ∈ R d , l ∈ R d : k 2 = l 2 = E}, E > 0.
(1.6)

In addition to f on M E , we consider also f Γ E , where

Γ E = {k = k E (p), l = l E (p) : p ∈ B2 √ E }, k E (p) = p 2 + ( E - p 2 4 ) 1/2 γ(p), l E (p) = - p 2 + ( E - p 2 4 ) 1/2 γ(p), (1.7) B r = {p ∈ R d : |p| < r}, Br = {p ∈ R d : |p| ≤ r}, r > 0, (1.8)
where γ is a piecewise continuous vector-function on R d such that

|γ(p)| = 1, γ(p)p = 0, p ∈ R d .
(1.9)

One can see that (1.11) where Λ ⊆ R + =]0, +∞[. We start with the following inverse scattering problems for equation (1.1) under assumptions (1.2): Problem 1.1. Reconstruct potential v on R d from its scattering amplitude f on some appropriate

Γ E ⊂ M E , dim Γ E = d, dim M E = 2d -2, E > 0, d ≥ 2. (1.10) Let M Λ = ∪ E∈Λ M E , Γ Λ = ∪ E∈Λ Γ E ,
M ′ ⊆ M R + . Problem 1.2. Reconstruct potential v on R d from its phaseless scattering data |f | 2 on some appropriate M ′ ⊆ M R + .
Note that in quantum mechanical scattering experiments (in framework of model described by equation (1.1)) the phaseless scattering data |f | 2 can be measured directly, whereas the complete scattering amplitude f is not accessible for direct measurements. Therefore, Problem 1.2 is of particular interest from applied point of view in the framework of quantum mechanical inverse scattering. However, in the literature much more results are given on Problem 1.1 (see [ABR], [B], [BAR], [ChS], [EW], [E], [ER], [F1], [F2], [G], [HH], [HN], [I], [IN], [Me], [Mo], [Ne], [N1]- [N7], [R], [S], [VW], [WY] and references therein) than on Problem 1.2 (see chapter X of [ChS] and recent works [K1], [K2] and references therein, where in [K1], [K2] some similar problem is considered).

In particular, for Problem 1.1 it is well known that the scattering amplitude f at high energies uniquely determines v via the formulas

v(k -l) = f (k, l) + O(E -1/2 ) as E → +∞, (k, l) ∈ M E , (1.12) v(p) = (2π) -d ∫ D e ipx v(x)dx, p ∈ R d ;
(1.13) see, for example, [F1], [N7].

On the other hand, for Problem 1.2 it is well known that the phaseless scattering data |f | 2 on M R + do not determine v uniquely, in general. In particular, we have that

f y (k, l) = e i(k-l)y f (k, l), |f y (k, l)| 2 = |f (k, l)| 2 , (k, l) ∈ M R + , y ∈ R d , (1.14)
where f is the scattering amplitude for v and f y is the scattering amplitude for v y , where

v y (x) = v(x -y), x ∈ R d , y ∈ R d ;
(1.15) see, for example, Lemma 1 of [N6].

In the present work, in view of the aforementioned nonuniqueness for Problem 1.2 we modify this problem into Problem 1.3 formulated below. Let . . . , n}, (1.16) where f is the initial scattering amplitude for v satisfying (1.2) and f j is the scattering amplitude for

S = {|f | 2 , |f j | 2 , j = 1,
v j = v + w j , j = 1, . . . , n, (1.17)
where w 1 , . . . , w n are additional a priori known background scatterers such that

w j ∈ L ∞ (R d ), supp w j ⊂ Ω j , Ω j is an open bounded domain in R d , Ω j ∩ D = ∅,
(1.18a)

w j ̸ = 0, w j 1 ̸ = w j 2 for j 1 ̸ = j 2 in L ∞ (R d ), j, j 1 , j 2 ∈ {1, . . . , n}. (1.18b)
Problem 1.3. Reconstruct potential v on R d from the phaseless scattering data S on some appropriate M ′ ⊆ M R + and for some appropriate background scatterers w 1 , . . . , w n .

Note also that Problems 1.1, 1.2, 1.3 can be considered as examples of ill-posed problems; see [LRS] for an introduction to this theory.

Problem 1.3 in dimension d = 1 was, actually, considered in [AS] for n = 1. However, to our knowledge, Problem 1.3 in dimension d ≥ 2 was not yet considered in the literature before the present work.

Results of the present work can be summarized as follows.

First, we give explicit formulas for solving Problem 1.3 in dimension d ≥ 2 for n = 2 and M ′ = Γ Λ defined by (1.7), (1.11) for any unbounded Λ ⊂ R + ; see Theorem 2.1, Remark 3.1 and Corollary 2.1 of Section 2. As an example of Λ for this result one can take

Λ = [E 0 , +∞[, E 0 > 0, or just Λ of Remark 2.1.
Second, we give a global uniqueness result for Problem 1.3 in dimension d ≥ 2 for n = 2 and M ′ = Γ Λ for any bounded infinite Λ ⊂ R + ; see Theorem 2.2 of Section 2. As an example of Λ for this result one can take

Λ =]E 0 -ε, E 0 + ε[, E 0 > 0, ε > 0, E 0 -ε ≥ 0, or just Λ of Theorem 2.2.
In addition, we indicate possible extensions of the aforementioned results to the case n = 1; see Propositions 2.1, 2.2 of Section 2.

The progress of the present work in comparison with the recent works [K1], [K2] includes explicit formulas for phaseless inverse scattering at high energies and no assumption that v ≥ 0. In addition, in the present work we consider inverse scattering from far field phaseless scattering data (and not from near field phaseless scattering data as in [K1], [K2]).

The main statements of the present work are presented in detail in the next section.

Main statements

2.1. Notations and related remarks. Let

û(p) = (2π) -d ∫ R d e ipx u(x)dx, p ∈ R d , (2.1)
where u is a test function on R d . In particular, we consider û = v, ŵj for u = v, w j , j = 1, . . . , n, where v, w j satisfy (1.2), (1.18). Note that if

u y (x) = u(x -y), x, y ∈ R d , (2.2a) then ûy (p) = e ipy û(p), p ∈ R d . (2.2b)
We represent v and ŵj as follows:

v(p) = |v(p)|θ(p), θ(p) = e iα(p) , ŵj (p) = | ŵj (p)|ω j (p), ω j (p) = e iβ j (p) , (2.3)
where p ∈ R d , j = 1, . . . , n.

We consider the following sets:

A y = {p ∈ R d : e 2ipy = 1}, y ∈ R d , (2.4) Z 0 = {p ∈ R d : |v(p)| = 0}, Z j = {p ∈ R d : | ŵj (p)| = 0}, j = 1, . . . , n, (2.5) Y j 1 ,j 2 = {p ∈ R d \(Z j 1 ∪ Z j 2 ) : (ω j 1 (p)) 2 = (ω j 2 (p)) 2 }, 1 ≤ j 1 , j 2 ≤ n, j 1 ̸ = j 2 . (2.6)
We have, in particular, that

A y is closed and M es A y = 0 in R d , y ̸ = 0. (2.7)
Assumptions (1.2) on v imply, in particular, that v is (complex-valued) real-analytic on R d . Therefore:

Z 0 is closed in R d ; M es Z 0 = 0 in R d if v ̸ ≡ 0.
(2.8)

Assumptions (1.18) on w j imply, in particular, that ŵj is (complex-valued) realanalytic on R d and ŵj ̸ ≡ 0, j = 1, . . . , n. Therefore, Z j is closed and M es Z j = 0 in R d , j = 1, . . . , n.

(2.9)

In addition, if

w j (x) = w 0 j (|x -y|), x ∈ R d , for some w 0 j , (2.10)
for some j and some y ∈ R d , then

Z j = {p ∈ R d : |p| ∈ R j }, (2.11)
where R j is a discrete set in R + without accumulation points (except +∞) and R j is independent of y.

In addition, taking into account (2.2), if

w j 2 (x) = w j 1 (x -y), x ∈ R d , j 2 ̸ = j 1 , (2.12)
for some j 1 , j 2 and some y ∈ R d \{0}, then

Y j 1 ,j 2 ⊆ A y .
(2.13)

Results on Problem 1.3 in dimension

d ≥ 2 for n = 2.
Theorem 2.1. Suppose that complex-valued v satisfies (1.2), complex-valued w j satisfies (1.18a), j = 1, 2, d ≥ 2. Then the following formulas hold: andc, ρ are defined by (3.10), (3.11). Suppose, in addition, that w 1 , w 2 satisfy (1.18b) and that

|v j (p)| 2 = lim p=k-l,(k,l)∈M E , E→+∞ |f j (k, l)| 2 for each p ∈ R d , j = 0, 1, 2, (2.14) ||v j (p)| 2 -|f j (k, l)| 2 | ≤ c(D j )N 3 j E -1/2 , p = k -l, (k, l) ∈ M E , E 1/2 ≥ ρ(D j , N j ), j = 0, 1, 2, (2.15) where v 0 = v, f 0 = f , D 0 = D, v j is defined by (1.17) and D j = D ∪ Ω j for j = 1, 2, ∥v j ∥ L ∞ (D j ) ≤ N j , j = 0, 1, 2,
M es Ȳ1,2 = 0 in R d , (2.16)
where Ȳ1,2 denotes the closure of Y 1,2 in R d . Then the following formula holds:

( cos α sin α ) = (sin(β 2 -β 1 )) -1 × ( sin β 2 -sin β 1 -cos β 2 cos β 1 ) ( (2|v|| ŵ1 |) -1 (|v 1 | 2 -|v| 2 -| ŵ1 | 2 ) (2|v|| ŵ2 |) -1 (|v 2 | 2 -|v| 2 -| ŵ2 | 2 )
) ,

(2.17)

α = α(p), |v| = |v(p)|, β j = β j (p), | ŵj | = | ŵj (p)|, j = 1, 2, p ∈ R d \(Z 0 ∪ Z 1 ∪ Z 2 ∪ Ȳ1,2 ),
where α, β 1 , β 2 are defined in (2.3). Theorem 2.1 is proved in Section 3.

Remark 2.1. Formulas (2.14), (2.15) of Theorem 2.1 remain valid with Γ E in place of M E , where Γ E is defined by (1.7). In addition, taking into account (2.9), (2.16) these formulas can be considered as explicit formulas for finding v on

R d from S = {|f | 2 , |f 1 | 2 , |f 2 | 2 } on Γ Λ and background w 1 , w 2 for any Λ = {E j ∈ R + : j ∈ N, E j → ∞ as j → ∞}, (2.18)
where Γ Λ is defined in (1.11).

Corollary 2.1. Let all assumptions of Theorem 2.1 on v and w 1 , w 2 be fulfilled. Let Λ be defined as in (2.18)

. Then S = {|f | 2 , |f 1 | 2 , |f 2 | 2 } on Γ Λ and background w 1 , w 2 uniquely determine v in L ∞ (R d ) via formulas (2.

14), (2.15) and the inverse Fourier transform.

In addition to results of Theorem 2.1, Remark 2.1 and Corollary 2.1 on the explicit reconstruction from phaseless scattering data at high energies, we have also the following global uniqueness result for the case of finite energies:

Theorem 2.2. Let v satisfy (1.2), w 1 , w 2 satisfy (1.18), (2.16), d ≥ 2, and v, w 1 , w 2 be real-valued. Let

Λ = {E j ∈ R + : j ∈ N, E j 1 ̸ = E j 2 for j 1 ̸ = j 2 , E j → E * as j → ∞}, E * > 0. (2.19) Then S = {|f | 2 , |f 1 | 2 , |f 2 | 2 } on Γ Λ and background w 1 , w 2 uniquely determine v in L ∞ (R d ).
Theorem 2.2 is proved in Section 4.

Results for the case n = 1.

Proposition 2.1. If complex-valued v satisfies (1.2), complex-valued w 1 satisfies (1.18a), d ≥ 2, then formulas (2.14), (2.15) hold for j = 0, 1. If, in addition,

w 1 ̸ = 0 in L ∞ (R d ), then cos(α -β 1 ) = (2|v|| ŵ1 |) -1 (|v 1 | 2 -|v| 2 -| ŵ1 | 2 ), α = α(p), |v| = |v(p)|, β 1 = β 1 (p), | ŵ1 | = | ŵ1 (p)|, p ∈ R d \(Z 0 ∪ Z 1 ), (2.20)
where α, β 1 are defined in (2.3). Proposition 2.1 is proved in Section 3.

Proposition 2.2. (A) There are not more than two different complex-valued potentials v satisfying (1.2) with given

S = {|f | 2 , |f 1 | 2 } on Γ Λ and background complex-valued w 1 satisfying (1.18a), w 1 ̸ = 0 in L ∞ (R d )
, where Λ is defined as in (2.18). (B) There are not more than two different real-valued potentials v satisfying (1.2) with given

S = {|f | 2 , |f 1 | 2 } on Γ Λ and background real-valued w 1 satisfying (1.18a), w 1 ̸ = 0 in L ∞ (R d ), where Λ is defined as in (2.19).
Proposition 2.2 is proved in Section 5.

Proofs of Proposition 2.1 and Theorem 2.1

3.1. Preliminaries. Let L ∞ σ (R d ) = {u ∈ L ∞ (R d ) : ∥u∥ σ < +∞}, ∥u∥ σ = ess sup x∈R d (1 + |x| 2 ) σ/2 |u(x)|, σ ≥ 0. (3.1) Note that v, w j , v j ∈ L ∞ σ (R d ) for each σ ≥ 0, (3.2)
where v, w j , v j , j ∈ {1, 2}, are the potentials of Proposition 2.1 and Theorem 2.1. We recall that

∥ < x > -s G + (k) < x > -s ∥ L 2 (R d )→L 2 (R d ) ≤ a 0 (d, s)|k| -1 , k ∈ R d , |k| ≥ 1, for s > 1/2, (3.3)
where G + (k) denotes the integral operator with the Schwartz kernel G + (x -y, k) of (1.4), < x > denotes the multiplication operator by the function (1 + |x| 2 ) 1/2 ; see [E], [J] and references therein. We will use the following detailed version of formula (1.12):

|f (k, l) -v(k -l)| ≤ 2(2π) -d a 0 (d, σ/2)(c 1 (d, σ)∥v∥ σ ) 2 E -1/2 , (k, l) ∈ M E , E 1/2 ≥ ρ 1 (d, σ, ∥v∥ σ ), σ > d, (3.4)
where a 0 (d, s) is the constant of (3.3),

c 1 (d, σ) = ( ∫ R d dx (1 + |x| 2 ) σ/2
) 1/2 , (3.5)

ρ 1 (d, σ, R) = max(2a 0 (d, σ/2)R, 1); (3.6) see formula (2.11) of [N7].

Proof of formulas (2.14), (2.15).

We have that

|v(k -l)| (3.1),(3.5) ≤ (2π) -d ∥v∥ σ (c 1 (d, σ)) 2 , (3.7) ||f (k, l)| 2 -|v(k -l)| 2 | = ||f (k, l)| -|v(k -l)||(|f (k, l)| + |v(k -l)|) ≤ |f (k, l) -v(k -l)|(2|v(k -l)| + |f (k, l) -v(k -l)|), (3.8)
where (k, l) ∈ M E , σ > d. Due to (3.4), (3.7), (3.8), we have that

||f (k, l)| 2 -|v(k -l)| 2 | ≤ 3(2π) -d ∥v∥ σ (c 1 (d, σ)) 2 |f (k, l) -v(k -l)| ≤ 6(2π) -2d a 0 (d, σ/2)((c 1 (d, σ)) 4 (∥v∥ σ ) 3 E -1/2 , (3.9) (k, l) ∈ M E , E -1/2 ≥ ρ 1 (d, σ, ∥v∥ σ ), σ > d. Formulas (2.14), (2.15) follow from (3.9) for v = v j , f = f j and from the possibility of choice of k = k E (p), l = l E (p) as in (1.7) for d ≥ 2. In addition, c(D) = 6(2π) -2d a 0 (d, σ/2)(c 1 (d, σ)) 4 (c 2 (d, σ)) 3 , (3.10) ρ(D, N ) = ρ 1 (d, σ, c 2 (D, σ)N ), (3.11)
for some fixed σ > d, where

c 2 (D, σ) = sup x∈D (1 + |x| 2 ) σ/2 .
(3.12)

Proof of formula (2.20).

We have that

|v 1 | 2 (1.17) = |v + ŵ1 | 2 (2.3) = ||v|e iα + | ŵ1 |e iβ 1 | = (|v| cos α + | ŵ1 | cos β 1 ) 2 + (|v| sin α + | ŵ1 | sin β 1 ) 2 = |v| 2 + | ŵ1 | 2 + 2|v|| ŵ1 |(cos α cos β 1 + sin α sin β 1 ) on R d .
(3.13) Formula (2.20) follows from (3.13).

3.4. Proof of formula (2.17). Using (3.13) and analogous formula for v2 = v + ŵ2 , we obtain the system

( cos β 1 sin β 1 cos β 2 sin β 2 ) ( cos α sin α ) = ( (2|v|| ŵ1 |) -1 (|v 1 | 2 -|v| 2 -| ŵ1 | 2 ) (2|v|| ŵ2 |) -1 (|v 2 | 2 -|v| 2 -| ŵ2 | 2 ) ) , (3.14) on R d \(Z 0 ∪ Z 1 ∪ Z 2 ).
Formula (2.17) follows from (3.14).

3.5. Final remark. Proposition 2.1 and Theorem 2.1 follow from formulas (2.14), (2.15), (2.20), (2.17) proved in Subsections 3.2, 3.3, 3.4.

Proof of Theorem 2.2

Let ∆ E 0 ,E = {(k, l) ∈ Γ E : k -l ∈ B 2 √ E 0 }, 0 < E 0 ≤ E, (4.1)
where Γ E , B r are defined by (1.7), (1.8). Theorem 2.2 follows from:

(1) the formulas of Theorem 2.1 with

∆ E * ,E in place of M E and |p| < 2 √ E * , (2) the fact that v on B 2 √ E * \(Z 1 ∪ Z 2 ∪ Ȳ1,2 ) uniquely determines v on R d (since v is real-analytic on R d ), and
(3) the results of Lemma 4.1 for v and for v = v + w j , j = 1, 2.

G + (x, k) = G + 0 (|x|, |k|), x ∈ R d , k ∈ R d \{0}, (4.3)
where G + is the function of (1.4) and G + 0 depends also on d;

(3) the properties that:

|k E (p)| = E 1/2 for E ∈]p 2 /4, +∞[, p ∈ R d ; E 1/2 is holomorphic in E ∈ C\] -∞, 0]; (E -p 2 /4) 1/2 is holomorphic in E ∈ C\] -∞, p 2 /4]; p ∈ R d ; G + 0 (r, κ) is holomorphic in κ ∈ C for odd d ≥ 3 and in κ ∈ C\] -∞, 0]
ψ + (•, k) ∈ L 2 (D) for each k ∈ R d \{0}
under our assumptions on v.

In connection with basic properties of function G + and basic properties of the Lippmann-Schwinger integral equation (1.4) we refer also to [BS], [F2], [Me]. R d ). Therefore, it remains to consider the case when |v| ̸ ≡ 0.

Proof of Proposition 2.2

Proof of part (A). Due to formulas (

2.14), (2.15) with Γ E in place of M E , we have that S = {|f | 2 , |f 1 | 2 } on Γ Λ uniquely determine |v|, |v 1 | on R d . If |v| ≡ 0, then v = 0 in L ∞ (
Due to (2.8), (2.9), j = 1, and continuity of v, ŵ1 , we can choose

p ′ ∈ R d , r ′ > 0 such that |v(p)| ̸ = 0, | ŵ1 (p)| ̸ = 0 for p ∈ B p ′ ,r ′ , (5.1) B p ′ ,r ′ = {p ∈ R d : |p -p ′ | < r ′ }. (5.2) Therefore, formula (2.20) for cos(α -β 1 ) holds for each p ∈ B p ′ ,r ′ . If cos(α -β 1 ) ≡ 1 on B p ′ ,r ′ , then α ≡ β 1 (mod 2π) on B p ′ ,r ′ . If cos(α -β 1 ) ≡ -1 on B p ′ ,r ′ , then α ≡ β 1 + π (mod 2π) on B p ′ ,

  specified by the following asymptotics as |x| → ∞:ψ + (x, k) = e ikx + c(d, |k|) e i|k||x| |x| (d-1)/2 f (k, |k|) = -πi(-2πi) (d-1)/2 |k| (d-3)/2 ,(1.3)

  for even d ≥ 2, where r > 0; (4) the result that (1.4) with k = k E (p) is a Fredholm integral equation of the second kind for ψ + (•, k) ∈ L 2 (D) with holomorphic dependence on the parameter E ∈ C\] -∞, p 2 /4] at fixed p ∈ R d ; (5) the result that (1.4) is uniquely solvable for

  |f (k E (p), l E (p))| 2 is real-analytic in E ∈]p 2 /4, +∞[ for fixed p ∈ R d , where k E (p), l E (p) are defined in (1.7 ); (b) |f | 2 on Γ Λ uniquely determines |f | 2 on ∆ E * ,E for each E ≥ E * ,where Γ Λ , Λ are defined in (1.11), (2.19). Statement (b) of Lemma 4.1 follows from statement (a) of Lemma 4.1 and the property that the accumulation point E * ∈]p 2 /4, +∞[ if p ∈ B 2 Let v satisfy (1.2) and be real-valued. Then f (k E (p), l E (p)) admits holomorphic extension in E to an open N in C, where ]p 2 /4, +∞[⊂ N , at fixed p ∈ R d .

	Lemma 4.1. Let v satisfy (1.2) and be real-valued. Then: E * . In turn, statement (a) of Lemma 4.1 follows from the presentation |f | 2 = f f and from Lemma 4.2. Lemma 4.2. Lemma 4.2 follows from: (1) the integral equation (1.4) for ψ + on D and the presentation (1.5) for f , where k = k E (p), l = l E (p); (a) √ (2) the property that	(4.2)

  r ′ . And in both cases v = |v|e iα is uniquely determined on B p ′ ,r ′ by |v|, |v 1 |, ŵ1 = | ŵ1 |e iβ 1 on B p ′ ,r ′ . Due to continuity of e iα , e iβ 1 on Bp ′ ,r ′ we can choose p ′′ ∈ B p ′ ,r ′ and r ′′ ∈]0, r ′ [ such that -1 < c min ≤ cos(α -β 1 ) ≤ c max < 1 on B p ′′ ,r ′′ (5.3)for some fixed c min , c max . Therefore, due to formula (2.20) for cos(α -β 1 ) and continuity of α, β 1 (mod 2π) on B p ′′ ,r ′′ we have that eitherα = β 1 + arc cos((2|v|| ŵ1 |) -1 (|v 1 | 2 -|v| 2 -| ŵ1 | 2 )) (5.4a) or α = β 1 -arc cos((2|v|| ŵ1 |) -1 (|v 1 | 2 -|v| 2 -| ŵ1 | 2 )) (5.4b) (mod 2π) on B p ′′ ,r ′′ , where arc cos takes values in [0, π]. Therefore, there are not more than two different v = |v|e iα on B p ′′ ,r ′′ with given |v|, |v 1 |, ŵ1 = | ŵ1 |e iβ 1 on B p ′′ ,r ′′ . In turn, v on B p ′′ ,r ′′ uniquely determines v on R d due to real analyticity of v. This completes the proof of part (A) of Proposition 2.2. Proof of part (B). Due to statement (b) of Lemma 4.1 (for v and for v = v + w 1 ), S = {|f | 2 , |f 1 | 2 } on Γ Λ uniquely determine S on ∆ E * ,E . Due to formulas (2.14), (2.15) with ∆ E * ,E in place of M E and |p| < 2 √ E * , we have that S on ∆ E * ,E uniquely determine |v|, |v 1 | on B 2 √ E * . Then in a completely similar way with the proof of part (A) of Proposition 2.2 we obtain that there are not more than two different v on B 2 √ E * with given |v|, |v 1 |, ŵ1 on B 2 uniquely determines v on R d due to real analyticity of v.This completes the proof of part(B).

	√	E * .
		Finally, v on B 2 √	E

*


Rupert G Perkins
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Johann Lavaud

Joao Serôdio

Jean-Luc Mouget

Paolo Cartaxana

Philippe Rosa

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Vertical cell movement is a primary response of intertidal benthic biofilms to increasing light dose

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Introduction

Microphytobenthic biofilms at the surface of intertidal estuarine sediments are highly productive [START_REF] Brotas | Spatio-temporal distribution of the microphytobenthic biomass in intertidal flats of Tagus Estuary (Portugal)[END_REF][START_REF] Macintyre | Microphytobenthos: the ecological role of the "Secret Garden" of unvegetated, shallow-water marine habitats. I. Distribution, abundance and primary production[END_REF][START_REF] Underwood | Primary production by phytoplankton and 30 microphytobenthos in estuaries[END_REF]. The regulatory mechanisms controlling the magnitude and periodicity of this productivity are partly understood as involving sun angle and tidal patterns [START_REF] Pinckney | Effects of tidal stage and sun angles on intertidal benthic microalgal productivity[END_REF] as well as changes in light dose exposure [START_REF] Kromkamp | Determination of microphytobenthos PSII quantum efficiency and photosynthetic activity by means of variable chlorophyll fluorescence[END_REF][START_REF] Serôdio | Fortnightly light and temperature variability in estuarine intertidal sediments and implications for microphytobenthos primary productivity[END_REF][START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF][START_REF] Jesus | Spatial dynamics of microphytobenthos determined by PAM fluorescence[END_REF]. For the latter it has been hypothesised [START_REF] Kromkamp | Determination of microphytobenthos PSII quantum efficiency and photosynthetic activity by means of variable chlorophyll fluorescence[END_REF][START_REF] Serôdio | Fortnightly light and temperature variability in estuarine intertidal sediments and implications for microphytobenthos primary productivity[END_REF][START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF]Jesus et al., 2006a) that cells optimise their position within the surface layers of a sediment biofilm, utilising sediment light attenuation to provide an optimal light environment; this is the concept of microcycling. The importance of vertical movement to regulate light exposure has recently been demonstrated thoroughly using chemical inhibition of movement (Cartaxana and Serôdio, 2008;Cartaxana et al. 2008). However, this is the first study to directly compare the roles of vertical movement, a behavioural form of photosynthetic down regulation (e.g. [START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF], with physiological down regulation in the form of non-photochemical quenching (NPQ; e.g. [START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Mechanisms, evolution and ecophysiology[END_REF].

Effectively diatom cells move vertically through the sediment matrix utilising extracellular polymers in response to changes in light environment: too much light, cells move downwards, not enough light, cells move upwards. This is a simplification however as the cumulative effect of light exposure over time modifies this response [START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF][START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]Jesus et al., 2006b). It should be emphasised as well, that this microcycling movement over short time scales is distinct from the bulk movements of cells as vertical migration [START_REF] Underwood | Primary production by phytoplankton and 30 microphytobenthos in estuaries[END_REF]; and see the review by [START_REF] Consalvey | The ups and downs of life in a benthic biofilm: Migration of benthic diatoms[END_REF] driven by tidal and sun angle driving forces as originally outlined by Pinkney and Zingmark (1991).

Why do diatom cells require an optimum light environment to maximise their photosynthetic potential? It is well known that excess light can lead to photodamage by production of free radicals and superoxides which may lead to protein breakdown in photosystem II reaction centres, e.g. the D1 dimer [START_REF] Olaizola | Non-photochemical fluorescence quenching and the diadinoxanthin cycle in a marine diatom[END_REF][START_REF] Materna | First induced plastid genome mutations in an alga with secondary plastids: psbA mutations in the diatom Phaeodactylum tricornutum (Bacillariohyceae) reveal consequences on the regulation of photosynthesis[END_REF]. Cells can hence prevent such damage through two processes, both of which effectively down regulate photosynthetic activity. Firstly, cells can migrate downwards away from high light that could result in a photodamaging light dose. This is effectively a behavioural form of down regulation [START_REF] Kromkamp | Determination of microphytobenthos PSII quantum efficiency and photosynthetic activity by means of variable chlorophyll fluorescence[END_REF][START_REF] Serôdio | Fortnightly light and temperature variability in estuarine intertidal sediments and implications for microphytobenthos primary productivity[END_REF][START_REF] Perkins | Responses of microphytobenthos to light: primary production and carbohydrate allocation over an emersion period[END_REF][START_REF] Mouget | Migration or photoacclimation to prevent photoinhibition and UV-B damage in marine microphytobenthic communities[END_REF]. Secondly cells can down regulate by diverting excess light energy away from PSII reaction centres via alternative energy pathways [START_REF] Ting | Photochemical and non-photochemical fluorescence quenching processes in the diatom Pheodactylum tricornutum[END_REF]Lavaud et al., 2002a;[START_REF] Goss | The importance of a highly active and ΔpH-regulated diatoxanthin epoxidase for the regulation of the PS II antenna function in diadinoxanthin cycle containing algae[END_REF][START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Mechanisms, evolution and ecophysiology[END_REF][START_REF] Cruz | Relationship of rapid light curves of variable fluorescence to photoacclimation and non-photochemical quenching in a benthic diatom[END_REF]. This physiological process of down regulation is often referred to as non-photochemical quenching (NPQ) as it quenches the energy using energy conversions with no photochemical output. The process utilises energisation of the thylakoid membrane by generation of a proton gradient, which induces de-epoxidation of diadinoxanthin to diatoxanthin (DT) known as the xanthophylls cycle (Lavaud et al., 2002a[START_REF] Lavaud | General features of photoprotection by energy dissipation in planktonic diatoms (Bacillariophyceae)[END_REF][START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Mechanisms, evolution and ecophysiology[END_REF][START_REF] Goss | The importance of a highly active and ΔpH-regulated diatoxanthin epoxidase for the regulation of the PS II antenna function in diadinoxanthin cycle containing algae[END_REF]. DT competes for light energy with chlorophyll pigments in the light harvesting complexes, hence diverting the energy away from the pathway that would lead to generation of harmful reducing agents created by over excitation of PSII reaction centres [START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Mechanisms, evolution and ecophysiology[END_REF][START_REF] Ruban | The super-excess energy dissipation in diatom algae: comparative analysis with higher plants[END_REF]. Diatoms are known to have highly effective xanthophyll cycle and are able to rapidly induce NPQ in response to increasing light levels [START_REF] Goss | The importance of a highly active and ΔpH-regulated diatoxanthin epoxidase for the regulation of the PS II antenna function in diadinoxanthin cycle containing algae[END_REF]Lavaud, 2004[START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Mechanisms, evolution and ecophysiology[END_REF][START_REF] Ruban | The super-excess energy dissipation in diatom algae: comparative analysis with higher plants[END_REF]Serodio et al., 2005[START_REF] Cruz | Relationship of rapid light curves of variable fluorescence to photoacclimation and non-photochemical quenching in a benthic diatom[END_REF] even so far as to induce short term photoacclimation through NPQ induction in the time required for 30 second rapid light curves, e.g. over a 4 minute period [START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF][START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF][START_REF] Cruz | Relationship of rapid light curves of variable fluorescence to photoacclimation and non-photochemical quenching in a benthic diatom[END_REF].

Diatom cells in surface biofilms can therefore respond to changing light environments, and hence accumulated historical light doses, through two mechanisms, vertical cell movement within the sediment matrix, or NPQ induction. These processes are now well understood, e.g. [START_REF] Consalvey | The ups and downs of life in a benthic biofilm: Migration of benthic diatoms[END_REF], [START_REF] Kromkamp | Determination of microphytobenthos PSII quantum efficiency and photosynthetic activity by means of variable chlorophyll fluorescence[END_REF], [START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF], [START_REF] Spilmont | Short-term variability of intertidal benthic community production during emersion and the implication in annual budget calculation[END_REF] and [START_REF] Mouget | Migration or photoacclimation to prevent photoinhibition and UV-B damage in marine microphytobenthic communities[END_REF] regarding light induced cell movement and [START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Mechanisms, evolution and ecophysiology[END_REF] and [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF] regarding NPQ. Also how does the down regulation effect net productivity and how does this vary in response to the light dose over a low tide emersion period? This study aimed to address these questions through manipulative experiments using engineered biofilms treated with chemical inhibitors, under two different light dose regimes. Chemical treatments comprised inhibition of cell motility using Latrunculin A which inhibits actin filaments involved in diatom movement without affecting photosynthetic activity (Cartaxana et al. 2008) and also the use of DL-dithiothreitol which inhibits the de-epoxidation of DD to DT, and hence inhibits NPQ induction (Lavaud et al., 2002b). These treatments were compared to controls over a 6 hour emersion period under two light treatments, ambient light and a constant low light environment. Thus the roles of cell movement and NPQ induction were compared as functions of the increasing photodose accumulated over the emersion period.

Methods

Surface mud to a depth of approximately 1 cm was collected on the 1 st July 2008 from Alcochete mudflat, located on the eastern shore of the Tagus Estuary (38 44' N, 9 08' W), composed of slightly gravelly mud (Jesus et al 2006c). All experimental measurements were carried out on the following day, 2 nd July 2008. The mud and surface biofilm was returned to the laboratory where a sub-sample was examined by light microscopy to determine the dominance of epipelic diatoms in the biofilm. The remainder of the surface mud was thoroughly mixed by hand and then evenly spread in trays to a depth of 5 cm. A shallow depth of site water (< 2 cm) was carefully added so as not to re-suspend the mud and the trays were left overnight in the laboratory. The following morning, at the start of the low tide emersion predicted for the original sample site, the shallow depth of site water was removed and a spectroradiometer (see below) was used to monitor the establishment of surface biomass in one of the sample trays. Plastic cores (2 cm × 2.5 cm diameter) were then carefully inserted into the mud to isolate minicore sediment samples in each sediment tray for the following chemical treatments: controls (addition of filtered site water only), Latrunculin A (Lat A, dissolved in site water) to inhibit cell motility and DL-dithiothreitol (DTT, in site water) to inhibit conversion of DD to DT and hence inhibit non-photochemical quenching (NPQ). Full details of these treatments are given below. Three replicates for each chemical treatment were used to provide independent samples for the following measurement: rapid light response curves using PAMfluorescence, spectroradiometry, sampling for pigment analysis using HPLC (minicore set 1) and 14 C radiolabelled measurement of primary productivity (minicore set 2). Hence 6 minicores were needed for each treatment for each time sampling point (n = 3, T1, 2 and 3 equally spaced 2 h apart). Finally the number of minicores was duplicated in a second sample tray to enable two light treatments to be investigated, ambient light and constant light (300 μmol m -2 s -1 ). Note that all light levels referred to were measured with a Licor cosine corrected light meter and refer to photosynthetically active radiation, 400 -700 nm. The constant light was provided by a quartz white light source (400W HPI-T Pro Philips). The experimental set up is summarized in Table 1. Ambient light treatment (Amb) and constant low light treatment (Con) were identical other than their respective light dose exposures calculated by integration over time of light measurements taken using a Licor cosine corrected light meter every 30 minutes during the experimental period. Finally, all treatments were applied once the biofilm had established at the sediment surface as assessed by the stabilization of the NDVI reflectance readings; hence chemical and light treatments were applied to established surface biofilms rather than prior to upward cell migration. Measurements using the following methodologies were taken at equal time intervals of 2 h at T1, T2 and T3, hence covering a 6 h exposure period typical for the original sample site. Experiments were carried out under ambient light on the roof of the Instituto de Oceanografia de Lisboa, Lisbon, Portugal. Engineered biofilm trays were incubated in temperature controlled water tanks to minimise potential over-heating (maximum temperatures measured at the sediment surface during the experimental period were 35C, comparable to those measured in situ). Light dose was calculated for each sampling point T1, T2, T3 by integrating the light measurements (using a Licor cosine corrected light meter) over the preceeding time period.

Chemical preparation and application

Controls -400 μL of filtered site water was added to all cores to mimic chemical treatments but without addition of DTT or Lat A (see below).

DTT -DL-dithiothreitol (Sigma) was prepared as a fresh stock on the morning of the experimental period. A stock solution of 20 mM (in ethanol) was diluted 100 times in freshly filtered site water to reach a final concentration of 200 µM. 400 μL of this solution were added to each core in order to cover the whole surface of the sediment.

Given the dimensions of the cores, the amount of DTT added in each core was 0.17 µmoles. This amount of DTT was previously determined to be sufficient to virtually fully inhibit the conversion of DD in DT in a 10 µg Chl a mL -1 suspension Phaeodactylum tricornutum (100% inhibition with 0.2 µmol DTT) (Lavaud et al., 2002b).

Latrunculin A -A concentrated Latrunculin A solution (1 mM) was prepared as a fresh stock on the morning of the experimental period by dissolving purified Lat A (Sigma-Aldrich) in dimethylsulfoxide. A solution of 12.5 M Lat A was prepared by dissolving the appropriate amount of the concentrated stock solution in filtered water collected at the sampling site. Small volumes of this solution (total of 300 L) were applied to undisturbed sediment samples by carefully pipetting directly onto the sediment surface, until forming a continuous thin layer that completely covered the sample. The amount of Lat A used was previously determined to be sufficient to virtually inhibit diatom migration in benthic biofilms (Cartaxana and Serôdio, 2008). The inhibitor was applied after the formation of the biofilm at the sediment surface during the period coinciding with the beginning of low tide at the sampling site.

Spectral reflectance

Spectral reflectance was measured with a USB2000 (Ocean Optics, Dunedin, USA) with a VIS-NIR optical configuration controlled by a laptop using OOIBase32™ software. The spectroradiometer sensor was positioned at a 45º angle pointing at the center of the minicore and measuring an approximate area of 1 cm 2 . Reflectance spectra of the target surface were calculated by dividing the upwelling spectral radiance from the sediment surface (L u ) with the reflectance of a clean white polystyrene plate (L d ) both spectra were corrected for dark noise (D n ) (electronic signal measured at total darkness):

(Equation 1)

Reflectance = (L u -D n )/(L d -D n ) (1)
The polystyrene plates differed less than 3% from a calibrated 99% reflectance standard plate (Spectralon) [START_REF] Forster | Field spectroscopy of estuarine intertidal sediments[END_REF]. The normalized vegetation index (NDVI) was calculated as follows:

(Equation 2)

NDVI = (InfraRed -Red)/(InfraRed + Red) (3)
where InfraRed is the average reflectance of the rage 748-752 nm and Red the average reflectance of the range 673-677 nm.

Reflectance derived indices are susceptible to background noise and are not sensitive enough to detect the didinoxanthin (DD) to diatoxanthin (DT) pigment conversion that occur during the xanthophyll cycle. Using diatom cultures [START_REF] Jesus | Detection of diatom xanthophyll cycle using spectral reflectance[END_REF] showed that the conversion of DD to DT causes a reflectance decrease at 508 that is proportional to DT content. However, this decrease was so small that only an index based on the second derivative spectrum was appropriate to detect it. Their DT index (DTI) used the second derivative peak at 508 nm normalized by the second derivative peak at 630 nm and showed very promising results in the determination of diatom DT content. Thus, DTI was used in the current study as a proxy for the DT present at the sediment surface.

The derivative spectra () were calculated using a finite approximation method [START_REF] Louchard | Derivative analysis of absorption features in hyperspectral remote sensing data of carbonate sediments[END_REF], after smoothing the reflectance spectra with a natural cubic spline function (60 nodes). The second derivative () was chosen because in theory it eliminates the background effects and strongly enhances minute changes in the reflectance spectra. This would be ideal in intertidal estuarine sediments where the background signal can be strongly influenced by organic matter, sediment type and moisture. The second derivative spectra were only calculated for the ambient light treatment due to the high noise spectra generated by the lamps used in the constant light treatment.

Rapid light response curves

Rapid light response curves were obtained using a Walz Water-PAM fluorimeter and following the methodology of [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF] except that 20 second light step increments were used rather than 30 seconds due to time constraints. Settings on the Water-PAM were as follows: saturating pulse at setting 10 (approximately 8,600 μmol m - 2 s -1 PAR) for 600 ms duration; light curve settings of 20 second light step duration covering 0 -1035 μmol m -2 s -1 PAR (previously determined as adequate to produce fully saturated light curves for biofilms from this site); due to time restrictions during the experimental period, increasing light level steps using the Water-PAM programming were used rather than preferred decreasing light steps using Win Control [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. Light curve measurements were taken in a random order between chemical treatments, however at each time point, ambient light measurements were made prior to constant light measurements. Once spectral reflectance and fluorescence measurements had been made, the same minicores were destructively sampled for pigment analysis (see below) with care to ensure that the area sampled was not that exposed to the light dose applied by the rapid light curve.

Analysis of rapid light curves also followed that described by [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF] with curve fitting following the iterative solution of [START_REF] Eilers | A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton[END_REF]

Pigment analysis

Approximately 50 mg of freeze-dried sediment were extracted in 95% cold buffered methanol (2% ammonium acetate) for 15 min at -20°C, in the dark. Samples were sonicated (Bransonic, model 1210) for 30 s at the beginning of the extraction period. Extracts were filtered (Fluoropore PTFE filter membranes, 0.2 μm pore size) and immediately injected in in a Shimadzu HPLC with photodiode array and fluorescence (Ex. 430 nm; Em. 670 nm) detectors [START_REF] Cartaxana | Effects of extraction on HPLC quantification of major pigments from benthic microalgae[END_REF]. Chromatographic separation was carried out using a C18 column for reverse phase chromatography (Supelcosil; 25 cm long; 4.6 mm in diameter; 5 m particles) and a 35 min elution programme. The solvent gradient followed [START_REF] Kraay | Separation of chlorophylls c 1 , c 2 , and c 3 of marine phytoplankton by reversed-phase C18 high-performance liquid chromatography[END_REF] with a flow rate of 0.6 mL min -1 and an injection volume of 100 L. Pigments were identified from absorbance spectra and retention times and concentrations calculated from the signals in the photodiode array detector or fluorescence detectors. Calibration of the HPLC peaks was performed using commercial standards from Sigma-Aldrich and DHI (Institute for Water and Environment). Samples were analysed for the xanthophyll pigments DD (the epoxidised form) and DT (the de-epoxidised form). The state of de-epoxidation (DEP in %) was calculated as DT/(DD+DT) x 100%.

Radiolabelled carbon uptake

Total primary productivity (μg C [μg Chl a] -1 h -1 ) was measured from subsamples of 14 C-labelled biofilm. Minicores were incubated in situ with labelled 14 C sodium bicarbonate. One mL (370 Bq) of label was added to each core and allowed to diffuse in the dark for 30 min. After dark diffusion (Smith and Underwood 1998) a 30 min incubation was carried out in both the ambient light and the constant light treatments, terminated by addition of 5% gluteraldehyde. The surface 2 mm depth (approximately) of each minicore was extracted and transferred to an Eppendorf. Sediment samples were later freeze-dried and had inorganic label driven off by addition of concentrated HCl for 24 h. After addition of scintillant cocktail (Optiphase Safe, Fisons, Loughborough, UK), carbon uptake rates were calculated from counts obtained from a Packard Tricarb460C scintillation counter (LKB, Cambridge, UK) with internal quench correction. Counts were corrected for self-quenching by the sediment using radiation standard curves with and without sediment addition. Self quenching reduced counts by 2 to 5%.

Statistical analysis

Significant difference was determined using two factor ANOVA with chemical treatment (Lat A, DTT or controls) nested within light treatment (ambient or constant light) nested within time (T1, 2, and 3). This resulted in 3 replicates for each of the 3 chemical treatments nested within 2 light treatments within 3 time points. Normality and homogeneity of variance of data were tested using the Kolmogorov-Smirnov test followed by Bartletts or Levenes test (for normal or non normal data respectively). If data did not have equal variance then a log transformation was applied [START_REF] Zar | Biostatistical Analysis[END_REF]. In all cases data were normal and non-parametric testing was not required. All tests were applied using Minitab V15 software.

Results

Accumulated light dose

The light dose calculated for ambient light and the constant light treatment (Table 2), showed a slightly higher accumulated dose for the constant light treatment at T1, which was reversed by T2. However it was not until T3 that the difference in light dose between the two treatments was significantly large, with a light dose under ambient light being 2.6 times that under constant light.

Migration

Visual observation of the biofilms showed clear downward migration of cells over the experimental time period except for the Lat A treatment which showed no difference in appearance (Authors pers. obs.). This was largely corroborated by the pigment data (Chl a) which showed clear declines in surface biomass by time T3 (Figure 1) for controls and DTT treatments under constant light (F 2,26 = 25.90, p < 0.01) and under ambient light (F 2,26 = 14.05, p < 0.01), but with no pattern of decline for the Lat A treatment. Migration monitored using the spectral reflectance NDVI index (Figure 2)

showed a similar result, with a decrease in surface biomass under ambient light for all three chemical treatments (F 2,26 = 23.4, p < 0.01) between T2 and T3, although the percentage decline for Lat A was only half that of the DTT treatment and the controls.

Under constant light, no data were obtained for T1, however between T2 and T3 there was a significant (F 2,17 = 18.6, p < 0.05) decline for the controls and DTT treatment, but no decline for the Lat A treatment. Overall, the Lat A clearly inhibited cell vertical migration compared to the other two treatments.

Fluorescence data

There was a significant decrease (F 2,26 = 8.403, p < 0.01) in rETR max over the experimental period for all treatments, although the magnitude of the decline was lower in treatments under constant light compared to those under higher ambient light (Figure 3). There was no significant difference in rETR max between treatments at time T1 or T2, however by T3 the Lat A treatment showed a significantly lower (F 2,26 = 7.444, p < 0.05) value than controls and the DTT treatment for both light treatments. The magnitude of this difference was clearly larger under ambient light compared to constant light. There was no significant difference between controls and the DTT treatment under either light environment.

Under constant light, α showed no significant pattern over time (Figure 4), although in general slight decreases (noticeable most for the DTT treatment) were observed. However under ambient light, α significantly decreased (F 2,26 = 6.281, p < 0.05) in all three treatments, with the decrease for the Lat A treatment being significantly greater (F 2,26 = 6.810, p < 0.05) than either controls or DTT treatments. The value for the Lat A treatment at T3 was essentially zero (0.0005 rel. units compared to an initial value of 0.25 real. units). The light saturation coefficient (E k ) followed exactly the same patterns as described above for rETR max , due to the magnitude of change in rETR max dominating the shape of the light response curves, rather than that of α (note E k = rETR max /α).

Productivity ( 14 C uptake rate)

Due to a high level of variation in values between replicates of the same treatment, no significant differences were observed between chemical treatments in either ambient light or constant light (Figure 5). There was also no significant difference between light treatments, however over time, all data showed a significant decrease (F 2,26 = 15.08, p < 0.01). Productivity did not correlate with rETR max uptake within chemical treatments, although the temporal decline for all data showed a significant correlation (r = 0.63, n = 27, p < 0.05) with rETR max (Figure 6). It should be noted that the 14 C has a lower resolution than the fluorescence methodology, with measurements effectively integrated over the surface 5 mm of the sediment rather than restricted to surface and near surface analysis for the latter method.

Non photochemical quenching (NPQ)

NPQ calculated from the change in maximum fluorescence yield (F m -F m ")/F m ", surprisingly showed negligible induction. In all cases the decline in quantum efficiency (ΔF/F m ") was the result of an increase in F" relative to F m " (Figure 7), F m " initially declined before showing an asymptotic increase. Such a pattern resulted in small values of NPQ (< 0.20) at low light, followed by a decrease to near zero, or often less than zero, at light levels at and above 320 μmol m -2 s -1 PAR (data not shown). DT measured by spectral analysis showed little change in the three treatments by T1 and T2 (Figure 8), however by T3 the DTI values were greater for the non-migratory Lat A treatment compared to the migratory biofilms in both controls and the DTT treatments. This method is under development, but clearly shows a treatment effect for the Lat A treatment regarding NPQ induction compared to the other two treatments. This overall pattern was corroborated by concomitant samples analysed by pigment analysis (Figure 9). Data for pigment analysis expressed as DD de-epoxidation (%), DT/Chl a and DT+DD/Chl a are shown in comparison with corresponding spectral derivative analysis.

These data showed little (non-significant) change under constant low light, however under high light, both DD de-epoxidation and DT/Chl a showed significantly higher values by T3 (F 2,26 = 157.67 p < 0.001) for the Lat A treatments compared to controls and the DTT treatment.

Discussion

These data clearly indicate that, for these biofilms at least, benthic diatoms principally employ vertical migration as their first main mechanism in response to increasing light dose exposure. This is concluded from the significant photoinhibition of the Lat A treated biofilms, with probable enhanced level of physiological down regulation through NPQ, when compared to the two migratory treatments, DTT treated biofilms and controls. In simple terms, cells migrated vertically in response to increasing light dose over time, but when vertical movement was inhibited by Lat A, NPQ induction increased, but not sufficiently to prevent photoinhibition. This is in agreement with the light induced vertical movement (microcycling) proposed by [START_REF] Kromkamp | Determination of microphytobenthos PSII quantum efficiency and photosynthetic activity by means of variable chlorophyll fluorescence[END_REF], [START_REF] Serôdio | Fortnightly light and temperature variability in estuarine intertidal sediments and implications for microphytobenthos primary productivity[END_REF] and [START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF], and also further emphasises the role of vertical movement demonstrated in other experiments using the same chemical inhibitors (Cartaxana and Serôdio 2008;Cartaxana et al. 2008).

Migration was significantly inhibited by the addition of Lat A (Figures 1 and2), in agreement with work by Cartaxana et al. (2008) and Cartaxana and Serôdio (2008).

This was apparent through analysis of Chl a pigment in the surface 2 mm (a comparatively low resolution method) and the surface chlorophyll proxy, NDVI. Both methods showed no major change over the experimental period, whereas for controls and the DTT treated biofilms, significant decreases in biomass were observed. It should be noted that there was no significant difference in biomass between DTT treatment and controls, indicating that DTT did not induce an increase or decrease in cell movement relative to controls. It should also be noted that patterns were largely the same between ambient light and constant light, thus the magnitude of the photodose did not enhance migration. This latter point could have two explanations. Firstly the magnitude of the vertical migration may have been predominantly determined by an endogenous tidal rhythm (e.g. [START_REF] Serôdio | Non destructive tracing of migratory rhythms of intertidal benthic microalgae using in vivo chlorophyll a fluorescence[END_REF]) rather than the light dose. Secondly micro-cycling of cells [START_REF] Kromkamp | Determination of microphytobenthos PSII quantum efficiency and photosynthetic activity by means of variable chlorophyll fluorescence[END_REF][START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF] may have resulted in similar light dose exposure, irrespective of the two light treatments. Thus the integrated light dose of cells cycling through the surface of the sediment was not significantly greater in the ambient light treatment (this being the product of light intensity and length of exposure) compared to cells at lower light level in the constant light treatment. It is likely that both processes played a role in the migratory pattern of the controls and DTT treated cells, however differentiation between these two driving functions was not an explicit aim of this study.

Also it should be noted that comparison of ambient light data at T2 and constant light data at T3, which related to biofilms that had been exposed to similar overall light dose, showed subtle differences in fluorescence values (rETRmax and α), demonstrating that light dose was not the sole driving function of the differences observed. Furthermore NPQ induction were investigated over the 6 h emersion period and hence the resolution of the measurements did not analyse short term patterns in NPQ induction. It is well known that diatoms may rapidly induce NPQ in response to short term (10s of seconds) changes in light environment (e.g. [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. The role of the comparatively long term light dose effect can be noted by the fact that it was not until T3, when the difference in light dose between the two treatments was greatest (Table 2), that differences between the chemical treatments were of highest magnitude.

Over the exposure period, relative maximum electron transport rate (rETR max ) decreased in all treatments (Figure 3). This may have been the result of an endogenous diel rhythm [START_REF] Underwood | Patterns in microphytobenthic primary productivity: Speciesspecific variation in migratory rhythms and photosynthetic efficiency in mixed-species biofilms[END_REF] and / or the effect of the increasing photodose. As the magnitude of the decrease was greatest under ambient light, compared to the lower photodose experienced under constant light, both an endogenous decrease and a photodose effect seem likely. The magnitude of this decrease in rETR max was greatest for Lat A treated biofilms, but only significantly so under ambient light, indicating the inhibition of cell vertical movement resulted in photoinhibition. This pattern was also indicated by the decline to effectively zero by the light use efficiency coefficient (α) for the Lat A treated biofilms under ambient light (Figure 4). Clearly this higher photodose induced photoinhibition (possibly photodamage) when cells were unable to migrate away from the sediment surface. It should be noted that no difference was observed between the DTT treated biofilms and the controls. Therefore it can be concluded that inhibition of NPQ (DTT treatment) had no significant impact whereas inhibition of migration (Lat A treatment) resulted in a reduction in both rETR max and α, but only when the light dose was sufficiently high compared to the constant light treatment.

It is unlikely that the decrease in photosynthetic activity over the experimental period was the result of increasing environmental stress in response to experimental conditions. In fact the use of the water bath may have reduced temperature stress relative to in situ temperature increases, and the biofilms showed no obvious drying out for any of the treatments. In situ warming and desiccation are likely to be equal to or greater than those experienced during this study, thus any temporal pattern is likely to occur under in situ conditions as well. In addition, as all chemical treatments were exposed to the same stress, albeit a lower warming under constant light, experimental induced stresses cannot explain the differences between the Lat A treatment and the controls and DTT treatment.

Productivity, when measured by 14 C uptake rate, showed no chemical or light treatment effects, indeed the only significant pattern observed was an overall temporal decline over the experimental period for the whole dataset. This decline correlated with that of rETR max (Figure 6) supporting the statement above that a combination of diel rhythm and light dose exposure resulted in a decrease in photosynthetic activity. The lack of any chemical treatment effect could be due to two reasons. Firstly the method effectively integrates the productivity measurement over the surface 5 mm depth of sediment, hence resulting in a weighted average value dependent upon the biomass distribution over this depth. Secondly the chemical treatments may not have been fully

active at depth despite the pre-measurement 30 minute incubation period, hence resulting in cell migration towards the surface of cells able to replenish the surface biofilm with photosyntheticaly active cells. The former seems more probable as an explanation as the latter would have resulted in a surface biomass enrichment in the Lat A treatment (i.e. cells would have migrated to the surface and then been unable to migrate back down due to the chemical treatment), which was not observed.

Analysis of the data indicating induction of non-photochemical quenching (NPQ)

is not so clear cut. In all biofilms, the quenching of the photochemical efficiency (ΔF/F m ") was the result of an increase in F' and not a quenching of the F m ' yield. This indicates a low level or even lack of induction of NPQ as indicated by the calculated values (NPQ = (F m -F m ')/F m '). For the migratory biofilms the data must be interpreted with care as downward migration between measurements of F m and F m ') results in an increase in the calculated value of NPQ solely due to the increased distance between the cells and the fluorimeter probe (e.g. [START_REF] Consalvey | PAM Fluorescence: A beginners guide for benthic diatomists[END_REF]Perkins et al. in press).

However this would have increased the magnitude in difference between the nonmigratory (Lat A) and migratory (controls and DTT) treatments. In comparison, both the spectral derivative (Figure 8) and the pigment analysis (Figure 9) for biofilms under ambient light indicated a greater level of NPQ induction in the Lat A treated biofilms.

Under constant light there was no difference between controls and DTT treated biofilms and no difference between chemical treatments. Thus a photodose effect was observed whereby the higher ambient light photodose induced a greater level of NPQ when cell vertical movement was inhibited. Diadinoxanthin de-epoxidation as well as the relative Diatoxanthin (DT) concentration (DT/Chl a) both showed the same patterns. Interestingly there was no increase in (DD+DT) concentration, indicating no de novo synthesis but a conversion of DD to DT as the primary NPQ mechanism. This is an expected result in response to high light exposure [START_REF] Lavaud | General features of photoprotection by energy dissipation in planktonic diatoms (Bacillariophyceae)[END_REF][START_REF] Schumann | Investigation of the quenching efficiency of diadinoxanthin in cells of Phaeodactylum tricornutum (Bacillariophyceae) with different pool sizes of xanthophyll cycle pigments[END_REF]). The lack of any significant effect of DTT treatment compared to controls may imply that the DTT In conclusion, this study has two main findings. Firstly optimization of photosynthetic activity in response to an increasing exposure to light (i.e. an accumulated light dose response) is largely due to vertical cell migration. Cells position themselves in the sediment surface layer such that the attenuation of light provides an optimal light environment for their photochemistry. This is in agreement of the microcycling and light induced vertical migration responses reported by Kromkamp et al. (1998), Serodio andCatarino (2000) and [START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF]. In addition, it goes towards explaining the fact that integrated biofilm light response curves examined in literature seem to saturate at 400 -800 μmol m -2 s -1 PAR (see [START_REF] Perkins | Can chlorophyll fluorescence be used to estimate the rate of photosynthetic electron transport within microphytobenthic biofilms?[END_REF][START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]Serôdio et al., 2003;[START_REF] Consalvey | PAM Fluorescence: A beginners guide for benthic diatomists[END_REF]Jesus et al., 2005, 2006 andothers), significantly lower than ambient light levels at the sediment surface on a sunny day. It seems logical then that cells would position themselves in a light environment nearer to 800 μmol m -2 s -1 PAR or lower, rather than expose themselves to the potentially photodamaging light intensities at the sediment surface. This cell migration may well be more energetically favorable than physiological down regulation processes such as NPQ induction. It is hypothesized from this data that NPQ is a secondary response to light dose and / or a response to more rapid changes in light environment rather than a longer term increase in light dose. Secondly these data suggest that a probable combination of vertical migration and physiological mechanisms result in a diel and/or tidal pattern of down regulation. [START_REF] Underwood | Patterns in microphytobenthic primary productivity: Speciesspecific variation in migratory rhythms and photosynthetic efficiency in mixed-species biofilms[END_REF] reported diel down regulation at the single cell level, and other studies suggest probable tidal patterns for integrated biofilm measurements [START_REF] Perkins | Responses of microphytobenthos to light: primary production and carbohydrate allocation over an emersion period[END_REF][START_REF] Jesus | Spatial dynamics of microphytobenthos determined by PAM fluorescence[END_REF]Jesus et al., , 2006)). Again it is logical that after adequate light exposure for photosynthate production, cells would down regulate their photosynthetic activity. Hence this diel pattern may be a response to integration of the light dose over time, rather than an endogenous rhythm. Hence this study has shown overall, the importance of cell vertical

  photochemical quenching (NPQ) was calculated as the change in maximum fluorescence

  dose was insufficient to inhibit NPQ induction. Certainly under ambient light, pigment data show an induction of NPQ in both these treatments relative to the constant light treatment. However the spectral derivate did not show this pattern, nor did fluorescence data indicate NPQ induction for any treatment. In addition the magnitude of NPQ induction in controls and DTT treatments was significantly less than in the Lat A treatment. Therefore the overall pattern in the combined datasets indicate that cell vertical movement was more important in optimizing photosynthetic activity, rather than NPQ induction.

Figure legends Figure 1 .

 legends1 Figure legends

Figure 2 .

 2 Figure 2. Surface biomass represented as the proxy of Normalised Difference Vegetation

Figure 3 .

 3 Figure 3. Maximum relative electron transport (rETR max ) rate as a proxy for productivity

Figure 4 .

 4 Figure 4. Light utilisation coefficient (α) measured using variable chlorophyll

Figure 5 .

 5 Figure 5. Productivity measured as the uptake rate of labelled carbon ( 14 C-NaHCO 3 ) for

Figure 6 .

 6 Figure 6. Maximum relative electron transport rate (rETR max ) presented as a function of productivity ( 14 C uptake rate) for the whole data set.

Figure 7 .

 7 Figure 7. Operational fluorescence yield (F) and maximum fluorescence yield (F m ")

Figure 8 .

 8 Figure 8. Diatoxanthin Index measured from the spectral second derivatives (508/630

Figure 9 .

 9 Figure 9. Pigment data of A: Diatoxanthin (DT) and the B: Percentage de-epoxidation

Table 1 .

 1 Overview of the experimental design showing the nesting of chemical treatments (Lat A = Latrunculin A to inhibit cell motility; DTT = DL-dithiothreitol to inhibit NPQ and controls) within light treatment (Amb = ambient, Con = constant) within time period (T1, 2, and 3) and the measurements made (Spec = spectroradiometry to measure NPQ induction and surface biomass as NDVI, RLC = rapid light curve by fluorescence, Pig = pigments including Chl a, DD and DT, 14 C = productivity measured as labelled carbon uptake rate). All measurements were made as triple replicates, i.e. 3 separate minicores.

	Time period	Light	Chemical	Measurement,	Measurement,
		Treatment	treatment	minicore set 1	minicore set 2
		Amb	Lat A	Spec, RLC, Pig	14 C
			DTT	Spec, RLC, Pig	14 C
	T1		Controls	Spec, RLC, Pig	14 C
		Con	Lat A	Spec, RLC, Pig	14 C
			DTT	Spec, RLC, Pig	14 C
			Controls	Spec, RLC, Pig	14 C
		Amb	Lat A	Spec, RLC, Pig	14 C
			DTT	Spec, RLC, Pig	14 C
	T2		Controls	Spec, RLC, Pig	14 C
		Con	Lat A	Spec, RLC, Pig	14 C
			DTT	Spec, RLC, Pig	14 C
			Controls	Spec, RLC, Pig	14 C
		Amb	Lat A	Spec, RLC, Pig	14 C
			DTT	Spec, RLC, Pig	14 C
	T3		Controls	Spec, RLC, Pig	14 C
		Con	Lat A	Spec, RLC, Pig	14 C
			DTT	Spec, RLC, Pig	14 C
			Controls	Spec, RLC, Pig	14 C

Table 2 .

 2 Accumulated light dose calculated from the product of light measurement and length of exposure at each sampling time (T1, T2 and T3) for the ambient and constant light treatments. Units of light dose are mole of photons m -2 .

	Sampling	Ambient treatment light dose	Constant treatment light dose
	Time
	T1	1.82	2.16
	T2	7.83	5.40
	T3	20.00	7.56