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# Molecular decision trees realized by ultrafast electronic spectroscopy

Contributed by R. D. Levine, August 9, 2013 (sent for review June 11, 2013)

## Significance

One possible way to reduce the physical dimensions of a computing node is to instruct a molecule to evaluate a complicated logic function. This is even more so if several such functions are processed in parallel. The interaction between light and matter is a suitable route because it is bilinear, depending on both the properties of the laser and of the molecule; the outcome depends on the initial state of the molecule and there can be more than one distinct path leading to the readout signal. Two-dimensional photon spectroscopy is shown to have four paths originating from each interaction, thereby enabling, as shown in *SI Text*, quaternary logic. In the main text, we discuss the simpler case of binary logic.

## Abstract

The outcome of a light–matter interaction depends on both the state of matter and the state of light. It is thus a natural setting for implementing bilinear classical logic. A description of the state of a time-varying system requires measuring an (ideally complete) set of time-dependent observables. Typically, this is prohibitive, but in weak-field spectroscopy we can move toward this goal because only a finite number of levels are accessible. Recent progress in nonlinear spectroscopies means that nontrivial measurements can be implemented and thereby give rise to interesting logic schemes where the outputs are functions of the observables. Lie algebra offers a natural tool for generating the outcome of the bilinear light–matter interaction. We show how to synthesize these ideas by explicitly discussing three-photon spectroscopy of a bichromophoric molecule for which there are four accessible states. Switching logic would use the on–off occupancies of these four states as outcomes. Here, we explore the use of all 16 observables that define the time-evolving state of the bichromophoric system. The bilinear laser–system interaction with the three pulses of the setup of a 2D photon echo spectroscopy experiment can be used to generate a rich parallel logic that corresponds to the implementation of a molecular decision tree. Our simulations allow relaxation by weak coupling to the environment, which adds to the complexity of the logic operations.

The need to further reduce the physical dimensions of a computing node is well recognized (1⇓⇓–4). A direct way to go below a nanometer scale is to use a molecule or an artificial atom—a quantum dot—as a switch (5⇓⇓⇓⇓⇓–11). Molecules can also respond in more interesting ways than switching. So for some time we have followed a program of seeking to implement an entire logic circuit on an atom or molecule and to concatenating such units. To do so, we used intramolecular dynamics resulting from the response of a molecule to a perturbation—the inputs to the computation—that can be electrical (12) or optical (13) or chemical (14), etc. In principle, such an approach can implement finite-state logic (15) because typically the response of a molecule depends on its initial state as well as on the applied perturbation.

In finite-state machines, the execution of the logic relies on transitions between states. The operation is inherently parallel (15). A simple situation is when a molecule relaxes after being perturbed by an optical (16, 17) or an electrical pulse (18, 19). In optical molecular implementations, several states can be simultaneously addressed, which leads to massively parallel linear finite-state machines (16, 17). The other mode of operation is to provide inputs at each machine cycle. If the molecule is switched between two states, the ability to encode a dependence on the initial state means that one can implement memristor logic (20). More elaborated memory integrated units like set–reset machines can also be implemented optically using a stimulated Raman adiabatic passage excitation pulse (21, 22) or can be based on molecular redox processes (23⇓–25). Here, we propose the realization of a molecular tree as a logic machine that accepts several inputs and processes them by performing bilinear operations that are dependent on the input and on the state of the machine.

One interpretation of a decision tree is that it provides a graphical representation of the truth table that defines a digital function. As an example, we show in Fig. 1*A* the truth table of the exclusive-OR (XOR) operation between two Boolean variables. The XOR function is an important two-input gate that performs the addition modulo two (denoted by the symbol ⊕) of the two Boolean variables. Its truth table provides the value of the function for each of the four combinations of inputs. The truth table can also be represented as a rooted binary tree (Fig. 1*B*) composed of three levels. Each nonterminal node (denoted by S) splits the XOR function by assigning a precise value to one of the input variable: starting from the root node, if is true (i.e., ), then we follow the tree down to the left to assign the value to the next input ; otherwise, is true (i.e., ), and we proceed down to the right. The second level performs an analog splitting according to the value of , thus generating four leaves (terminal nodes) labeled by the result of the XOR operation on the specific pairs of inputs corresponding to each path.

We here propose and illustrate molecular realizations of decision trees based on nonlinear spectroscopy. Among these spectroscopic techniques, four-wave mixing, in particular, two-dimensional photon echo (2D-PE), is a powerful tool that provides information on dissipation, dephasing, solvation, and (electronic or vibrational) interstate coupling mechanisms in various multistate material systems (26⇓⇓⇓–30). When the system dynamics is characterized in terms of the time-dependent density matrix, four-wave–mixing spectroscopy monitors populations and coherences in real time.

We take advantage of the richness of the information provided by 2D-PE (31) for implementing molecular decision trees. An advantage of weak-field spectroscopy is that the dynamics reflects the detailed structure of the chromophoric molecules that are used because the laser field as used is not strong enough to distort the molecule beyond recognition. Another advantage is that we can trace the different paths that connect the initial state of the molecule to the readout. This enables us to relate each path to a path down a decision tree (DT) and, because more than one path connects the initial state of the molecule to the readouts, to implement computations in parallel. In the language of optical spectroscopy, we can say that each such computational path corresponds uniquely to a Feynman diagram as discussed for 2D-PE spectroscopy, for example in refs. 32 and 33 (Fig. S1). Ideally, we would like to resolve the separate paths in a given experiment. This ideal is not possible with current implementation of 2D electronic spectroscopy, which limits the range of functions that can be computed. The number of paths that can be simultaneously resolved can be increased using polarization (34) or 3D Fourier transform spectroscopy (35).

For the purpose of logic, we find it useful to describe the time evolution in the so-called Heisenberg picture. In this way of thinking, an elementary act of perturbation of the system is represented as a bilinear operation. It is bilinear because the interaction consists of taking the commutator of the perturbation with the density matrix of the system. So the operation is linear in the state of the system and linear in the perturbation. The application of the commutation relation at each photon–molecule interaction process in a 2D-PE experiment generates a commutator tree as shown in Fig. 2.

For a weak optical perturbation, it is realistic to allow the system to have only a finite number of states that can be accessed by the photons. In third-order nonlinear spectroscopy of a bichromophoric molecule, there will be four states. This suggests that we use a Lie algebraic approach based on the group SU (4), which are norm-conserving, hence the S, unitary transformations among four states. One can use such a group to implement exact dynamics (36⇓–38). To mimic a weak-field experiment such as 2D-PE, we only need to implement perturbation theory to third order, third corresponding to three photon processes. The structure of the Lie algebra and the nature of the inputs is then imprinted on the structure of the dynamics and thus on the logic. In particular, we show how a decision tree is generated by the evolution of an excitonic-dimer system probed by 2D-PE experiment.

The first section explains the Lie algebraic background of the present work with an emphasis on the analogy between the bilinear operations of the logic and the bilinear operation (i.e., the commutator) of the algebra that is implied by the quantum mechanical Heisenberg equation of motion. In the second section, we recast the evolution of the density matrix during the PE sequence of laser pulses and the corresponding detected signal in a form that makes the connection with the implemented logic more transparent. In the third section, we give specific examples of the implementation of decision trees in the 2D-PE spectra calculated for a bichromophoric dimer system. Concluding remarks and perspectives are given in the last section.

## The Lie Algebraic Description of Photon Echo Spectroscopy

The 2D-PE spectroscopy of coupled chromophores has been previously modeled at different levels of sophistication (32, 33, 39, 40). We consider here the model of an excitonically coupled heterodimer and describe its coupling to environment within the Redfield approach. Our technical purpose is to calculate the evolution of the density matrix during the pulse sequence defining the PE experiment and the resulting signal. This is summarized in the next section by emphasizing the underlying Lie algebraic structure, while full details are given in *SI Text*, *Section 1* and *Section 2*.

The first step is to introduce the Hamiltonian describing the two interacting chromophores and their coupling with the external field showing that they can be written as a linear combination of elements of the SU (4) Lie algebra. We note that it thereby follows (36⇓–38) that the time evolution of any observable relevant to the system can be so described in closed form. In particular, the evolution of the density matrix of the bichromophore can be represented as a linear combination of the elements of the algebra. This structure of the density matrix is preserved also when we include relaxation described by a Redfield equation in the Bloch limit.

The dimer system is composed by two interacting chromophoric units that we describe in the excitonic representation, whose level structure is shown in Fig. 2*B*. The electronic Hamiltonian in the site basis is standard (Eqs. **S10**–**S13** in *SI Text*). It includes two states for each chromophore whose energies are corrected for the solvent contribution and the coulombic interaction that couples the excited state of each chromophore and leads to exciton basis.

In the exciton basis, the four states shown in Fig. 2*B* give rise to 16 operators, , where , that form a set that is closed under commutation:This closure is central for our purpose because we will draw an analogy between the commutation of two operators and the bilinear operations of the logic. In the excitonic basis, the set of 16 operators is complete, meaning that any operator of interest can be written as a linear combination of these elementary operators. The excitonic Hamiltonian, , is clearly of the required form. The full Hamiltonian is with being the external electric field. Each pulse is characterized by its frequency , wavevector , and an envelope, , centered at time , respectively. The field–matter interaction is described by the action of the transition dipole operator, , which in term of the basis operators reads as follows:where the are the transition dipoles. The key technical point is that throughout the time evolution the density matrix, , of the system is also a linear combination of the basis operators:with time-dependent expansion coefficients. The proof is a direct consequence of the closure property (Eq. **1**). The expansion above for the density matrix is clearly valid at the initial time when . To propagate the density matrix forward in time, we need to commute it with the Hamiltonian, for example:where and are the coefficients of the operators and in the density matrix and the dipole operator, Eqs. **3** and **2**. The complete equation of motion is as follows:where the first term on the right is the quantum mechanical evolution of the chromophore, whereas the Redfield tensor, *R*, describes the dissipative dynamics induced by the coupling with the environment. The tetradic relaxation matrix, , is the transfer rate from to and its explicit form is given in *SI Text*, *Section 2*. In the so-called Bloch limit, the only nonzero elements of the Redfield tensor are population decay from state *i*, , population transfer from *i* to *j* (*i* ≠ *j*), , and coherence dephasing . It is clear by inspection that including the Redfield relaxation does not violate the closure meaning that the rate of change of the density matrix, , remains a linear combination of the elementary operators, .

## Evolution of the Density Matrix and the 2D-PE Signal

During the excitation sequence (Fig. 2*A*), the system interacts with three ultrashort laser pulses. A polarization is created, which generates the signal that can be probed in several (phase-matching) directions. Because the interactions are weak, we use the standard perturbational approach based on the expansion of the density matrix in orders of the electric field, leading to the following expression for the total third-order polarization (33): being the transition dipole operator (Eq. **2**). The final spectrum is equivalent to a double Fourier transform of the time-dependent signal:where the pulse delay times *τ* and *T* are defined in Fig. 2*A*. With the convention used in Eq. **6**, the rephasing signal emitted in the direction appears at positive frequencies .

To calculate the time evolution of the third-order density matrix, which defines the signal through Eq. **6**, we apply the rotating wave approximation (RWA) and assume the impulsive limit for short laser pulses. In this case, the time ordering of the interactions is well defined and the (positive) propagation times start at the midpoint for the corresponding -pulse. The three interactions with the fields result in three commutation operations according to Eq. **4**, whereas the time evolution operator in the absence of the external field is defined by the evolution Eq. **5** as . In the Heisenberg picture for the dipole operator (i.e., ), the third-order density matrix can be written as a function of the propagation times as follows:Switching to a Liouville space notation (33), Eq. **7** takes a more compact form as follows:with being the operator generating the action of the transition dipole. The time propagation between the laser pulses is generated by the operator, , . Eq. **8** gives an intuitive understanding of the dynamics of the density matrix during the PE experiment: starting from the initial state the system experiences instant interactions with the external field and subsequent free evolution in the time intervals between them. The “history” of each element of the density matrix during the experiment is imprinted in the time-dependent expansion coefficients of the rhs of Eq. **8**. It is understood that Eqs. **7** and **8** are valid in the impulsive limit of the pulse–matter interaction; beyond such approximation a threefold convolution with the pulse profiles has to be performed (*SI Text*, *Section 1*).

The structure of the transition dipole matrix reflects the allowed transitions showed in the scheme of the energetic levels given in Fig. 2. Because at the initial time, *t* =0, before any interaction, the system is in its stationary ground state, i.e., , the first interaction excites the one exciton transitions (i.e., , *j* = 1, 2) (Fig. 2*C*). According to Eq. **5**, the time evolution during *τ* is a damped oscillation with the exciton frequencies and a decoherence rate determined by the corresponding element of the Redfield matrix. The second interaction transforms these coherences in population states (, *j* = 0, 1, 2), or electronic coherences (, ). The transitions between the ground and the double exciton states (, ) do not survive the RWA and thus do not contribute to the observed spectrum. The evolution during *T* includes the decay of the electronic coherences and the population transfer between the two excitonic states promoted by the coupling with the bath (FRET). Finally, the third interaction brings back the system into coherence states (, , , , *j* = 1, 2), which evolve during the detection time, *t*, generating the emitted signal. Because each pulse excites all of the allowed transitions in its spectral bandwidth, each specific sequence of the states of the system driven by a specific sequence of interactions with the three fields constitutes a particular path. In Fig. 2, for example, we highlighted one of the possible paths by selecting a specific sequence of transitions among the elements of the density matrix. The possible paths deriving from the evolution of the density matrix during the PE sequence are more commonly represented as double-sided Feynman diagrams (33). The Feynman diagram corresponding to the highlighted sequence of interactions is shown in Fig. 2*D*. In the language of pump-probe, it corresponds to the ground-state bleaching contribution to the diagonal peak (, ) of the rephasing spectrum. For a dimer system, all of the paths contributing to the PE spectrum have been specified and analyzed previously (see, e.g., refs. 39 and 41) and their corresponding Feynman diagrams are reported in Fig. S1.

In the time evolution of the density matrix, following the commutation relation (4) and the structure of the dipole matrix in the exciton basis, each interaction with the field connects one element with four other elements , defining the commutator tree. The complete commutator tree, only sketched in Fig. 2*C*, is shown in Fig. S2. The number of connections induced by each interaction is determined by the structure of the dipole operator for the bichromophoric model.

## Logic Decision Trees

The simultaneous occurrence of the different paths contributing to the PE signal is appealing because it represents a realization of parallel logic operation at the molecular level. To build a bridge between the physical evolution of density matrix during the 2D-PE experiment and the classical logic of a decision tree, we first introduce some basic concepts that generalize the example discussed in Fig. 1. A switching (Boolean) function of *n* variables is a map . It is completely characterized by its truth table, which lists the value of the function for each of the 2^{n} possible combination of inputs. Any switching function can be decomposed according to Shannon decomposition rule (42) as follows:where are known as the negative and the positive Shannon cofactor, respectively. By applying the Shannon expansion recursively to a Boolean logic function, we can represent the function by a binary decision tree (BDT). Fig. 1 above shows the Shannon tree corresponding to the two variable function XOR. If we label each edge with a literal of a variable as in Fig. 1, the product of the literals from the root node to a terminal node along a specific path represents a minterm. For example, the leftmost path of the diagram in Fig. 1 represents the product . The tree is thus equivalent to the minterm expansion of the function *F*; for the XOR function of Fig. 1, we have the following:where the second equality gives the sum-of-product (SOP) form of the function *F* in the XOR representation. Notice that the function and its minterm expansion are unchanged if the “exclusive or” is replaced by the OR operation. This is because two different minterms are never both true for the same input assignment. Generalized Shannon expansion and SOP forms for multivalued logic function have also been developed (43⇓⇓–46). For the sake of simplicity, we will first give an example of connection between the physical measurement in 2D-PE and the decomposition of a logic function within the simpler framework of the two-valued Boolean logic. However, the complexity of the physical signal reflected in the commutator tree that represents the evolution of the full density matrix (Fig. S2) invites an analysis in terms of multivalued logic. We then discuss an example of multivalued logic by implementing a decision diagram for an integer function. A different way of exploiting the possibility of encoding multivalued variable is given in *SI Text*, *Section 4*, and Fig. S3 with the realization of a quaternary decision tree.

From the analysis of the dynamical evolution of the density matrix during the interaction with the three laser pulses presented in the previous section (Fig. 2), each interaction with the field triggers a branching of the path followed by the system according to the structure of the dipole operator. Let us now map the components of the 2D-PE signal derived in the previous section into a BDT logic data structure. To this end, we consider a subset of all of the possible paths from the ground state of the dimer, , , to the final coherence state, which generates the signal in . Different subsets can be extracted. Here, we choose as example the subset of eight different paths shown in Fig. 3*A* in the form of a decision tree. These paths contribute to the rephasing signal emitted by the dimer system in the phase-matching direction . In Fig. 3*A*, each path starting from the root node and ending in a leaf node of the tree corresponds to a Feynman diagram. The notation of the nodes follows from the commutator tree as shown in Fig. 2. The notation used for each path is given in Fig. S1.

The first level of the tree, corresponding to the assignment of , can be translated to the proposition “the interacting photon has energy .” In the path leading from to , is realized, whereas in the path leading from to , not(), is realized, because the transition corresponds to the absorption of . Then the system evolves during until the second interaction occurs, which assigns in a similar way the value of . The interaction with the second pulse drives the system in an excited population state ( or ) or in a coherent superposition of exciton states ( and ). During the second evolution period, corresponding to the evolution during the population time, *T*, dephasing and population transfer occur until the system interacts with the third laser pulse. The assignment of corresponds to the proposition “the interacting photon is adsorbed by the system.” The branches for which is realized correspond to the contribution of the excited-state absorption Feynman diagrams (*Rb* and *Rg*), whereas the branches in which is realized correspond to the stimulated emission contributions (*Ra* and *Rh*). The Feynman diagrams corresponding to the ground state bleach (*Rc* and *Rd*) are not used in the mapping of the logic function.

Each Feynman path contributing to the signal plays the role of a minterm in the logical plane. The next step is to “read” the coefficients of the minterm decomposition from the measured spectrum. In the Shannon decomposition (Eq. **10**), these coefficients are simply the Boolean value of the function for the corresponding input string.

The spectroscopic observables are not directly the Feynman paths but rather the intensities of the four peaks, C12, C21, D1, and D2, in the 2D spectrum shown schematically in the *Inset* of Fig. 3*A*. Because we use classical Boolean logic, where the variables values are 0 or 1, we use for assigning the output the power spectrum in the phase-matching direction . All of the paths are included in the computed spectrum so that it corresponds to an experimental measurement. The four intensities computed for a given value of the relaxation time *T* are mapped to a Boolean value that is equal to 1 when the intensity is higher than a threshold value and 0 otherwise. The spectral position to which each path contributes is reported in Fig. 3*A*. All of the paths contributing to the same spectral position are assigned the same Boolean value, depending if the peak is high or low for the relaxation time, *T*, considered. This logic assignment is dictated by the fact that, in currently performed 2D-PE experiments, one cannot resolve each Feynman path individually. It restricts the class of function of three Boolean variables that can be implemented to functions for which two set of three minterms have the same value: , , and , which contribute to the spectral position C_{12} and , , and , which correspond to the spectral position C_{21}. See *SI Text*, *Section 3*, for more details. Advances in experimental schemes toward the unraveling of the different pathways, as e.g., the use of polarization control (34) or 3D spectra (35), could lead in the future to the capability of representing all of the 256 functions of three Boolean variables.

The Shannon decomposition (Eq. **9**) implemented at each cycle, is bilinear in the variable and the cofactor of the function. Physically, the decomposition is implemented by the commutator, (Eq. **4**), which is bilinear in the states of the machine, , and the dipole operator. The implementation of the decision tree that we propose above can therefore be interpreted as a nonlinear three-cycle finite-state machine, as schematically shown in Fig. 3*B*. The machine receives two inputs, *x*_{i} and its negation , at each cycle *i* = 1, 2, 3. In the first step, the logic unit performs the cofactoring of the function with respect to the first input, , generating the two new functions, and . This is a bilinear logic operation between the variable and the functions *f*_{0} and *f*_{1} of the two remaining variables: . The functions *f*_{0} and *f*_{1} are stored in the memory register M_{1} and will be cofactored with respect to *x*_{2} in the second cycle. In the second cycle, *f*_{0} and *f*_{1} are decomposed to obtain the four functions, *f*_{00}, *f*_{01}, *f*_{10}, *f*_{11} (state M_{2}). In the final cycle, the decomposition with respect to *x*_{3} generates the vector of the function values, O in Fig. 3*B*, which are outputted in the form of the corresponding spectral position to be read.

Which function is computed can be designed by adjusting the value of the relaxation time, *T*. We report in Fig. 4 the 2D-PE spectra calculated for different values of the population time *T*: *T* = 23 fs (Fig. 4*A*), *T* = 46 fs (Fig. 4*B*), and *T* = 138 fs (Fig. 4*C*). The periodic intensity variation of the two cross-peaks (“quantum beats”) that is clearly visible in the spectra of Fig. 4 *A* and *B* is a manifestation of the electronic coherence terms, which are present in the paths *Rh* and *Rg*. For these paths, the system is in an exciton-coherence state ( or ) during the population time *T*, and the path contributions oscillate with a period determined by the difference of the one exciton frequencies, i.e., . The chosen *T* values for the spectra of Fig. 4 *A* and *B* correspond to and , respectively. Parameters which give a large exciton splitting (a short modulation period) have been taken from ref. 40 to emphasize the periodic behavior. In the spectrum corresponding to the longest population time (Fig. 4*C*), the intensity of the D1 diagonal peak decreases due to the population transfer to the lower energy exciton. Together with the three spectra, we report the truth tables of the corresponding 3-bit functions determined according to the binary tree structure of Fig. 3. The intensities of the four peaks of a single spectrum determine the values of the function for all of the possible combinations of inputs. Because these values are also the coefficients of the Shannon decomposition of the function, it is easy to rewrite the functions in their canonical (sum of product or sum of minterm) form, as it is shown in Fig. 4 under each spectrum.

Because the peaks in the spectrum are characterized by well-defined relative intensities, they are well suited to be mapped into integer values and not only to the binary outputs (0, 1). An integer number represents a string of binary values through its binary encoding (e.g., the binary representation of the integer 5 is 101 because 5 = 1 × 2^{2} + 0 × 2^{1} + 1 × 2^{0}). In many cases, representations of Boolean functions at the integer level (denoted in the computer representation as the word level) are convenient because more compact. For example, *m* output switching functions each of *n*-variables, can be represented by the integer valued function, , whose values at are . The example that we discuss here is that of word-level decision diagrams. They admit integer values in the terminal nodes and can thus be used to represent multiple-output functions. By using the same BDT as Fig. 3 but assigning to the spectrum a four-valued reading, we can represent a three input–two output function. An example is reported in Fig. 5, where we show the spectrum computed for a population time of 115 fs in Fig. 5*A*. By assigning the relative intensity of the four peaks a value among (0, 1, 2, 3), we can read the integer valued function reported in the truth table of Fig. 5*B*, where the last column is the binary translation that corresponds to the two output functions .

## Concluding Remarks

Our proposal can be scaled up in two different ways: by increasing the number of the interactions or more simply by increasing the number of accessible transitions. The first method would increase the number of levels of the DT (the so-called depth of the DT), whereas the second will increase the number of outgoing edges for each node (the so-called width of the DT). Although fifth-order optical spectroscopies are still technologically challenging, the studies of large aggregates with more chromophoric units has been performed by 2D-PE and seems to us the most straightforward generalization of the idea illustrated here for a dimer system. The number of different transitions triggered by each interaction with the external field defines the set of logical values that each variable can assume (i.e., the radix of the logic), so that the binary implementation presented here can be generalized to multivalued logic. As an example in *SI Text*, *Section 4*, we sketch the realization of a quaternary decision tree by considering the whole set of Feynman diagrams contributing to the signal emitted by the dimer system in all of the four phase-matching directions. Notice that, within this interpretation of the rephasing and not-rephasing spectra, we have access to 2^{6} = 64 evaluations of a switching function simultaneously performed by the evolution of the system during the 2D-PE experiment.

## Acknowledgments

This work is supported by the European Commission Future and Emerging Technologies Proactive Project MULTI (317707). F.R. is a Director of Research with Fonds National de la Recherche Scientifique (Belgium). B.F. thanks the University of Liège for a postdoctoral fellowship. The work of D.H. and R.D.L. is also supported by The James Franck Program for Laser–Matter Interaction. E.C. appreciates the support of European Research Council Starting Grant “Quantum-Coherent Drive of Energy Transfer Along Helical Structures by Polarized Light.”

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: rafi{at}fh.huji.ac.il.

Author contributions: B.F., R.D.L., and F.R. designed research; B.F., D.H., R.D.L., and F.R. performed research; B.F., D.H., E.C., R.D.L., and F.R. analyzed data; and B.F., E.C., R.D.L., and F.R. wrote the paper.

The authors declare no conflict of interest.

See Commentary on page 17167.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1314978110/-/DCSupplemental.

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