Robert Alicki

Filippo Giraldi

A quantum model of almost perfect energy transfer

Keywords: numbers: 03.65 Yz, 03.67.-a, 05.60 Gg

The Wigner-Weisskopf-type model describing the energy transfer between two centers mediated by a continuum of energy levels is studied. This work is motivated by the recent interest in transport phenomena at nanoscale in biology and quantum engineering. The analytical estimation for the energy transfer efficiency is derived in the weak coupling regime and the conditions for the almost perfect transfer are discussed. The embedding of the standard tight-binding model into Wigner-Weisskopf one which includes the environmental noise is presented.

Introduction

The almost perfect energy transfer (APET) in biologically relevant molecular systems attracts attention of physicists in the recent years [START_REF] Giraldi | Charge and Energy Transfer Dynamics in Molecular Systems[END_REF]. Similar phenomena should be observed also in the engineered systems relevant for information processing like arrays of quantum dots or Josephson junctions [START_REF] Keating | [END_REF]. Most of the theoretical analysis is performed using tight-binding Hamiltonian for an interacting N-body system in the single excitation regime. The influence of environment is usually modeled by Markovian master equations with phenomenological decay parameters. The numerical computations show the interplay between quantum propagation, quantum localization, decoherence and dissipation [4,3,5]. The aim of this paper is to present a simple, essentially exactly solvable model which describes a generic class of such phenomena and allows to derive analytical bounds on the efficiency of energy transfer processes. The motivation for the applied formalism comes from the particular model of two identical atoms placed in the focuses of two parabolic mirrors separated by a distance ℓ, see Fig. 1. We assume that the dipole moments of both atoms are parallel to the symmetry axis of the system. If initially the first atom is excited and the second is in the ground state we expect that roughly after time t ≃ ℓ/c + O(τ ), where τ is a mean life-time of the excited state, the energy quantum carried by a photon is almost perfectly transfered to the second atom. Obviously APET demands fine-tuning conditions: the atoms should be identical, placed exactly at the focuses of perfectly aligned mirrors and the dipoles should be parallel to the symmetry axis.

The simple description of such a system can be given in terms of the Wigner-Weisskopf (W-W) Hamiltonian (compare the case of a single mirror and a single atom [START_REF] Stobinska | [END_REF]). The conditions which allows for APET are encoded entirely in the structure of the Hamiltonian which is determined by the details of the model. Treating W-W model as a generic one of a large class of energy transfer phenomena between two localized centers one can try to find a general bound for the efficiency of such process and determine the sufficient conditions for APET.

Wigner-Weisskopf model

In order to introduce a 2-state W-W model we consider a quantum system of two 2-level atoms interacting with an electromagnetic field which can be described by the standard Hamiltonian in the rotating wave approximation

H = ω 1 2 σ z 1 + ω 2 2 σ z 2 + d 3 k ω(k) a † (k)a(k) + σ + 1 a(g 1 ) + σ - 1 a † (g 1 ) + σ + 2 a(g 2 ) + σ - 2 a † (g 2 ). (1)
The operators a(g) and a † (g) are smeared annihilation and creation operators for the electromagnetic field given by a † (g refer to 2-level atoms. In the case of identical atoms with relative position described by the vector r the formfactors differ by the relative phase

) = d 3 k g(k)a † (k), g j (k), j = 1,
L(k) = k • r g 2 (k) = e -iL(k) g 1 (k). ( 2
)
The crucial property of the Hamiltonian (1) is the commutation with the excitation number operator defined as

N ex = d 3 k a † (k)a(k) + (σ z 1 + 1/2) + (σ z 2 + 1/2). (3)
The total Hilbert space of the system possesses a tensor product structure

H tot = C 2 ⊗ F ph ⊗ C 2 (4)
with the bosonic Fock space F ph describing an electromagnetic field. As a consequence of the commutation [H, N ex ] = 0 the subspace corresponding to the eigenvalue 1 of N ex , which can be called single exciton Hilbert space is invariant under the evolution. It possesses a direct orthogonal sum structure, i.e., is spanned by the following vectors

|1 ≡ | ↑; 1 ⊗ |vac ⊗ | ↓; 2 , |2 ≡ | ↓; 1 ⊗ |vac ⊗ | ↑; 2 , |f ≡ | ↓; 1 ⊗ a † (f )|vac ⊗ | ↓; 2 for any wave -packet f. (5)
Here, | ↑; j and | ↓; j denote excited and ground state of the j-th atom, respectively, and |vac is the vacuum state of an electromagnetic field. This mathematical construction, which has been used frequently in quantum optics [START_REF] Agarwal | Quantum statistical theories of spontaneous emission and their relation to other approaches[END_REF], will be called the 2-state W-W model. The 2-state W-W model is suitable for the following generic physical situation. We restrict ourselves to physical systems which can be described in terms of the singleexciton Hilbert space H ex . In this Hilbert space we choose two orthogonal vectors |1 , |2 which are determined by the initial state preparation and the measurement procedures. Namely, the initial state ("donor") of the system at time t 0 = 0 is denoted by |1 , while the measurement at time t > 0 after preparation is a von Neumann projection on the state |2 ("acceptor").

The Hilbert space of the model system is decomposed into a direct sum (compare with (5))

H ex = C ⊕ L 2 (Ω) ⊕ C (6)
where the one-dimensional subspaces C are generated by the states |1 , |2 and the Hilbert space L 2 (Ω) is their orthogonal supplement. For the convenience we represent this Hilbert space as the space of wave packets f (k), g(k), k ∈ Ω denoted by |f , |g with the scalar product

f |g = Ω f (k)g(k) dk . (7)
One should stress that the notation of above is used for convenience only. One can always replace the continuous variable k by a joint set of continuous variables ω and discrete quantum numbers m such that |k is replaced by |ω, m and

ω, m|ω ′ , m ′ = δ(ω -ω ′ )δ mm ′ ,
and

Ω dk → m dω. (8)
The 2-state W-W Hamiltonian, which can be seen as the restriction of ( 1) to the singleexciton space, reads

H = H 0 + V, H 0 = ω 1 |1 1| + ω 2 |2 2| + Ω dk ω(k)|k k|, V = (|1 g 1 | + |g 1 1|) + (|2 g 2 | + |g 2 2|) (9)
where g 1,2 ∈ L 2 (Ω). All details of the model are hidden in the form of formfactors g 1,2 and the spectral resolution of the part of the Hamiltonian denoted here by Ω dk ω(k)|k k|.

Remarks The presented model should be treated as an open quantum system where the system consists of donor and acceptor and the continuous modes form a reservoir. Notice that strictly speaking for the 2-atom model presented in the Introduction the cavity formed by the mirrors is finite and hence the modes are discrete. Therefore the W-W Hamiltonian applies to the case of a large enough cavity such that the Heisenberg time t H = 1/δω ( δω -characteristic energy level splitting of modes) is much longer than any relevant time scale. Alternatively, the continuous spectrum can be used if the modes themselves are unstable, due to environmental effects, and have finite life-times comparable to t H .

Efficiency of energy transfer

The system begins its evolution in the excited state |1 and after time t can be found in the state |2 with the probability

P 12 (t) = |A 12 (t)| 2 , A 12 = 2|e -iHt |1 . ( 10
)
The APET holds if ω 1 ≃ ω 2 and for a certain time t 0 the transfer probability P 12 (t 0 ) ≃ 1.

To find the sufficient conditions for APET in terms of the Hamiltonian (9) notice that the (complex) probability amplitude A 12 (t) is a scalar product of two states, e -iHt/2 |1 and e iHt/2 |2 which evolve forward and backward in time, respectively. For t ≫ τ where τ is the life-time of the excited states one expects that the following approximation holds

e -iHt/2 |1 ≃ e -iH 0 t/2 |f 1 , e iHt/2 |2 ≃ e iH 0 t/2 |f 2 (11)
where f 1 (k), f 2 (k) are certain wave packets from the Hilbert space component L 2 (Ω) describing intermediate excitonic states. The estimation (11) is equivalent to the existence of wave operators defined as

W + = lim t→+∞ e iH 0 t e -iHt , W -= lim t→+∞ e -iH 0 t e iHt (12)
which holds under mild conditions on the formfactors g 1,2 and exciton's dispersion relation ω(k), at least in the weak sense of convergence of matrix elements. Combining (11) with ( 12) one obtains

f 1 (k) = k|W + |1 , f 2 (k) = k|W -|2 (13)
and finally

A 12 (t) = k∈Ω dk e -iω(k)t k|W + |1 k|W -|2 . ( 14
)
The advantage of the Wigner-Weisskopf model is the fact that the matrix elements 1|W -|k and 2|W + |k can be exactly computed using Laplace transforms (see the next Section).

The computation of matrix elements

The basic tools used for the computation of the probability amplitude (14) are: the identity

e -iHt = e -iH 0 t -i t 0 ds [e -iω 1 (t-s) |1 g 1 | + |g 1 (t -s) 1| + e -iω 2 (t-s) |2 g 2 | + |g 2 (t -s) 2|]e -iHs (15)
where |g j (t) ≡ e -iH 0 t |g j , and the Laplace transform

f (z) = ∞ 0 e -zt f (t) dt. ( 16
)
Introducing the notation (j, j ′ = 1, 2):

S jj ′ (t) = j|e -iHt |j ′ F jj ′ (t) = g j |e -iHt |j ′ G jj ′ (t) = g j |e -iH 0 t |g j ′ . ( 17
)
and using the identity (15), the definition of the wave operators (12), the notation (17) and the Laplace transform we have

f 1 (k) = k|W + |1 = -i g 1 (k) S11 (-iω(k)) + g 2 (k) S21 (-iω(k)) f 2 (k) = k|W -|2 = i g 1 (k) S21 (-iω(k)) + g 2 (k) S22 (-iω(k)) . ( 18
)
Combining the identity (15) with the definitions (17) we obtain a series of equations

S 11 (t) = e -iω 1 t -i t 0 e -iω 1 (t-s) F 11 (s) ds, S 22 (t) = e -iω 2 t -i t 0 e -iω 2 (t-s) F 22 (s) ds, S 12 (t) = -i t 0 e -iω 1 (t-s) F 12 (s) ds, F 11 (t) = -i t 0 G 11 (t -s)S 11 (s) ds -i t 0 G 12 (t -s)S 21 (s) ds, F 22 (t) = -i t 0 G 22 (t -s)S 22 (s) ds -i t 0 G 21 (t -s)S 12 (s) ds, F 12 (t) = -i t 0 [G 11 (t -s)S 12 (s) + G 12 (t -s)S 22 (s)] ds, (19)
which can be converted into equations for Laplace transforms

S11 (z) = 1 z + iω 1 [1 -i F11 (z)], S22 (z) = 1 z + iω 2 [1 -i F22 (z)], S12 (z) = -i z + iω 1 F12 (z), S21 (z) = -i z + iω 2 F21 (z), F11 (z) = -i G11 (z) S11 (z) -i G12 (z) S21 (z), F22 (z) = -i G22 (z) S22 (z) -i G21 (z) S12 (z), F12 (z) = -i[ G11 (z) S12 (z) + G12 (z) S22 (z)] F21 (z) = -i[ G21 (z) S11 (z) + G22 (z) S21 (z)]. ( 20
)
The system of equations (20) can be solved yielding the relevant Laplace transforms

S11 (z) = 1 z + iω 1 + G11 (z) 1 + G12 (z) G21 (z) W (z) S12 (z) = -G12 (z) W (z) S21 (z) = -G21 (z) W (z) S22 (z) = 1 z + iω 2 + G22 (z) 1 + G12 (z) G21 (z) W (z) (21)
where

W (z) = (z + iω 1 + G11 (z))(z + iω 2 + G22 (z)) -G12 (z) G21 (z). ( 22
)
Combining ( 21) with ( 18) and ( 14) one can obtain the exact final formula for the transition amplitude which can be used for numerical calculations. The formulas can be essentially simplified if one neglects all terms containing G12 (z), G21 (z). This approximation means that the interaction between donor and acceptor mediated by the reservoir is neglected as well as the collective contribution to their relaxation processes.

In the case of two atoms in a large cavity it means that the dipole-dipole interaction and the "superradiance/subradiance" effects [10] are neglected.

Markovian approximation

In order to continue analytical analysis of the problem we consider the case of weak coupling or Markovian approximation and neglect the collective effects as discussed above. The further approximation concerns Laplace transforms Gjj (z). The real part of Gjj (-iω), taken at the physical frequency is the standard Fermi Golden Rule approximation for the decay rate of the state |j (j = 1, 2) given by

γ j = π Ω dk |g j (k)| 2 δ(ω(k) -ω ren j ), (23)
while the imaginary part is a "radiative correction" to the bare frequencies ω j yielding a physical renormalized frequency

ω ren j = ω j - Ω dk |g j (k)| 2 P 1 ω(k) -ω ren j . (24)
Notice, that the equation ( 24) has a self-consistent character because ω ren j appears on both sides of it. It is due to the fact that we cannot expect that the "radiative correction" is small and hence it is not justified to put the bare value ω j on the RHS of (24) and (23).

Summarizing, the Markovian approximation means that

S21 (-iω(k)) ≃ 0 , Sjj (-iω(k)) ≃ [i(ω ren j -ω(k)) + γ j ] -1 , γ j ≪ ω j . (25)
Therefore using Eqs. 18) and (25) one obtains

f 1 (k) = g 1 (k) ω(k) -ω ren 1 + iγ 1 , f 2 (k) = g 2 (k) ω(k) -ω ren 2 -iγ 2 (26)
and

A 12 (t) = k∈Ω dk e -iω(k)t g 1 (k)g 2 (k) (ω(k) -ω ren 1 + iγ 1 )(ω(k) -ω ren 2 + iγ 2 ) ( 27
)
One can estimate the upper bound for P 12 (t) using ( 27) to get

P 12 (t) = |A 12 (t)| 2 ≤ f 2 2 f 1 2 . ( 28
)
The equality

∞ -∞ γ[x 2 + γ 2 ] -1 dx = π implies for example f 1 2 = Ω |f 1 (k)| 2 dk = Ω dk |g 1 (k)| 2 (ω ren 1 -ω(k)) 2 + γ 2 1 ≃ 1 π ∞ 0 γ 1 (ω ren 1 -ω) 2 + γ 2 1 ≃ 1 - γ 1 πω 1 ( 29
)
what finally gives the bound (compare with [8])

P 12 (t) ≤ 1 - γ 1 πω ren 1 1 - γ 2 πω ren 2 < 1.
(30)

In the weak coupling regime the upper bound is close to one.

Conditions for APET

In order to approach the bound (30) and achieve APET certain matching conditions implied directly by the formula ( 27) must be satisfied. To present them in a transparent form we introduce certain additional assumptions in a convenient parametrization.

There are satisfied by the two-mirror system discussed in the Introduction.

Assume that the intermediate Hilbert space L 2 (Ω) is spanned by the basis |ω, m and the corresponding part of the Hamiltonian reads

H 1 = m dω ω |ω, m ω, m|. ( 31
)
The formfactors possess the following structure

|g 1 = dω g(ω)|ω, m 0 , |g 2 = dω e -iL(ω) g(ω)|ω, m 0 . ( 32
)
where g(ω) is a smooth enough formfactor and L(ω) accounts for the spatial separation of donor and acceptor similarly to [START_REF] Keating | [END_REF]. Now the mechanism leading to APET can be illustrated. We have to put the resonance condition

ω ren 1 = ω ren 2 ≡ ω 0 , (33)
which by (23) implies also the equality of decay rates

γ 1 = γ 2 ≡ γ. ( 34
)
Then we have the approximative expression

A 12 (t) ≃ 1 π dω γ (ω -ω 0 ) 2 + γ 2 ω -ω 0 -iγ ω -ω 0 + iγ e i(L(ω)-ωt) . ( 35
)
Further approximations valid for |ω -

ω 0 | ≤ γ ω -ω 0 -iγ ω -ω 0 + iγ ≃ -e i2(ω-ω 0 )/γ , L(ω) ≃ L(ω 0 ) + L ′ (ω 0 )(ω -ω 0 ) (36) lead to |A 12 (t)| ≃ 1 π | dω γ (ω -ω 0 ) 2 + γ 2 exp{i[L ′ (ω 0 ) + 2τ -t](ω -ω 0 )}| (37)
where τ = 1/γ. The transition probability is maximal and close to one if the oscillating factor in the integral of above vanishes what happens for the time

t 0 = L ′ (ω 0 ) + 2τ (38)
which can be called the optimal transfer time.

Transport in quantum networks

The standard tight-binding model of energy transport in quantum networks which is applicable both to molecular systems and engineered ones consist of N 2-level "atoms" described by the Hilbert space C 2 N and the Hamiltonian written in terms of Pauli matrices

H N = N k=1 ω k σ z k + N k<l=2 (h kl σ + k σ - l + h.c.) (39)
where {ω k } are energies of the sites and {h kl ; k < l} are hopping amplitudes. Similarly to the example in the Section 2 the exciton number operator

N ex = N k=1 (σ z k + 1/2) (40)
commutes with H N . Again a single exciton N -dimensional Hilbert space C N is invariant with respect to the dynamics and the corresponding restriction of the Hamiltonian H N is given by the N × N hermitian matrix [h kl ] with the diagonal elements h kk = ω k . We attribute to the donor site an index "1" and to the acceptor site an index "2". The famous example of such a model is the standard description of the exciton transfer in the Fenna-Matthews-Olson (FMO) complex which involves N = 7 centers and the experimentally determined 7 × 7 single-exciton Hamiltonian [11,12]. To describe the influence of environment different schemes have been proposed mainly based on the Markovian master equations for the reduced density matrix of the exciton system [3,4,5,9]. Here we propose a different approach based on the W-W model where only donor and acceptor form an open system while the other sites are included into a bath.

We assume that the direct hopping 1 ↔ 2 is negligible, i.e. h 12 = h 21 = 0. We have to add also the site N + 1 with the energy ǫ N +1 which represents a sink coupled only to the acceptor by the matrix element h 2(N +1) . Under these conditions the extended Hamiltonian can be recast into the discrete version of the W-W Hamiltonian

H (d) = ω 1 |e 1 e 1 | + ω 2 |e 2 e 2 | + (|e 1 g 1 | + |e 2 g 2 | + h.c.) + H (d) 1 . (41)
Here

|e 1 = [1, 0, ..., 0], |e 2 = [0, 1, ..., 0] , |g 1 = [0, 0, h 13 , ...h 1N , 0] ,|g 2 = [0, 0, h 23 , ...h 2N , h 2(N +1) ] and H (d)
1 is a submatrix of H (d) with indices k, l = 3, 4, ..., N, N + 1.

H (d) 1
can be written in its spectral decomposition form

H (d) 1 = N +1 α=3 ǫ α |α α| (42)
where |N + 1 = [0, ...0, 1]. The energies ǫ α are chosen in a decreasing order what is consistent with the intuition that the energy of a sink ǫ N +1 should be the lowest one.

Notice that (41),(42) can be treated as a discrete version of the continuous spectrum Hamiltonian (9). In the next step we embed the discrete model into a continuous one. Physically, it means that the interaction with other (environmental) degrees of freedom is taken into account. This interaction transforms the eigenstates |α into resonances with finite spectral widths. The Hilbert space spanned by {|α } is replaced by L 2 (R) and the Hamiltonian (42) by the multiplication operator by ω which can be formally written as H 1 = dω ω |ω ω|. Consider the correlation functions characterizing the interaction of the donor-acceptor system with the bath consisting of other sites including a sink site

G (d) jj (t) = g j |e -iH 1 t |g j = N +1 α=3 | α|g j | 2 e -iǫαt , j = 1, 2. (43)
In order to include the interaction with the other degrees of freedom, e.g. vibrational modes for molecular systems like FMO, we assume that the discrete modes |α are unstable and the correlations decay with the decoherence rates Γ α . The modified correlation functions corresponding to the continuous model read

G jj (t) = N +1 α=3 | α|g j | 2 e -iǫαt e -Γα|t| , j = 1, 2. ( 44
)
One can replace the discrete formactors α|g j by the suitable wave functions g j (ω) from

L 2 (R) satisfying Re ∞ 0 e iωt G jj (t) dt = N +1 α=3 | α|g j | 2 Γ α (ω -ǫ α ) 2 + Γ 2 α = π|g j (ω)| 2 . (45)
The condition (45) determines only the modulus of g j (ω). The relative phase L(ω) should be constructed as an interpolation between discrete relative phases for α|g 1 and α|g 2 defined at the points {ω = ǫ α }.

Energy transfer controlled by decoherence

The search for the condition which should be satisfied to achieve APET in quantum networks can be treated as a control problem for quantum open systems. One of the possible settings is to treat the tight-binding Hamiltonian (39) as given and optimize the choice of decoherence rates {Γ α }, acceptor-sink coupling h 2(N +1) and the sink energy ǫ N +1 to maximize the transition probability P 12 (t 0 ). The advantage of the presented model is a deeper insight into the mechanism of APET based on the discussion in the Section 6 where analytical formulas are used. The first condition for APET is a fine-tuning one, namely the resonance condition (33). It can be discussed using the following formula for the renormalized frequencies obtained by putting (44) into (24) One can expect that this condition can be satisfied generically for the widths of the resonances comparable to the typical nearest neighbor energy spacing.

ω ren j = ω j - N +1 α=3 | α|g j | 2 ǫ α -ω
The optimal transfer time estimated from the formula (38) reads

t 0 ≃ π ǫ α 0 -ǫ α 0 +1 + 2 π|g 1,2 (ω 0 )| 2 (47)
where ǫ α 0 ≥ ω 0 ≥ ǫ α 0 +1 .

Concluding remarks

The 2-state Wigner-Weisskopf model provides a versatile mathematical tool to study, within the single exciton approach, the transport properties in complex molecular systems or engineered networks relevant for quantum information processing, both in the weak coupling (Markovian) and strong coupling (non-Markovian) regimes. The preliminary results for the FMO model (work in progress) show that the presented here APET conditions can be achieved under realistic assumptions concerning the decoherence rates and sink parameters. The necessity of fine-tuning to achieve APET in disordered molecular networks has been already noticed in [9] where sampling over many molecular configurations has been performed. However, the presented here formalism allows to design the optimal parameters and shows the details of the physical mechanisms leading to APET. In the case of engineered networks the fine-tuning conditions imply a careful design while for biologically relevant systems one can imagine that the natural selection mechanism plays a crucial role.

Figure 1 .

 1 Figure 1. The APET for two atoms places in the focuses of two parabolic mirrors. Radiating dipoles are parallel to the axis.

  2, are suitable formfactors, and the quantum fields satisfy [a(k), a † (k ′ )] = δ(kk ′ ) (we omit for simplicity the polarization degrees of freedom). The standard 2 × 2 Pauli matrices σ ±,z 1,2

  from (34) which implies that |g 1 (ω)| and |g 2 (ω)| should essentially coincide in the neighborhood of ω 0 of the width γ ≃ π|g 1 (ω 0 )| 2 ≃ π|g 2 (ω 0 )| 2 .

Acknowledgments The authors thank Ilya Sinayskiy for the assistance. R.A. is supported by the Polish Ministry of Science and Higher Education, grant PB/2082/B/H03/2010/38.


Núria Cerdà-Costa

Jaume Bonet

M Rosario Fernández

Francesco X Avilés

Baldomero Oliva

Sandra Villegas
email: sandra.villegas@uab.cat

Prediction of a new class of RNA recognition motif

Keywords:

The observation that activation domains (AD) of procarboxypeptidases are rather long, compared to pro-regions of other zymogens, points out the possibility that they could play additional roles apart from precluding enzymatic activity within the proenzyme and helping in its folding process. In the present work, we have compared the overall pro-domain tertiary structure with several proteins belonging to the same fold in SCOP by using structure and sequence comparisons. The best score has been obtained between the activation domain of the human procarboxypeptidase A4 (ADA4h) and the human U1A protein from the U1 snRNP.

Structural alignment has revealed the existence of RNP1-and RNP2-related sequences in ADA4h. After modeling ADA4h on U1A, the new structure was used to extract a new sequence pattern characteristic for important residues at key positions. The new sequence pattern allowed scanning protein sequences to predict the RNA-binding function for 32 sequences undetected by PFAM. Unspecific RNA EMSA experimentally supported the prediction that ADA4h binds an RNA motif similar to the U1A binding-motif of stem-loop II of U1 small nuclear RNA. The experiments carried out with ADA4h in the present work

Manuscript

Click here to download Manuscript: JMMO1391R2_final.doc suggest the sharing of a common ancestor with other RRMs. However, the fact that key residues preventing activity within the proenzyme are also key residues for RNA binding might induce the activation domains of procarboxypeptidases to evolve from the canonical RNP1 and 2 sequences.

Introduction

Carboxypeptidases (CPs) are proteases that hydrolyze C-terminal peptide bonds from peptides and proteins. A relatively large number of different carboxypeptidases exist, covering a variety of specificities and complementing each other when acting in the same location [START_REF] Vendrell | Metallocarboxypeptidases. Handbook of Metalloproteins[END_REF].

The most used classification of proteases can be found at the MEROPS database [START_REF] Rawlings | MEROPS: the peptidase database[END_REF], in which metallocarboxypeptidases belong to the M14 family. The M14 family can be divided into two main different subfamilies according to their sequence homology, the N/E subfamily and the A/B subfamily.

The A/B subfamily has been the most studied one, with the bovine carboxypeptidase A (CPA) as the reference model. It comprises enzymes produced as zymogens, having an N-terminal activation segment (or pro-region), about 95 residues long, which inhibits the enzyme's activity until it is released due to limited proteolysis. They are usually synthesized in the pancreas and transported to the duodenum where they participate in food processing; nonetheless, some members, like CPU and CPA3, originate in other tissues and act in alternative processes such as blood coagulation [START_REF] Hendriks | Carboxypeptidase. Handbook of Proteolytic Enzymes[END_REF] or anaphylactic and inflammatory responses [START_REF] Springman | Mast cell procarboxypeptidase. A Molecular modeling and biochemical characterization of its processing within secretory granules[END_REF], respectively.

The activation segment, or pro-region, found in the A/B subfamily is remarkably long, constituting one-fourth of the entire proenzyme. It is composed of an activation domain (AD), with a globular fold independent of the catalytic domain, and a connecting -helix that covalently holds them together.

The sequence similarity among different CPs pro-regions is quite low (20% to 50%), the residues involved in the enzyme inhibition being the most conserved ones; however, they share an almost identical + topology formed by two -helices and four -strands arranged in an open-sandwich conformation. As a result, all of the activation domains with a resolved 3-D structure belong to the same family in the Structural Classification Of Proteins (SCOP) database (Pancreatic carboxypeptidase, activation domain) [START_REF] Murzin | SCOP: a structural classification of proteins database for the investigation of sequences and structures[END_REF]; http://scop.mrclmb.cam.ac.uk/scop/).

The fact that the activation segments are relatively long, as compared to pro-regions of other zymogens, brought up some questions about the possibility that they could play other roles apart from precluding enzymatic activity within the proenzyme. Folding studies performed in ADA2h showed an extremely fast two-state kinetics [START_REF] Villegas | The activation pathway of procarboxypeptidase B from porcine pancreas: participation of the active enzyme in the proteolytic processing[END_REF], suggesting that ADs could assist in the proenzyme folding. Supporting this hypothesis, recombinant expressions of various forms of CPs in the methylotrophic yeast Pichia pastoris have been proved unsuccessful, whereas the proenzyme forms can be produced at high yields.

One of the first set of experiments studied the possible Ca 2+ binding properties, since some structural similarity with the EF-hand protein family had been found [START_REF] Vilanova | Analysis of the conformation and ligand-binding properties of the activation segment of pig procarboxypeptidase A[END_REF]. No significant results regarding Ca 2+ binding were obtained and, later on, the similarity was also discarded using bioinformatic sequential and structural approaches, although some homologies with complement factor B were proposed [START_REF] Vendrell | Procarboxypeptidase A activation segment compared to structures of other proteins[END_REF].

In the present work, the overall pro-domain tertiary structure has been compared with several proteins belonging to the same fold in SCOP. The sequence alignment derived from the structure comparison was used to extract a sequence pattern characteristic for important residues at key positions. The sequence pattern was further used to scan protein sequences to detect RNA-binding proteins. Subsequently, the RNA-binding capability of ADA4h was tested experimentally in order to confirm our prediction.

Methods

Bioinformatics

The search of remote homologs of ADA4h was performed with three iterations of PSI-BLAST [START_REF] Altschul | Gapped BLAST and PSI-BLAST: a new generation of protein database search programs[END_REF] searching in Uniprot [START_REF] Garcia-Castellanos | Detailed molecular comparison between the inhibition mode of A/B-type carboxypeptidases in the zymogen state and by the endogenous inhibitor latexin[END_REF], using the BLOSUM45 matrix and standard parameters.

After the first iteration, the position-specific scoring matrix (PSSM) included the first hit sequence not belonging to the procarboxypeptidase family in order to find remote homologs of ADA4h. A second search was performed on PDB [START_REF] Berman | The Protein Data Bank[END_REF] (using the same parameters) searching remote homologs with structure.

The 3-D structure of ADA4h was extracted from the structure of the human procarboxypeptidase A4 (code 2BOA [START_REF] Garcia-Castellanos | Detailed molecular comparison between the inhibition mode of A/B-type carboxypeptidases in the zymogen state and by the endogenous inhibitor latexin[END_REF]) retrieved from the PDB [START_REF] Berman | The Protein Data Bank[END_REF]. ADA4h belongs to the pancreatic carboxypeptidase, activation domain family, in the ferredoxin-like fold, from SCOP [START_REF] Murzin | SCOP: a structural classification of proteins database for the investigation of sequences and structures[END_REF]. All proteins with a similar fold were retrieved from PDB, and all protein domains with these folds were extracted from them. The Structural Time Series Analyser Modeller and Predictor (STAMP) program [START_REF] Russell | Multiple protein sequence alignment from tertiary structure comparison: assignment of global and residue confidence levels[END_REF] was used to superimpose AD folds and to compare their structures by means of the root mean square deviation (RMSD).

The capability of ADA4h of changing the orientation of some of its loops to simulate the function of a different family of the same ferredoxin-like fold was tested via comparative modeling using MODELLER [START_REF] Sali | Comparative protein modelling by satisfaction of spatial restraints[END_REF] and the selected templates from the ferredoxin-like fold.

Inside this fold there are many proteins with functions unrelated to the carboxypeptidase precursor. Among them, many belong to several superfamilies with the nucleotide-recognition function. Particularly interesting is one superfamily whose function is related to RNA binding and that can be identified by an RNA Recognition Motif (RRM). This motif is defined in PROSITE [START_REF] Hulo | The PROSITE database[END_REF] by means of two sequence signatures called as RNP1 and RNP2, separated by about 30 residues, and it is supposed to bind single-stranded RNA. Due to the structural similarity among ADA4h and the proteins of the families that contained this motif, we were able to identify similar sequence patches (RNP1 and RNP2) in ADA4h, also separated by approximately 30 residues. Previous work had shown the possibility to infer functional relationships within the same SCOP fold by distant homology [START_REF] Aloy | Structural similarity to link sequence space: new potential superfamilies and implications for structural genomics[END_REF] and the importance of functional loops to imprint a particular function [START_REF] Espadaler | Identification of function-associated loop motifs and application to protein function prediction[END_REF]. Therefore, we hypothesize a new motif based on the RNP1 and RNP2 sequences of RRM that could also include the sequence patches of ADA4h. The new motif was tested by searching proteins containing the new motif in the Uniprot database with ScanProsite [START_REF] De Castro | ScanProsite: detection of PROSITE signature matches and ProRule-associated functional and structural residues in proteins[END_REF].

Materials

The MEGAscript ® T7 Kit for transcription, the RNase inhibitor SuperaseIn and the RNase decontamination solution RNaseZAP ® were obtained from Ambion. The RNA purification columns Micro Bio-Spin were purchased from Bio-Rad, the [-32 P]UTP from GE Healthcare, and the C-UVette ® DNase RNase Protein-free from Eppendorf. Primers were synthesized by Roche Molecular Biochemicals. All other chemicals were obtained from SIGMA.

Cloning and expression

ADA4h was cloned into the pET-30 Xa/LIC vector from Novagen resulting in an N-terminal His-tagged protein. The recombinant vector was then transformed into the Escherichia coli strain BL21 (DE3), and the obtained clones were screened for positives.

Cells were grown and induced as previously described [START_REF] Viguera | Favourable native-like helical local interactions can accelerate protein folding[END_REF] and ADA4h was purified from the intracellular soluble fraction using nickel sepharose. A 60-mM imidazole wash was applied once the sample was loaded and the column was re-equilibrated, followed by an elution step with 300 mM imidazole. The elution fractions containing His-tagged ADA4h were dialyzed overnight against binding buffer, and then were again applied to nickel sepharose in a batch manner. Nickel sepharose-bound protein was digested overnight at room temperature with the necessary amount of Factor Xa, in order to eliminate the His-tag. The protein was recovered in the supernatant and kept at 0ºC in 50 mM Tris-HCl 0.5 M NaCl, pH 8.0. The identity of the protein was confirmed using MALDI-TOF mass spectrometry and N-terminal sequencing.

In vitro transcription of RNA

An RNA probe carrying two copies of the U1A-binding motif from stem-loop II of U1 snRNA hairpin [START_REF] Allain | Structural basis of the RNA-binding specificity of human U1A protein[END_REF] was synthesized by in vitro transcription and named bipolar-stem-loop: gggAgggUUAACAUUgCACUCCgUUgUCCAUCCCAAAAAAAAAAAgggUUAACAUU gCACUCCgUUgUCC.

The online program OligoAnalyzer 3.0 from Integrated DNA Technologies  (http://www.idtdna.com/analyzer/Applications/OligoAnalyzer/Default.aspx) was used to test the secondary structure.

In order to obtain the afore mentioned RNA probe, a direct primer carrying the T7 RNA polymerase promoter was designed: 5'-TAATACGACTCACTATAGGGAGGGTTAACATTGCACTCCGTTGTCCATCCCAAAA AAAAA-3' partially overlapping with the reverse one, 5'-GGACAACGGAGTGCAATGTTAACCCTTTTTTTTTTTGGGATGGACAACGG-3'. Both primers, 1 nmol of each, were incubated with 4 U of Klenow and 0.6 mM dNTPs at 37ºC for 1h, and the enzyme was inactivated afterwards at 65ºC for 10 min. The DNA was then analyzed in a 2% agarose gel and quantified.

In vitro transcription reactions were performed essentially as indicated in the MEGAshortscript ® T7 Kit. The reaction was performed in 10 l, using a 1:10 ratio for UTP, with respect to other NTPs, adding 40 Ci of [-32 P]UTP, and incubating at 37ºC for 2h. A second in vitro transcription in the absence of [-32 P]UTP was carried out as indicated in the kit instruction manual and used as the cold probe. The amount of DNA template used in both reactions was 125 nM. In both cases, after the transcription reaction, 10 U of RNase-free DNaseI were added and the mixture was incubated for 15 min at 37ºC. The RNA obtained was then purified using Bio-Rad Micro Bio-Spin columns P-30 Tris RNase-Free. The unlabeled RNA was quantified by UV-spectrometry using C-UVette ® DNase RNase Proteinfree and the radioactive probe by scintillation counting.

RNA electrophoretic mobility shift assays (EMSA)

Twenty-five fmol of labeled RNA were incubated in a final volume of 10 l for 20 min at room temperature with the indicated amounts of protein in a buffer containing 10 mM Hepes, pH 7.4, 25 mM potassium acetate, 2.5 mM magnesium acetate, 0.5% Igepal CA 630, 5%

glycerol, 1 mM dithiothreitol and 1 U/l of SuperaseIn [START_REF] Tang | Identification of zeta-crystallin/NADPH:quinone reductase as a renal glutaminase mRNA pH response element-binding protein[END_REF]. Yeast tRNA was also added in competition assays (0.5 g of tRNA; probe:competitor ratio 1:2 x10 7 (w:w)).

RNA probes were previously heated at 95ºC in order to disrupt their secondary structure and left 5 min on ice so that their proper conformation was finally achieved. EMSA were performed using one-phase 5% polyacrylamide gels with TBE buffer (90 mM Tris-HCl, pH 8.5, 110 mM boric acid, 2 mM EDTA) as running buffer and were left at 60-70 V for 90 min.

The gel was then fixed with 20% ethanol, 10% acetic acid, for 30 min, introduced into a plastic-bag and visualized using a Fluor-S TM MultiImager (Bio-Rad).

Results

ADA4h particularities

First of all, a computational characterization of the available activation domains (ADs) was carried out using online tools such as ProtParam [START_REF] Gasteiger | Protein Identification and Analysis Tools on the ExPASy ServerProtein Identification and Analysis Tools on the ExPASy Server[END_REF] from the Expasy webpage (http://www.expasy.org/tools/protpar-ref.html). All domains show a conserved length, although their composition is quite different. The pI for all domains is below neutrality; nonetheless, ADA4h presents a fairly basic pI (9.30) compensated by a highly acidic connecting helix. Thus, the entire activation segment retains an acidic pI, similar to the other procarboxypeptidases. Apart from ADA4h, only the ADs of the recently described procarboxypeptidases A5 and A6 and the AD from PCPU approach neutrality [START_REF] Pereira | Human procarboxypeptidase B: three-dimensional structure and implications for thrombin-activatable fibrinolysis inhibitor (TAFI)[END_REF][START_REF] Wei | Identification and characterization of three members of the human metallocarboxypeptidase gene family[END_REF].

Structural search

A significant sequence alignment was found (e-value 1e-17) using Position-Specific Iterated Basic Local Alignment Search Tool (PSI-BLAST) between ADA4h and the U1 SNP1associating protein (Q03714), a U1-type yeast-protein involved in mRNA splicing.

The structure of ADA4h was assigned to the SCOP family pancreatic carboxypeptidase, activation domain within the ferredoxin-like fold. However, the structure of U1 SNP1associating protein (Q03714) is not available in the Protein Data Bank (PDB). Therefore, a PSI-BLAST search on PDB was performed in order to find a structural template for Q03714.

We found a significant alignment between Q03714 and several domains from de canonical RNA-Binding-Domain (RBD) family of SCOP (psi-blast e-values between 7e-33 and 3e-19), suggesting some evolutive relationship between Q03714 and this SCOP family.

Consequently, the domain of U1 SNP1-associating protein was assigned to the canonical RNA-binding-domain (RBD) SCOP family, also in the ferredoxin-like fold. We selected as template for Q03714 the human U1 small nuclear ribonucleoprotein A (P09012) from the members of this family, with code 1M5Kc [START_REF] Rupert | Crystal structure of a hairpin ribozyme-inhibitor complex with implications for catalysis[END_REF] in the PDB. This protein was crystallized in interaction with an RNA molecule. However, instead of modeling the structure of Q03714 with 1M5Kc, we straight compared ADA4h with 1M5Kc. The structural superposition between ADA4h pro-domain and 1M5Kc produces a sequence alignment that proves a 14% identity and covers 58% of sequence similarities according to the BLOSUM45 matrix (Fig. 1a).

This superimposition allowed us to allocate the RNA molecule, co-crystallized with U1A in 1M5Kc, and to assume the putative binding between RNA and the activation domain of human procarboxypeptidase A4 (Fig. 1b). Although the superimposition shows a putative binding region between ADA4h and RNA with two sequence patterns similar to the RNA Recognition Motif (RRM) of U1A (Fig. 1a), there are conformational restrictions to accommodate the backbone chain of ADA4h to perform similar binding. The major differences between both backbone conformations are found in loops 1 and 3. We postulate that the orientation of these loops would be dependent on the partner molecule that interacts with ADA4h, where the actual conformation of ADA4h binds the catalytic domain within procarboxypeptidase A4 while on the interaction with RNA it would adopt a conformation similar to U1A. In agreement with this hypothesis, the main motions of the binding region studied in a previous work [START_REF] Gargallo | Molecular dynamics simulation of highly charged proteins: comparison of the particle-particle particle-mesh and reaction field methods for the calculation of electrostatic interactions[END_REF] shows the correlation between the RNP1 and RNP2 sequences patches and loops 1 and 3. In order to test this hypothesis, ADA4h was modeled using the conformation of U1A as template and the structural alignment of the superimposition where the putative RNA-binding regions found on ADA4h corresponded to the RRM patterns of U1A (Fig. 2). The possibility that the interaction with RNA was due to the positive charge of ADA4h was also assessed. The comparison with U1A suggested that ADA4h had less positive charges on its surface than U1A (Fig. 3).

The new conformation of ADA4h (ADA4hm) was evaluated with the pseudo-energies of ProsaII [START_REF] Sippl | Recognition of errors in three-dimensional structures of proteins[END_REF], the Z-score based on split-potentials recently developed in our group [START_REF] Aloy | Splitting statistical potentials into meaningful scoring functions: testing the prediction of near-native structures from decoy conformations[END_REF],

QMEAN [START_REF] Benkert | QMEANclust: estimation of protein model quality by combining a composite scoring function with structural density information BMC[END_REF], Gromos [29] and ANOLEA [30] (see supplementary material). According to ProsaII statistical potentials, the modified orientation of loops 1 and 3, whose flexibility has been described before [31], was energetically possible but not favorable (Fig. 4a,4b). First, the comparison with the energies of the template U1A (Fig. 4a) shows the conflictive regions of the model with respect to the conformation of the original sequence that produces this fold;

second, the comparison between modified and unmodified conformations of ADA4h (Fig. 4b)

shows that the conformation can endure the re-orientation of the first helix and of loops 1 and 3. Furthermore, to check if the energetic conflicts of this conformational change were due to the side-chain packing or to the backbone conformation, we used the split-potentials recently developed in our group [START_REF] Aloy | Splitting statistical potentials into meaningful scoring functions: testing the prediction of near-native structures from decoy conformations[END_REF]. Z-scores based on split-potentials of ADA4h and ADA4hm were calculated using pair-residue distances between C carbons (Fig. 4c) and the minimum distance between the atoms of each two residues (Fig. 4d). The location of the energy differences between the conformations of ADA4h and ADA4hm, calculated with energies that take into account the side-chain packing (Fig. 4d) or only the backbone conformation (Fig. 4c), were similar. This indicates that the conformational conflicts detected in the modified conformation of ADA4hm are mainly due to the change of the backbone, but not to the new residue packing. We further compared the disposition of the side-chains of ADA4h in the binding region with the RRM residues of U1A to check whether they might perform similar binding with RNA (Table 1, Fig. 5). Among the key residues for RNA binding by U1A, the most important are Arg52, Gln54 and Phe56 belonging to RNP1 and Tyr13 belonging to RNP2. On ADA4h, these residues are substituted by Arg45, Pro46 and Asp48 for RNP1, and Arg11 for RNP2 (Table 1).

The interactions of Arg52 in U1A are mostly produced by Arg45 in ADA4h (Fig. 5b). Also, Arg11 (ADA4h) produces a π interaction similar to that of Tyr13 (U1A) with C40 of the RNA chain. This interaction is improved in the modeled structure (Fig. 5d).

The aromatic stacking of Phe56 (U1A) with A41 of the RNA [32] would be weakened if the interaction were carried out by Asp48 of ADA4h (Fig. 5c). Finally, Gln54 orientation cannot be emulated by Pro46 of ADA4h (Fig. 5a). Both interactions could occur with a different RNA substrate (see further discussion).

Pattern discovery

From the structural alignment, a pattern was identified performing the RNA-recognition function. The RRM identified in 1M5Kc has been described as a combination of two regions (RNP1 and RNP2) in PROSITE. According to the sequence comparison from the structural alignment, these two regions present variants of the original sequences of U1A in the ADA4h sequence (Fig. 1a). Furthermore, both regions are structurally located in the same position according to the structural alignment (Fig. 1b and1c). When structurally compared to other known RRM domains, the new conformation of ADA4h displayed the same disposition for its RNP motifs (Fig. 6). Consequently, we propose a new RRM pattern that would characterize a putative binding of RNA.

The new RRM pattern was defined by variations in the sequences of the RNP1 and RNP2 regions (Fig. 1a). A PROSITE SCAN [START_REF] De Castro | ScanProsite: detection of PROSITE signature matches and ProRule-associated functional and structural residues in proteins[END_REF] search was performed in the UniProt database to find all sequences matching the new motif. Table 2 shows the distribution of functions of the 70 new sequences found in Uniprot using the new RRM motif but not the original PROSITE RRM motif.

A total of 48 sequences out of these 70 were aligned with the RRM_1 motif from PFAM [33] under an e-value threshold of 10 -5 . Moreover, the RRM_1 motif aligned with our motif only for 35 of these 48, while for the remaining 13 the alignment was located on a different position of the sequence. Consequently, the new pattern here described predicts the RNA-binding function for 32 sequences undetected by PFAM and may imply a different binding behavior for 13 known RNA-binding proteins, while 50% confirmed the expected function of our new pattern.

Additionally, 52 sequences out of these 70 perform functions involving nucleic-acid binding according to the information extracted from Swissprot. This means that we have found an explanation for 4 sequences that could not be described by using PFAM. Finally, of the remaining 18 proteins, 5 contributed to functions unrelated to nucleotide binding and 13 had an unknown function. ADA4h was among the 5 proteins unrelated to RNA binding, according

to Uniprot labeled functions. The corroboration of our prediction would imply that: 1) a new RRM-motif type for RNA binding would be possible, and 2) ADA4h would have a moonlighting, undetected function.

RNA electrophoretic mobility shift assays (EMSA)

The purified ADA4h protein was tested for RNA-binding activity using the U1A bindingmotif of stem-loop II of U1 small nuclear RNA (snRNA) [START_REF] Allain | Structural basis of the RNA-binding specificity of human U1A protein[END_REF]. A 70-mer RNA probe, called the bipolar-stem-loop, was designed for carrying two copies of the target RNA motif. The presence of the designed secondary structure was checked by using the online program OligoAnalyzer 3.0 from Integrated DNA Technologies  . A free energy of -18.9 kcal mol -1 is calculated for this secondary structure.

The recently-found yeast -crystalline (ZTAIp), an RNA-binding protein [34], and bovine serum albumin (BSA) were used as positive and negative controls, respectively (Fig. 7). A small amount of radioactivity was found in the shifted band (Fig. 7a), which concomitantly increases with the concentration of protein (Lanes 3-5) and was neither present in the negative control (Lane 2) nor when the cold probe was added as competitor in a ten-fold excess (Lane 1).

Another experiment with competitor tRNA was performed in order to check the ADA4h specificity for the bipolar-stem-loop (Fig. 7b), and, although the shifted band is not as clear as in the EMSA without competitor tRNA (Fig. 7), some faint shift can still be observed (see . The shifting also decreases for yeast -crystalline as expected (Lane 5).

These results confirm that ADA4h has RNA-binding capability, although it does not show a strong specificity for the bipolar-stem-loop designed in this work. The search for structures related to the activation domains of procarboxypeptidases, but not to the carboxypeptidase function, has yielded as the best score the U1A RNP (1M5Kc) with the activation domain of human procarboxypeptidase A4. RNP2 of U1A fits perfectly in the consensus with the IYINNL sequence. In the case of ADA4h, the RNP2-aligned sequence corresponds to: LRI-NV. The differences, with respect to the consensus sequence, are labeled in italics and underlined. Albeit these differences, the sequence conserves four out of six residues and is quite similar to the RNP2 consensus sequence.

Discussion

The RNP1 sequence of U1A is RGQAFVIF, where Gln does not correspond either to Phe or Tyr. This particular Gln, Gln54, stacks on G42 of the Polyadenylation Inhibition Element (PIE) RNA in the crystal structure of the complex, performing the same function as any Phe

or Tyr [START_REF] Allain | Structural basis of the RNA-binding specificity of human U1A protein[END_REF]. In the case of ADA4h, the corresponding sequence is R-PVDVLV, showing the lack of residue 2. Although just three residues out of eight are strictly conserved, it could be considered that Val47 and Asp48 are not so different from [GA]-[FY] of the consensus sequence. In fact, an equivalent Val is found in the PABP and in the U2AF35 RNP1 sequences, and also an equivalent Asp is found in the ALY RNP1 sequence [35]. Finally, the residue at position 8 is not conserved, as is the case of the polypyrimidine-tract-binding protein (PTB) RNP1 sequences A and C, the Nucleolin RNP1 sequence B, the LA and the PABP sequences [35]. Taking all of these criteria into account, the sequence of RNP1 of ADA4h approaches the consensus except for the lack of Gly in position 2.

It is important to note three different general features of RNA-binding domains. Firstly, although some conserved aromatic residues are always found at the interface of the complex, the topology of the bound RNA is characteristic of each complex solved and the sequencespecificity cannot easily be predicted. Secondly, RRMs that bind RNA show a wide spectrum of affinities. And, finally, some RRMs are specific for protein binding rather than for RNA binding, suggesting that some RRM might have evolved from RNA to protein recognition [39].

Previous works have shown that Gly is a relevant residue for several nucleotide recognition motifs, such as RRM [40,41]. Despite its importance, it is clear that the fact that Gly has no side-chain indicates that this residue has to be involved in the flexibility and length of the pattern, improving interaction or facilitating specificity. Although our proposed RNP1-like motif lacks the presence of this Gly, it aligned several times (50% correctly aligned) with the RRM PFAM domain. This may prove that the interaction with RNA can still be plausible by the conservation of other key residues.

Binding specificity of U1A

The U1A protein tertiary structure has been obtained in solution in complex with one of its specific RNA, the PIE from its pre-mRNA [START_REF] Allain | Structural basis of the RNA-binding specificity of human U1A protein[END_REF]42]. The determinants for binding have been analyzed in detail and compared with those of the previously known crystal complex with stem-loop II from U1-snRNA [START_REF] Oubridge | Crystal structure at 192 A resolution of the RNA-binding domain of the U1A spliceosomal protein complexed with an RNA hairpin[END_REF]. Afterwards, the complex with stem-loop II was also solved by nuclear magnetic resonance (NMR). Intermolecular recognition of both RNAs required not only conformational changes in the protein, shifting loop 3 and the C-terminal extension, but also in the RNA, kinking it by induced fitting.

The general fold for U1A is a four-stranded -sheet flanked on one side by two -helices. The complex for PIE shows the -sheet surface interacting with the bases, whereas the phosphate backbone points towards the solvent, away from the protein. This implies that electrostatic interactions are not determinant for binding and that hydrophobic contact must be regarded as the main force fitting together both molecules. This is also the case of stem-loop II.

PIE and stem-loop II have a 7-nucleotide sequence that interacts with the -sheet and the Cterminal extension of U1A. The RNP1 and RNP2 sequences of the -sheet interact with this ssRNA in loop conformation. These interactions are also shown by other RNA protein complexes, and are mainly responsible for unspecific contacts. The fact that ADA4h contains RNP1-like and RNP2-like sequences could explain the experimentally observed binding to the bipolar-stem-loop constructed in this work. Differences between U1A RNP and ADA4h

RNP-like motifs are summarized in Table 3. Finally, two hydrophobic interactions are important for U1A, these by Ser46 and Lys88; Pro40 and Glu75 of ADA4h align to them, respectively.

Concerning the contacts with the dsRNA structure, loops 1 and 3 are the ones interacting and have been pointed out as the key residues for specificity. An important region in U1A is an unusual cluster of six basic residues. Lys20 is found aligned to Asn16 of ADA4h but is preceded by Arg15, a residue more basic than Lys. Lys22 and Lys23 have no basic counterpart in ADA4h. Although neither Arg47 nor Lys50 align with Lys38 of ADA4h, they are all located in loop 3 near the RNA-binding interface. Finally, the most important basic residue in U1A, and characteristic of RNP1, Arg52, perfectly aligns to Arg45 in ADA4h.

Specificity in these loops is not a common feature of all RRM, since their size and distribution of basic amino-acids is quite variable [START_REF] Burd | Conserved structures and diversity of functions of RNAbinding proteins[END_REF]. Thus, it is fairly possible that ADA4h, with a cluster of three basic residues in loops 1 and 3, would use this mechanism for molecular discrimination to a lesser extent than U1A does.

Loop 3 is the most variable segment between RRM, and in the case of the U1A complex it is determinant for specificity. However, important residues in U1A have their counterpart in ADA4h: Arg52 (Arg45 in ADA4h) contributes with several hydrogen bonds and packs against Leu49 (Phe43 in ADA4h) and the CG pair closing the structure of the RNA. The residues positioning the dsRNA portion are Ser46 (Pro40 in ADA4h), Ser48 (Ser42 in ADA4h), Leu49 (Phe43 in ADA4h) and Arg52 (Arg45 in ADA4h). This can account for the inability of 0.5 g of competitor tRNA to totally abolish the complex between ADA4h with 25 fmol of bipolar-stem-loop RNA, a ratio of 2 x 10 7 :1 (w:w), in the EMSA.

Proposed moonlighting function for ADA4h

Apart from the relevance in assisting the folding process, the known function for the activation domains of carboxypeptidases is the inhibition of the enzyme in the proenzyme form. The contacts in the crystal structure between ADA4h and its enzyme moiety involve one residue of RNP2 (Arg11), three residues corresponding to -strand 2 (Asn35, Phe36, and Trp37) and one located in loop 3 (Lys38) of the domain [START_REF] Garcia-Castellanos | Detailed molecular comparison between the inhibition mode of A/B-type carboxypeptidases in the zymogen state and by the endogenous inhibitor latexin[END_REF]. The fact that important residues for inhibition are located in the same region determinant for RNA binding might induce the ADA4h domain to evolve from the canonical RNP1 and 2 sequences. This hypothesis is based on the suggestion that some RRM might have evolved from RNA to protein recognition

[39], as previously mentioned. It is likely that among the activation domain of procarboxypeptidases, ADA4h is the one that best conserves RNA-binding properties. Apart from the structural approach presented in this work pointing out ADA4h as being related to RNA-binding proteins, it is noteworthy that ADA4h shares about a 20% homology with U1A, whereas ADA2h and ADB only share 11% and 3%, respectively, as determined by using the CLUSTALW multiple sequence alignment program [START_REF] Chenna | Multiple sequence alignment with the Clustal series of programs[END_REF].

hPCPA4 is a procarboxypeptidase mainly expressed in cancerous prostate, brain and during development. Its gene is subject to genomic imprinting and its regulation is due to histone hyperacetylation, as observed in the sodium butyrate treatment of androgen-independent prostate-cancer PC-3 cell lines [START_REF] Huang | Carboxypeptidase A3 (CPA3): a novel gene highly induced by histone deacetylase inhibitors during differentiation of prostate epithelial cancer cells[END_REF], which lead to growth inhibition. Very low expression has been found in the pancreas or other normal tissues. Recently, some proteins involved in different processes such as metabolism have been proved to bind RNA, participating in regulatory mechanisms [START_REF] Ciesla | Metabolic enzymes that bind RNA: yet another level of cellular regulatory network?[END_REF]. Although further experiments should be performed, this could also be the case of ADA4h, as pointed out in this work, where fold homology has been revealed as a suitable tool for predicting protein function.

Conclusions

Bioinformatic tools have been proven useful in the prediction of both the possible function of ADA4h as an RNA-interacting protein and the affinity of such interaction. Our results seem to indicate that activation domains of procarboxypeptidases would have lost part of their ability to bind RNA during evolution as a consequence of the optimization of the inhibition of the catalytic domain.

Figure captions

  Motif (RRM) is one of the most abundant in eukaryotes, and is estimated to be present in 2% of the human gene products [35]. Initial biochemical studies on poly(A)-binding protein (PABP) and hnRNP protein C defined a consensus RNA-binding domain of about 90 residues containing a central sequence of eight conserved residues [36, 37]. The [RK]-G-[FY]-[GA]-[FY]-[ILV]-X-[FY] sequence was called RNP1 when a second consensus sequence, [ILV]-[FY]-[ILV]-X-N-L, was found in the N-terminus [38]. RNP2 is less conserved and shorter than RNP1. RNP1 contains mainly aromatic and basic residues, whereas RNP2 is rather hydrophobic.

Fig. 1 Fig. 3 18 Fig. 4 Fig. 5 Fig. 6 Fig. 7

 13184567 Fig. 1 Structural superposition of ADA4h and U1A. (a) Sequence alignment derived from the structural superposition. The RNP regions of U1A are emphasized in yellow. RNP1 and 2 regions stand out in red and blue for ADA4h. The PROSITE pattern description is shown below the alignment and modifications from the original pattern are underlined. The secondary structure of U1A is displayed under the alignment. (b) Superposition of structures obtained from the structural alignment of the activation domain of human procarboxypeptidase A4 (in blue) with the RNA-binding protein U1A (in red). RNP regions of both structures are highlighted in yellow (RMSD 2.16). (c) Structural alignment of RNA-binding protein U1A (in red) and the structural model of ADA4h (in blue) using the structure of U1A in 1M5Kc as template (RMSD 1.70). RNP regions of both structures are highlighted in yellow

Figure 1

 1 Figure 1Click here to download high resolution image

Figure 2

 2 Figure 2 Click here to download high resolution image

Figure 3

 3 Figure 3Click here to download high resolution image

Figure 4

 4 Figure 4Click here to download high resolution image

Figure 5

 5 Figure 5 Click here to download high resolution image

Figure 6

 6 Figure 6Click here to download high resolution image

Figure 7

 7 Figure 7Click here to download high resolution image

Table 1 Comparison of the RNA interactions by U1A and ADA4h.

 1 Comparison between the interactions performed by U1A and ADA4h, and the proposed conformational change of

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Table 2 Functional distribution of the new sequences found with the modified RNA recognition motif.

 2 The first six functions, excluding the procarboxypeptidase precursor, are related to nucleotide recognition. It must be noted that 75% of the new sequences found with the modified RRM are related to this function

	Function found	Frequency (%)
	Carboxypeptidase precursor	2
	Nucleolin	34
	Splicing factor	10
	RNA-binding	16
	Nucleotide binding	13
	DNA helicase	2
	Polysaccharide biosynthesis	1
	Dioxygenase	4
	Transmembrane component	1
	Hypothetical protein

Acknowledgments

NCC was supported by an AGAUR FI fellowship (2003 FI 00388, Generalitat de Catalunya).

JB and BO acknowledge support from MEC (BIO08-0206). F.X.A acknowledges financial support from the CICYT (BIO04-2046). SV acknowledges funding from the Fondo de Investigación Sanitaria (FIS)/Ministerio de Sanidad y Consumo (PI070148), and from the Fundación Mutua Madrileña (FMM-08) and SGR 2009-00761.

Supplementary material available 23