Patrícia F N Faísca

Cláudio M Gomes

On the relation between native geometry and conformational plasticity

Keywords: numbers: 87.15.Cc, 87.15.ak, 87.15.hm, 87.15.hp mutagenesis, φ-value analysis, folding pathways, long-range contacts

In protein folding the term plasticity refers to the number of alternative folding pathways encountered in response to free energy perturbations such as those induced by mutation. Here we explore the relation between folding plasticity and a gross, generic feature of the native geometry, namely, the relative number of local and non-local native contacts. The results from our study, which is based on Monte Carlo simulations of simple lattice proteins, show that folding to a structure that is rich in local contacts is considerably more plastic than folding to a native geometry characterized by having a very large number of long-range contacts (i.e., contacts between amino acids that are separated by more than 12 units of backbone distance). The smaller folding plasticity of 'nonlocal' native geometries is probably a direct consequence of their higher folding cooperativity that renders the folding reaction more robust against single-and multiple-point mutations.

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A C C E P T E D

M A N U S C R I P T

I. INTRODUCTION

During the last 15 years significant progress has been achieved towards a complete understanding of the kinetics and mechanisms of protein folding. The synergistic link between computer simulations and in vitro experiments has proven particularly fruitful in this endeavour [1,2]. Much of our current knowledge on protein folding has been gathered by studying small (i.e., with less than 100 amino acids) monomeric proteins, epitomized by the likes of the 64-residue protein Chymotrypsin Inhibitor 2 (CI2). Indeed, their two-state folding kinetics [START_REF] Jackson | How do small single-domain proteins fold?[END_REF] renders them particularly suitable models to investigate, both in vitro and in silico, the rather complex phenomenon that is the folding 'reaction'.

A major and challenging task in studying two-state proteins is the structural characterization of the folding transition state (TS), located on the top of the free energy barrier that separates the native fold from the ensemble of unfolded conformations. Indeed, due to its fleeting nature, the commonly available biophysical tools have revealed inappropriate to probe the TS' structure. Thus, experimental studies of the TS have remained predominantly rooted in the use of a particular class of protein engineering methods, the so-called φ-value analysis, developed by Fersht and coworkers back in the late 1980s [START_REF] Fersht | Structure and Mechanism in Protein Science: A guide to enzyme catalysis and protein folding[END_REF]. In the φ-value analysis a non-disruptive mutation (i.e., a mutation that does not change the structure of the native state, and does not alter the folding pathway either) is made at some position in the protein sequence [START_REF] Fersht | φ-value analysis and the nature of protein-folding transition states[END_REF]. The change in the activation energy of folding of the mutant with respect to that of the wild-type (WT) protein, denoted by ∆∆G T S-D , is measured together with the change in the free energy of folding, ∆∆G N -D , caused by the mutation.

The corresponding φ value is then defined as the ratio between these quantities, namely, φ = ∆∆G T S-D /∆∆G N -D . A φ value of unity means that the energy of the TS is perturbed

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upon mutation exactly as much as that of the native state, which has "traditionally been interpreted as if the protein structure is folded at the site of mutation in the TS. Conversely, residues that are unfolded in the TS exhibit φ-values of zero. The concept of conformational/folding plasticity [START_REF] Munoz | Folding plasticity[END_REF], or folding malleability [START_REF] Lindberg | Malleability of protein folding pathways: a simple reason for complex behaviour[END_REF], refers to the number of alternative folding pathways encountered in response to free energy perturbations such as those induced by solvent changes or mutation; the larger that number, the broader the TS, and the larger the plasticity. Although it is experimentally very difficult to investigate in a direct manner the number of folding pathways leading to the native state, evidence for the existence of conformational plasticity was found in a few cases by means

The traditional interpretation of fractional

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of protein engineering experiments. For example, single point mutations in the N-terminal domain of the monomeric λ 6-85 -repressor revealed a broad TS [START_REF] Burton | The energy landscape of a fast-folding protein mapped by Ala-Gly substitutions[END_REF]. The use of circular permutation methods -a less milder protein engineering technique where the protein's backbone is cut-up at some point, and its original N-and C-terminal parts are linked together in a process that leads to minimal alteration of the native fold -revealed considerable structural changes in the TSs of both the α-spectrin SH3 domain [START_REF] Viguera | Different folding transition states may result in the same native structure[END_REF] and src SH3 domain [START_REF] Grantcharova | Circularization Changes the Folding Transition State of the src SH3 Domain[END_REF], and even more dramatic modifications in the TS of the ribosomal protein S6 [START_REF] Lindberg | Complete change of the protein folding transition state upon circular permutation[END_REF], a finding that was corroborated by computer simulation studies [START_REF] Hubner | Common motifs and topological effects in the protein folding transition state[END_REF][START_REF] Hubner | Simulation, experiment, and evolution: Understanding nucleation in protein S6 folding[END_REF][START_REF] Matysiak | Optimal combination of theory and experiment for the characterization of the protein folding landscape of S6: How far can a minimalist model go?[END_REF]. The plasticity exhibited by these proteins has been taken has evidence for the existence of multiple folding pathways [START_REF] Munoz | Folding plasticity[END_REF].

In a recent study, Klimov and Thirumalai have studied the influence of the distribution of secondary structural elements along the protein sequence using an off-lattice coarse grained model and Langevin Molecular Dynamics simulations. The major result that came out of their study is that a symmetric distribution of α helices and β sheets with respect to sequence midpoint favours multiple folding pathways [START_REF] Klimov | Symmetric connectivity of secondary structure elements enhances the diversity of folding pathways[END_REF].

A study of the validity of the φ-value analysis as a tool for identifying critical Our analysis shows that folding to the native geometry that is rich in nonlocal longrange (LR) contacts is considerably less plastic (i.e., more conformationally robust) than folding to the model protein that has predominantly local contacts, and that this is possibly a direct consequence of the more cooperative folding transition exhibited by the non-local geometry. This reinforces the results of [START_REF] Faisca | Identifying critical residues in protein folding: Insights from φ-values and P fold analysis[END_REF], according to which the picture of the TS emerging from the φ-value analysis is more reliable when applied to target proteins having a distinctively large number of non-local native contacts.

Therefore, if φ-value mutational data is to be interpreted in the traditional way, which is so far the most commonly adopted view, it is important to select native folds where the number of LR native contacts is sharply dominant; they show a higher robustness against mutation and, at the coarse level of contact cluster, they tend to fold in a Levinthal-like manner, i.e., as single route folding proteins.

This article is organized in the following manner. After the introductory section, the protein models and simulation methodologies are described. Then, we present and discuss the results from simulations. Finally, in the last section we draw some concluding remarks.

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We consider a simple three-dimensional lattice model of a protein molecule with chain length N=48. In such a minimalist model amino acids, represented by beads of uniform size, occupy the lattice vertices and the peptide bond, which covalently connects amino acids along the polypeptide chain, is represented by sticks with uniform (unit) length corresponding to the lattice spacing.

To mimic protein energetics we use the Gō model [START_REF] Go | Respective roles of short-and long-range interactions in protein folding[END_REF]. In the Gō model the energy of a conformation, defined by the set of bead coordinates { r i }, is given by the contact Hamiltonian

H({ r i }) = N i>j ∆( r i -r j ), (1)
where the contact function ∆( r ir j ), is unity only if beads i and j form a non-covalent native contact, i.e., a contact between a pair of beads that is present in the native structure, and is zero otherwise. The Gō potential is based on the idea that the native fold is very well optimized energetically. Accordingly, it ascribes equal stabilizing energies (e.g., = -1.0) to all the native contacts and neutral energies ( = 0) to all non-native contacts. The motivation to use the Gō model is based on the well accepted finding that for small, single domain two-state proteins, the geometry of the native fold is the major determinant of folding kinetics [START_REF] Plaxco | Topology, stability, sequence and length: Defining the determinants of two-state protein folding kinetics[END_REF][START_REF] Gromiha | Comparison between long-range interactions and contact order in determining the folding rate of two-state folders: Application of long-range order to folding rate prediction[END_REF][START_REF] Zhou | Folding Rate Prediction Using Total Contact Distance[END_REF].

In order to mimic the protein's relaxation towards the native state we use a standard Monte Carlo (MC) algorithm together with the kink-jump move set [START_REF] Landau | A guide to Monte Carlo Simulations in statistical physics[END_REF]. Accordingly, local random displacements of one or two beads (at the same time) are repeatedly accepted or

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rejected in accordance with the standard Metropolis MC rule [START_REF] Metropolis | Equation of state calculations by fast computing machines[END_REF]. A MC simulation starts from a randomly generated unfolded conformation and the folding dynamics is monitored by following the evolution of the fraction of native contacts, Q = q/L, where L is number of contacts in the native fold and q is the number of native contacts formed at each MC step. The number of MC steps required to fold to the native state (i.e., to achieve

Q = 1.0)
is the first passage time (FPT) and the folding time, t, is computed as the mean FPT of 100 simulations. Except otherwise stated folding is studied at the so-called optimal folding temperature, the temperature that minimizes the folding time [START_REF] Oliveberg | Negative activation enthalpies in the kinetics of protein folding[END_REF][START_REF] Gutin | Temperature dependence of the folding rate in a simple protein model: Search for a "glass" transition[END_REF][START_REF] Cieplak | Scaling of folding properties in simple models of proteins[END_REF][START_REF] Faisca | Thermodynamic control and dynamical regimes in protein folding[END_REF]. The folding transition temperature, T f , is defined is the temperature at which denatured states and the native state are equally populated at equilibrium. In the context of a lattice model it can be defined as the temperature at which the average value < Q > of the fraction of native contacts is equal to 0.5 [START_REF] Abkevich | Impact of local and non-local interactions[END_REF]. In order to determine T f we averaged Q, after collapse to the native state, over MC simulations lasting ∼ 10 9 MCS.

B. Target geometries

Two native folds, which are amongst the 'simplest' (geometry 1) and the most 'complex' (geometry 2) cuboid geometries found through lattice simulations of homopolymer relaxation, were considered in this study (Figure 1). A non-local contact is considered long-range (LR) if the two beads participating in the contact are separated by more than 12 units of backbone distance [START_REF] Gromiha | Comparison between long-range interactions and contact order in determining the folding rate of two-state folders: Application of long-range order to folding rate prediction[END_REF]. Accordingly, 33% of the native contacts in geometry 1 are LR, while geometry 2 has 74% LR contacts.

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III. RESULTS

A. A picture of the folding reaction

The mechanistic equivalent of the experimental φ-value of residue i at time t, φ mec i (t) , is given by the number of native contacts q Γ i (t), the residue establishes in conformation Γ, normalised to the number of contacts it establishes in the native fold, q native , i.e., φ mec i = q Γ i (t)/q native [START_REF] Vendruscolo | Three key residues form a critical contact network in a protein folding transition state[END_REF][START_REF] Hubner | Commitment and nucleation in the protein G transition state[END_REF][START_REF] Allen | Transition states for protein folding using molecular dynamics and experimental restraints[END_REF]; a residue is fully native if φ mec = 1. Thus, φ mec provides a direct measure of structure formation.

In what follows we use φ mec to obtain a picture of the folding reactions leading to geometry 1 and geometry 2, and we use the fraction of native contacts as a progress variable. In particular, for fraction of native contacts, Q, we compute the number of times each residue is fully native (i.e., φ mec = 1) and normalize it to the total number of times Q is counted in the course of a folding event. The probability thus computed is then averaged over an ensemble of 100 independent folding simulations. We remark that although the probability to fold, P fold , is the most accurate progress variable in lattice simulations of protein folding [START_REF] Du | On the transition coordinate for protein folding[END_REF], its use for the purposes of the present goal is computationally prohibitive as it would have to be evaluated for every conformation sampled in a folding event. This is why we use instead the fraction of native contacts, which was recently shown to measure correctly the degree of closeness to the native fold [START_REF] Cho | Structure reaction coordinates capture protein folding on smooth landscapes[END_REF] for proteins with smooth energy landscapes (i.e., single exponential kinetics) [START_REF] Gillespie | Non-glassy kinetics in the folding of a simple, single domain protein[END_REF][START_REF] Clementi | Prediction of Folding Mechanism for Circularpermuted Proteins[END_REF], like those considered here.

Not surprisingly, the folding pattern of geometry 1 (Figure 2, top left), i.e., the conformational changes leading to the native fold, is readily distinct from that exhibited by geometry 2 (Figure 3, top left). For example, from early to mid folding (i.e., 0.26 < Q < 0.53), there are more residues in geometry 2 with a higher probability of being in their native environ-

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ment. These probabilities decrease sharply immediately prior to collapse into the native fold (i.e., 0.79 < Q < 1.0). On the other hand, for the vast majority of the residues belonging to geometry 1, the probability of being fully native increases in a rather continuous way from early to late folding.

B. Native geometry and conformational plasticity

Within the context of the Gō model a single-point mutation is equivalent to replace the set of native contacts established by the mutated residue with neutral contacts, i.e., contacts to which zero energy is ascribed. In a recent study we have mutated every single residue in each protein model and measured the change in the folding time of the mutant relative to the wild-type (WT) sequence (Figure 4); an extensive number of double point mutations was also performed [START_REF] Faisca | Identifying critical residues in protein folding: Insights from φ-values and P fold analysis[END_REF]. We now ask how does the folding pattern of both protein models that reported for geometry 1, is observed for geometry 2 (Figure 3). Indeed, in this case, the folding pattern is considerably more robust, which suggests that the search for the native fold is constrained to follow a fixed sequence of conformational changes, i.e., that this geometry has a smaller conformational plasticity.

C. Long-range contacts, the structure of the transition state and the cooperativity of the folding transition

As discussed below, the large number of LR contacts in geometry 2 protects folding from perturbation, which decreases the conformational plasticity, in two ways: i) by increasing the robustness of the transition state' structure against mutation and ii) by increasing the cooperativity of the folding transition.

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Figure 5 (straight line) reports the probability that each protein residue is fully native in the TS (Q ≈ 0.5) of geometry 1 (left) and geometry 2 (right), and how this probability changes upon mutation. The structure of the TS of geometry 2 is clearly more robust against mutation than that of geometry 1. Indeed, most mutations in geometry 1 lead to a large structural consolidation between residues 12 and 33. In geometry 2, the most deleterious double-point mutations disrupt networks of non-local LR contacts that form the core structure of the TS [START_REF] Faisca | Identifying critical residues in protein folding: Insights from φ-values and P fold analysis[END_REF]. Since the formation of LR contacts greatly restricts the number of conformations available to the folding chain, the establishment of LR contacts is generally entropically costly. Because the LR contacts are clearly dominant in the native fold of geometry 2 it is highly unlikely that upon the mutation of one or more of its residues, folding can occur through an energetically competitive TS, as this would imply changing the TS structure by increasing substantially its content in local (i.e., less entropically costly)

native contacts [START_REF] Baker | A surprising simplicity to protein folding[END_REF][START_REF] Dokholyan | Identifying the protein folding nucleus using molecular dynamics[END_REF]. Thus, for the non-local geometry 2, the robustness of its TS is a direct consequence of the fold's high content in LR contacts.

In protein folding the formation (and breaking) of native contacts in a non-independent manner (i.e., cooperatively) results into the depletion of partially folded conformational states, which translates into a kinetic behaviour that fits remarkably well a two-state model.

Indeed, at the transition midpoint of a two-state folding 'reaction', half of the protein molecules in the test tube are folded and half of them are coil. Thus, microscopically, a cooperative folding transition is characterized by a bimodal distribution of protein molecules over energy, fraction of native contacts, Q, or any other observable parameter [START_REF] Makhatadze | Energetics of protein structure[END_REF]. Data reported in Figure 6 (top) shows that at the transition temperature, T m , the population distribution of the fraction of native contacts for geometry 2, P (Q), is more strongly bimodal than that of geometry 1, showing that for geometry 2 the folding transition is clearly

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more cooperative [START_REF] Abkevich | Impact of local and non-local interactions[END_REF]. The stronger cooperative stabilization of the native state of geometry 2 relative to that of geometry 1 is more evident from the corresponding free energy curve, F (Q) = -kT ln P (Q), where a very high free energy maximum separates the native conformation from the of ensemble of unfolded conformers (Figure 6, bottom) [START_REF] Go | Respective roles of short-and long-range interactions in protein folding[END_REF]. Mechanistically, the strong cooperative folding of geometry 2 restricts the number of allowed conformational changes, and as a consequence, the number of alternative folding trajectories (i.e., the conformational plasticity) is smaller for this more complex geometry.

IV. CONCLUSION

According to the statistical or landscape view of protein folding the native state can be reached from a myriad of microscopic parallel pathways. This 'new' view contrasts the socalled traditional view that envisages folding as a Levinthal-like search, where the elements of the native structure assemble in a well-defined order. Thus, identifying the number of folding pathways accessible to a protein chain is a crucial question in molecular biology. leading to two distinct protein geometries. More specifically, we have investigated how the probability that each residue is fully native (i.e., that is has all its native contacts formed) evolves during the folding process, and how this evolution responds to mutation. Our findings suggest that folding to native geometries which are distinctively rich in long-range native contacts is more robust against mutation than folding to native geometries where the number of local native contacts is dominant. Indeed, 'local' geometries seem to be able to find alternative folding routes in response to mutation while non-local geometries appear to behave in a Levintahl-like manner, their folding reaction being constrained to follow a fixed sequence of conformational changes, when monitored at the coarse level of contact cluster (i.e., the set of native contacts established by each residue). In other words, the conformational plasticity is expected to be small for target proteins where the number of non-local native contacts is sharply larger than the number of local contacts. These findings are supported by experimental results reported for the λ 6-85 -repressor (an α-protein that is rich local contacts) and for protein CI2, and src SH3 domain (α/β proteins rich in non-local contacts); while the TSs of the latter were found to be largely invariant against single and double point mutations [START_REF] Itzhaki | The structure of the transition-state for folding of chymotrypsin inhibitor-2 analyzed by protein engineering methods: evidence for a nucleationcondensation mechanism for protein-folding[END_REF][START_REF] Grantcharova | Long-range order in the src SH3 folding transition state[END_REF], substantial changes were found for the former. Mechanistically, the lower folding plasticity of the more complex native geometries is most possibly a direct consequence of their more cooperative folding transition, which results from a large content in long-range native contacts. In agreement with this explanation, experimental evidence for high conformational plasticity, resulting from a weak cooperative behaviour, was recently reported. Indeed, a change in the folding mechanism was found for two members of the cks family as a result of weakening the cooperativity of the core of the protein [START_REF] Seeliger | Weak cooperativity in the core causes a switch in folding mechanism between two proteins of the cks family[END_REF].

Recent molecular dynamics

In our previous work [START_REF] Faisca | Identifying critical residues in protein folding: Insights from φ-values and P fold analysis[END_REF] Each square in the contact map represents a native contact. For structures that like ours are maximally compact cuboids with N = 48 residues there are 57 native contacts. A non-local contact between two residues i and j is defined as LR if their sequence separation is at least 12 units, i.e., |i -j| ≥ 12 [START_REF] Gromiha | Comparison between long-range interactions and contact order in determining the folding rate of two-state folders: Application of long-range order to folding rate prediction[END_REF]. Accordingly, the number of LR (white squares) contacts in geometry 1 is 19 and in geometry 2 is 42.

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  φ values is, however, not straightforward as they might indicate the existence of multiple folding pathways or a unique TS with genuinely weakened interactions [8]. Moreover, the so-called nonclassical φ-values (φ > 1 or φ < 0 ) are difficult to interpret in the traditional φ-value model. Recently, Weikl and co-workers have proposed a new model that is able to capture and interpret nonclassical φ-values [9, 10]. The model assumes that cooperative structural elements (e.g., and α-helix or a β-hairpin) are fully formed in the TS, and that a mutation on a single residue affects the whole structural element where the native contacts established by the mutated residue are located. Likewise, the activation energy of folding in the definition of the φ-values contain explicit free energy contributions from different substructural elements of the protein. This model was recently applied to study two small βsheet domains, namely the FBP and PIN WW domains, which were assumed to fold via two distinct folding pathways (and transition states), and it was able to reproduce φ-values in good agreement with those reported experimentally [11].

(

  i.e., nucleating) residues in protein folding and providing a structural characterization of the TS was recently presented in [42]. This study involved measuring the folding time (i.e., the inverse of the folding rate) of all possible single point mutations and many double point mutations in two model proteins with different native geometries. Here we propose to explore the link between conformational plasticity and a gross geometric trait of the native structure, the relative number of local and non-local native contacts, by using as starting point the mutational data reported in [42]. For each native geometry, we selected a number of mutations with different kinetic effects. We investigate the relation between folding A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT plasticity and native geometry by studying the impact of these mutations on the folding 'reaction' from the point of view of the conformational changes it encompasses. This is done by monitoring for each mutant the degree of nativeness of each residue along the folding process and comparing it with the wild type protein.

  previously described alters upon mutation? To answer this question we have selected single point mutations that produce different kinetic effects. For both geometries we have investigated the effect on conformational plasticity of the most deleterious and of the milder forms of single point mutations. We have also considered the impact on the conformational changes occurring during folding of the most deleterious double point mutations. Thus, for geometry 1 we have looked into the folding patterns obtained upon mutating residue 29 (which leads to the largest folding time increase of 200%), residues 5 and 36 (neutral mutations located below and above the sequence midpoint respectively, and also residue 8 (which decreases the WT protein folding time by 16.2%) (Figure 4). A simultaneous mutation on the residues 20 and 30, which is the most deleterious double point mutation for geometry 1, increases the folding time of the WT protein by almost two A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT orders of magnitude [42]. We have also investigated its impact on conformational plasticity. The pattern of the folding reaction associated with any one of these mutations is sharply different from that exhibited by the WT protein (Figure 2), which is indicative that the less complex geometry has access to several folding pathways. For the more complex geometry 2, one has selected the mutation on residue 36 (which produces the largest increase in folding time of 600%), on residue 24 (which is the only mutation for which a vanishingly small change of 1.2% in the folding time is observed), and also on residue 17 (which leads to a mild increase of 27% the WT protein's folding time). We have also investigated the effects of two double point mutations, namely on residues 7 and 34, and on residues 35 and 36. These mutations increase the WT protein's folding time by more than two orders of magnitude [42]. A scenario, considerably different from

  simulations of the unfolding of CI2 at high temperature showed that the observation of sequential or multiple unfolding pathways depends on the 'resolution', i.e. level of structural detail, at which one observes the unfolding process. Indeed, at the level of individual contact formation the unfolding of CI2 happens through highly parallel folding pathways while at the coarser level of contact clusters sequential folding events emerge [22]. Experimentally, however, the observation of multiple folding pathways has proven a much more challenging task, and indirect evidence for alternative routes to the native state has been reported for engineered proteins only and the term folding or conformational plasticity was coined to denote the multiplicity of folding A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT pathways identified for the perturbed (i.e., mutated) system. Folding plasticity for small perturbations (i.e., mild mutations) is taken as an indication that multiple folding routes exist for the wild-type protein. Here we have investigated the effect of singleand double-point mutations on the robustness of the folding reaction (and TS structure)

Interestingly, the more

  complex native fold, i.e., the one that is rich is non-local native contacts is clearly less symmetric than the more local geometry, suggesting that a relation similar to that found by Klimov and Thirumalai between the symmetry of the tertiary structure and the folding plasticity [26] may also hold A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT for lattice proteins.

FIG. 1 :

 1 FIG.1: Three dimensional representation of geometry 1 and 2 (top row) and corresponding contact maps (bottom row). Each square in the contact map represents a native contact. For structures that like ours are maximally compact cuboids with N = 48 residues there are 57 native contacts. A non-local contact between two residues i and j is defined as LR if their sequence separation is at least 12 units, i.e., |i -j| ≥ 12[START_REF] Gromiha | Comparison between long-range interactions and contact order in determining the folding rate of two-state folders: Application of long-range order to folding rate prediction[END_REF]. Accordingly, the number of LR (white squares) contacts in geometry 1 is 19 and in geometry 2 is 42.

FIG. 2 :FIG. 3 :FIG. 4 :FIG. 5 :

 2345 FIG.2: Folding patterns of geometry 1: probability that each residue is in its native environment along the reaction coordinate fraction of native contacts, Q, for the wild type (top left) and mutated sequences in geometry 1.

we have concluded that native folds having a dis- tinctively large number of non-local native contacts are more suitable targets for TS studies based on the use of

φ-value analysis.

The results reported here strengthen this conclusion; indeed, due to their smaller folding plasticity, model proteins with a distinctively large number of LR contacts are preferable targets for TS studies based on the traditional interpretation of

	V. ACKNOWLEDGMENTS
	P.F.N.F. thanks Fundação para a Ciência e Tecnologia (FCT) for financial support
	through grants SFRH/BPD/21492/2005 and POCI/QUI/58482/2004.
	[1] Dobson C. M., Karplus M. The fundamentals of protein folding: bringing together theory and
	experiment, Curr. Opinion Struct. Biol. (1999) 9:92-101.
	[2] Weihua G., Shea J. E., Berry S. The physics of the interactions governing folding and associ-

φ-values. ation of proteins, Ann. N. Y. Acad. Sci. (2005) 1066:34-53.


Kamila Środa

Krystyna Michalak

Jadwiga Maniewska

Grzegorz Grynkiewicz

Wieslaw Szeja

Jadwiga Zawisza

Andrzej B Hendrich
email: hendrich@biofiz.am.wroc.pl

Wiesław Szeja

Wiesław Szeja

Genistein derivatives decrease liposome membrane integrity -calcein release and molecular modeling study

Keywords: genistein derivatives, calcein release, molecular modeling, membrane permeability

Introduction

Peroxidation of lipids, as well as other membrane components, by reactive oxygen species -ROS -leads to the impairment of membrane structure and functions, and in consequence may cause many serious diseases like atherosclerosis, stroke, cardiovascular disease or cancer [START_REF] Middleton Jr | The effects of plant flavonoids on mammalian cells: implications for inflammation, heart disease, and cancer[END_REF], [START_REF] Hendrich | Flavonoid-membrane interactions: possible consequences for biological effects of some polyphenolic compounds[END_REF]. Detrimental effects of oxidative damage can be reduced by the presence of different molecules possessing antioxidant, ROS scavenging properties -flavonoids belong to the best known representatives of this group of substances. When considering the molecular mechanism of flavonoid antioxidant activity exerted on lipid membranes it seems obvious that interaction of flavonoids with lipids must be involved [3 -6]. Since the position of flavonoids within lipid bilayer seemed to be essential for their activity several attempts were made to determine the localization of membrane-bound flavonoid molecules. In the early calorimetric study Saija et al. [START_REF] Saija | Flavanoids as antioxidant agents: importance of their interaction with biomembranes[END_REF] suggested that quercetin, hesperetin, naringenin and rutin superficially interact with lipid bilayer. On the other hand penetration of hydrophobic bilayer core by a group of flavonoids and isoflavonoids was deduced by Arora et al. [START_REF] Arora | Modulation of liposomal membrane fluidity by flavonoids and isoflavonoids[END_REF] on the basis of fluorescence spectroscopic experiments. Intrinsic fluorescence properties of fisetin were used by Sengupta et al. [START_REF] Sengupta | Investigations on the binding and antioxidant properties of the plant flavonoid fisetin in model biomembranes[END_REF] to show that this molecule was localized and rigidly bound at the polar/apolar interfacial region of lipid bilayer. In a series of 1 H magic angel spinning NMR experiments it was shown that in fact flavonoid molecules are broadly distributed along the membrane normal with a maximal probability of localization close to polar/apolar interface [START_REF] Scheidt | Investigation of the membrane localization and distribution of flavonoids by high-resolution magic angle spinning NMR spectroscopy[END_REF]. After intercalation of flavonoids into membrane the antioxidant effects can be achieved by the two complementary mechanisms -i) direct scavenging of ROS by flavonoid molecules and/or ii) reduction of the ROS access to the interior of membrane caused by flavonoidinduced alteration of membrane fluidity. This last effect was recorded in the studies on the impact of different flavonoid types on biophysical properties of lipid bilayers: green tea catechins [START_REF] Erlejman | The interaction of flavonoids with membranes: potential determinant of flavonoid antioxidant effects[END_REF][START_REF] Tsuchiya | Effects of green tea catechins on membrane fluidity[END_REF][START_REF] Tsuchiya | Takeuchi Membrane-rigidifying effects of anti-cancer dietary factors[END_REF], flavonols [START_REF] Erlejman | The interaction of flavonoids with membranes: potential determinant of flavonoid antioxidant effects[END_REF] or isoflavones [START_REF] Arora | Modulation of liposomal membrane fluidity by flavonoids and isoflavonoids[END_REF][START_REF] Erlejman | The interaction of flavonoids with membranes: potential determinant of flavonoid antioxidant effects[END_REF]. The consequences of flavonoid-

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ACCEPTED MANUSCRIPT 4 membrane interactions are not limited to the antioxidant action of these drugs: alteration of bilayer properties may affect also the activity of integral proteins and thus influence transport and other membrane-related processes [START_REF] Hendrich | Flavonoid-membrane interactions: possible consequences for biological effects of some polyphenolic compounds[END_REF][START_REF] Michalak | The role of the membrane actions of phenothiazines and flavonoids as functional modulators[END_REF]. Such effects were found for erythrocyte membrane proteins [START_REF] Pawlikowska-Pawlęga | The study of the quercetin action on human erythrocyte membranes[END_REF][START_REF] Pawlikowska-Pawlega | Modification of membranes by quercetin, a naturally occurring flavonoid, via its incorporation in the polar head group[END_REF] and also for gramicidin ion channels [START_REF] Hwang | Genistein can modulate channel function by a phosphorylation-independent mechanism: importance of hydrophobic mismatch and bilayer mechanics[END_REF].

Calcein leakage from liposomes is a simple but efficient assay often used in the studies on the influence of various factors on membrane permeability or integrity [15 -17]. This method was also applied to monitor the effects exerted by different flavonoids on vesicle [START_REF] Ollila | Characterization of flavonoidbiomembrane interactions[END_REF] and giant unilamellar liposome membranes [START_REF] Tamba | Single GUV method reveals interaction of tea catechin (-)-epigallocatechin gallate with lipid membranes[END_REF]. In the present paper the calcein leakage assay was used to study the impact of a group of newly synthesized genistein derivatives on the liposome membrane permeability. The results of these experiments are discussed in context of different descriptors of the flavonoids molecular structure obtained by the computer-assisted molecular modeling.

Materials and Methods

Chemicals

Genistein and its derivatives were synthesized at Department of Organic, Bioorganic

A C C E P T E D M A N U S C R I P T

Mesaurement of calcein release from liposomes was performed using the method developed by Gubernator et al. [START_REF] Gubernator | Dual effect of alkylresorcinols, natural amphiphilic compounds, upon liposomal permeability[END_REF]. In brief, the samples containing chloroform solution of lipid were dried under the stream of nitrogen to prepare a thin lipid film on the wall of a test-tube.

Residual amounts of organic solvent were removed under vacuum for at least 2 hours. The The permeability of liposome membranes was calculated using the following formula:

[ ] % 100 0 0 ⋅ - - = ∞ F F F F P t A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 6
where P is liposome membrane permeability, F t is calcein fluorescence intensity in the sample, F 0 is initial fluorescence, and F ∞ is maximal fluorescence of the sample after lysis by Triton X-100.

Since in our experiments DMSO was used to obtain stock solution of flavonoids we additionally checked if there was any significant influence of this compound on liposome permeability. We found that DMSO added alone at the same amount as in the rest of experiments caused only slight increase of calcein leakage (P was increased by less than 5%).

Molecular modeling

Theoretical calculations were performed using Titan 1.0.8 software (Wavefunction, Inc., Irvine, USA & Schrodinger, Inc., Portland, USA). The properties of studied isoflavonoids were modeled using AM1 semi-empirical molecular orbital method.

Results & Discussion

In present paper we describe the results of the studies on the effect of a group of newly the other hand the lowest increase of permeability (70%) was found for Lac-3. Genistein itself caused the weakest effect since at saturation the permeability increase was 45%-50% and the saturation was observed at higher concentration (25 μM) than in case of its studied herein derivatives. The reasons for the saturation of calcein release observed at higher concentrations of studied flavonoids are unclear. This effect might be caused by the limited interactions of flavonoids with lipid membranes which may result from the self-aggregation of flavonoid molecules at high enough concentrations. The aggregation effect was found for morin [START_REF] Liu | Interaction of morin with CTMAB: aggregation in micellar[END_REF] and intuitively is supported by our observations of very poor solubility of studied flavonoids in water.

Since the most profound changes of membrane permeability were observed in narrow flavonoid concentration range of 0. This indicates that fluorine in the IFG74 aromatic system may play an important role in interaction of this flavonoid with lipid molecules. This suggestion seems to be confirmed by the example of another flavonoid IFG73, which structurally is similar to IFG74 but substituted by -CN instead of fluorine at 4-th position of additional benzyl ring. As shown in Fig. 2B IFG73 at concentration 1.0 μM induces less intensive dye efflux from liposomes (24.37%) than IFG 74 at the same concentration (54.81%).

To study more deeply the role of flavonoid structure in interactions of these derivatives with the liposome membrane we calculated several molecular descriptors that characterize the studied compounds. The values of these descriptors are presented in Table 1. Comparing the order of genistein benzyl derivatives permeabilizing abilities with order of certain descriptor values it is easy to find that among the ovality, octanol/water partition coefficient (Log P), molecular polarizability, HOMO and LUMO energies, and molecular dipole moment only this last property seems to be somehow related with experimentally determined flavonoids activity. This conclusion is based on observation that the order of membrane permeabilization (IFG74 > IFG73 > IFG71 > IFG62) represents the inversed order of molecular dipole moments.

Dipole moment is one of the molecular descriptors that are often used in different QSAR studies, which were performed among others to predict antiparasitic properties of thieno[2,3b]pyridine derivatives [START_REF] Bernardino | Design, synthesis, SAR, and biological evaluation of new 4-(phenylamino)thieno[2,3-b]pyridine derivatives[END_REF], cyclooxygenase-2 inhibition by 3-phenoxypyran-4-one [START_REF] Shekhawat | Quantitative structure-activity relationship study of orally active cyclooxygenase-2 (COX-2) inhibitors of derivatives of 3-phenoxypyran-4-one[END_REF], and also anticancer effects of flavonoids [START_REF] Singh | Development of QSAR models to predict estrogenic, carcinogenic, and cancer protective effects of phytoestrogens[END_REF]. Experimental results show, however, that genistein is much less active than any of its benzyl derivatives. Therefore we conclude that the ability of genistein benzyl derivatives to interact with liposome membrane and permeabilize it is governed by the dipole moment of the additional fragment of the molecule.

Ram-3 and Lac-3 are of course excluded from the above way of reasoning because their sugar moieties are highly hydrophilic and they could not intercalate into lipid bilayer like benzyl derivatives. Despite of this they are able to destabilize the liposome membrane and permeabilize it as efficiently as benzyl derivatives.

In the present work we have shown that benzyl and glycosylated derivatives of genistein are able to interact with liposome membrane and permeabilize it. Comparing the experimental

A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T A C C E P T E D M A N U S C R I P T

  Chemistry and Biotechnology of Silesian Technical University, their purity was checked by HPLC and NMR. The chemical structures of the isoflavones used in the present paper are shown in Fig. 1. Since studied flavonoids were almost insoluble in water, in experiments their DMSO stock (5 mM) solutions were used. Egg yolk phoshphatidylcholine (EYPC) and fluorescent label -bis[N,N-bis(carboxymethyl) aminomethyl]fluorescein (calcein) were purchased from SIGMA (Poznań, Poland). Calcein before use was purified by chromatography on Sephadex G-50 column.

  lipid film was hydrated with 35 mM calcein solution in buffer (10 mM Hepes, pH 7.4). The obtained multilamellar liposomes were calibrated by extrusion (seven times) through 200 nm nucleopore polycarbonate filters (Whatman Millipore, USA). Afterwards, the liposome suspension was purified from non-encapsulated fluorescent dye by gel filtration on a Sepharose 4B column (1 cm x 20 cm) equilibrated with Hepes buffer (10 mM Hepes, 150 mM NaCl, 1 mM EDTA, pH = 7.4). Stock solutions of studied compounds was prepared in DMSO (5 mM). Flavonoids were added to calcein-loaded liposomes (10 μM of lipid per sample) to achieve the appropriate range of concentrations for the planned experiments. The liposomes containing calcein were incubated with studied flavonoids for 15 minutes in darkness at room temperature. The degree of calcein release was determined with Perkin Elmer spectrofluorimeter LS 50B. The excitation wavelength was 490 nm and the emission wavelength was 520 nm. The total amount of calcein in liposomes was assessed by measuring the fluorescence of liposome lysate obtained by addition of Triton X-100 to a final concentration of 0.1%.

  synthesized flavonoids on the membrane permeability. The alteration of calcein fluorescence intensity was used to monitor the bilayer destabilization induced by increasing flavonoid concentrations. Calcein was entrapped in unilamellar liposomes composed of egg yolk phosphatidylcholine (EYPC), which in the temperature of experiments (25 o C) was in a liquid-crystalline phase. The initial concentration of calcein in liposomes was high enough to ensure the self-quenching of its fluorescence. Decrease of the dye concentration occurring when the fluorescence probe leaked from the vesicles into the surrounding buffer caused the increase of its fluorescence. All flavonoids (except genistein) were tested in the same concentration range (from 0.5 μM up to 61 μM), their influence on liposome permeability is A C C E P T E D M A N U S C R I P T shown in Fig. 2A. As follows from this figure, we observed similar pattern of relation between fluorescence intensity and flavonoid concentration for all tested compounds. For low concentrations (0.5 μM -5 μM) we recorded the rapid increase of the calcein fluorescence, while for higher concentrations this effect was saturated and almost constant values of fluorescence were observed. This pattern of permeability-drug concentration dependence is in keeping with our previous studies [20] in which we have observed that the silybin -major biologically active flavanolignan extracted from milk thistle (Sylibum marianum) -was able to increase the permeability of EYPC liposomes for calcein in a similar way. The saturation level of the permeability increase varied for different studied flavonoids. The highest permeability was observed for liposomes modified with IFG74 and IFG62 (almost 80%), on

5 - 5

 55 μM, then Fig. 2B shows how particular genistein derivatives differed in the induction of calcein leakage from liposomes. Analyzing the dependence of liposome permeability on flavonoid concentration one may divide the studied compounds into three groups: IFG62 and IFG71 showing only moderate dynamics of changes, IFG73 and Ram-3 characterized by medium activity, Lac-3 and IFG74 presenting A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8 the most dynamic effects. As shown in Fig 2B IFG 62 and IFG71 at 0.5 μM concentration did not induced a significant calcein leakage but at 5 μM concentration the compounds induced much intensive dye efflux from liposomes: 70% and 65% for IFG62 and IFG71, respectively. The effects of IFG 73 and Ram-3 were stronger than those of IFG62 and IFG71 but weaker than IFG74 and Lac-3. Significant increase of calcein leakage was observed after the addition of IFG74 even at concentration lower than 1 μM. On the other hand, the second of the most active compounds (Lac-3) caused the saturation of calcein leakage at relatively low concentration -2.5 μM. Generally, comparing the influence of genistein benzyl derivatives on liposome membrane permeability one can classify them in the following order: IFG 74 > IFG 73 > IFG 71 > IFG 62. When comparing the effect of glycosylated derivatives on membrane permeability it was recognized that for Lac-3 increase of permeability appeared at lower concentrations but the maximal increase (saturation level) of the calcein leakage observed at high flavonoid concentrations was stronger for Ram-3. The experimental results described above should be discussed concerning the chemical structure of studied compounds. The case of the most effective genistein benzyl derivative -IFG 74 seems to be particular. This compound has a fluorine atom at the position 4 of additional benzyl ring. The molecules containing in their structure fluorine, the most electronegative halogen, display a number of physical properties that are unmatched by any other organic molecules of medicinal interest. The electronegativity and size of fluorine orbitals are similar to those of carbon and therefore the carbon-fluorine bond is one of the most energetic in which a carbon atom can participate. Due to the big difference in electronegativity between carbon and fluorine the bond between these atoms is characterized by a large dipole moment. Presence of this dipole moment together with the distribution of charge in a specific molecule, may significantly enhance the molecule's ability to engage in intramolecular interactions. This way of reasoning holds also for aromatic systems, where the A C C E P T E D M A N U S C R I P T presence of fluorine changes the electrostatic distribution of the molecular surface. It may lead to the enhancement of the ability of aromatic fluorine to bind to different molecules [22].

  Benzyl genistein derivatives studied in present work differed only by one fragment of the molecule (by group substituting additional benzene ring). This prompted us to ask a question if the dipole moment of entire molecule is important for observed effect or, maybe the dipole moment of molecule's fragment has bigger impact on the A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 10 interaction of flavonoids with lipid bilayer. To answer this question we have calculated the dipole moment and its spatial components also for the "head" of IFG74 (benzene ring substituted by methyl group and fluorine in para position). Considering the vectors of dipole moments of genistein, IFG74 and IFG74 head we concluded that small value of IFG74 dipole moment results presumably from the summing-up of the genistein and IFG74 head dipole moment vectors (schematically presented in Fig. 3). This conclusion is in keeping with the presented above discussion of the role of fluorine atom. Further calculation and spatial analysis of the dipole moment vectors of the "heads" of other genistein benzyl derivatives (data not shown) have demonstrated that they are oriented in space similarly to IFG74 head and the order of their values repeats the order of permeabilizing effects exerted by these compounds on liposome membranes. Final argument confirming the role of local dipole moment of the molecule's fragment is provided by the comparison of the dipole moments and liposome pemeabilizing activity of parent genistein and its benzyl derivatives. According to the value of the dipole moment of the entire molecule genistein (1.25 Debye) should be only slightly less effective than IFG74 (0.71 Debye) and more effective than IFG73 (1.98 Debye).

Fig. 1 .

 1 Fig. 1. Chemical structures of genistein and its derivatives.

Fig. 2 .

 2 Fig. 2. The dependence of liposome permeability on the concentration of genistein (ο) and its

Fig. 3 .

 3 Fig. 3. Dipole moment vector of IFG74 as a sum of genistein and IFG74 head dipole

Figure 1 A

 1 Figure 1

Table 1 .

 1 Molecular descriptors of genistein and its benzyl derivatives. Log P is a logarithm of octanol-water partition coefficient, Polariz. -the electronic polarizability of molecule, HOMO -highest occupied molecular orbital energy, LUMO -lowest unoccupied molecular orbital energy, M.D x , M.D y , M.D z , M.D. -spatial components and total dipole moment, respectively.Spatial components of IFG74 head were calculated assuming that x-y plane lies in the plane of its benzene ring and do not include fact that this ring is twisted with the respect to the plane of genistein molecule.

	Flavonoid	Ovality	Log P	Polariz.	HOMO	LUMO	M.D x	M.D y	M.D z	M.D.
				[Å 3 ]	[eV]	[eV]	[Debye]	[Debye]	[Debye]	[Debye]
	Genistein	1.46	1.74	58.31	-8.80	-0.70	0.60	0.27	-1.06	1.25
	IFG62	1.67	3.61	67.90	-8.73	-0.60	0.76	-2.21	-1.16	2.60
	IFG71	1.65	4.30	66.90	-8.77	-0.66	1.80	-1.21	-0.62	2.26
	IFG73	1.65	3.77	67.31	-8.83	-0.77	1.05	-0.69	1.52	1.98
	IFG74	1.63	3.90	66.19	-8.79	-0.70	0.63	0.30	0.08	0.71
	IFG74head	-	-	-	-	-	-0.03	0.0	-1.86	1.86

Acknowledgements

This work was supported by a grant 2 P05A 090 20 from Polish Ministry of Science and Higher Education.

results obtained for benzyl derivatives with theoretical calculations we suggested also that local dipole moment of additional benzene ring (with its substitutions) might play an important role in flavonoid-membrane interactions. This suggestion may be of some importance for QSAR studies in which mostly the dipole moments of entire studied molecules are taken into account.

A C C E P T E D

M A N U S C R I P T


Rubin Gulaboski
email: rubingulaboski@excite.com

Milivoj Lovrić

Valentin Mirceski

Ivan Bogeski

Markus Hoth

A New Rapid and Simple Method to Determine the Kinetics of Electrode Reactions of Biologically Relevant Compounds from the Half-Peak Width of the Square-Wave Voltammograms

Keywords: charge transfer kinetics, standard rate constant, square-wave voltammetry, half-peak width, theoretical calculations, protein-film voltammetry, redox reactions

come

A New Rapid and Simple Method to Determine the Kinetics of Electrode Reactions of Biologically Relevant Compounds from the Half-Peak Width of the Square-Wave Voltammograms

Introduction

The electron (or charge) transfer reactions are central to the function of proteins and other redox active compounds in many biological processes. This is well known in bioenergetics: photosynthesis and respiration realize energy conversion through a complex sequence of electron transfer reactions. However, electron transfer also takes place in many other biological processes ranging from cell defense to gene control. The rate of electron transfer from a donor D to an acceptor A is a key parameter that determines biological function, and much effort has been made to relate the rate of electron transfer to structural and thermodynamic features of the compounds of interest.

In the last 20 years the square-wave voltammetry (SWV) emerged as an inevitable instrumental tool in numerous research laboratories dealing with biochemical, physiological, physical, chemical, environmental and medical issues [START_REF] Osteryoung | Square-Wave Voltammetry[END_REF][START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF][START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Smith | Redox properties of several bacterial ferredoxins using square-wave voltammetry[END_REF][START_REF] Scholz | Electrochemistry of Immobilized Particles and Droplets[END_REF][START_REF] Hu | Direct electrochemistry of redox proteins or enzymes at various film electrodes and their possible applications in monitoring some pollutants[END_REF]. The electrochemical analysis of various biologically relevant redox-active compounds with square-wave voltammetry attracted significant attention, mainly because of the ability of SWV to gain detailed insight into mechanisms of chemical reactions coupled to electron transfer, as well as due to appropriateness of the technique for sensory biotechnology [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Wu | Biosensors based on direct electron transfer in redox proteins[END_REF][START_REF] Armstrong | Bioelectrochemistry of Biomacromolecules[END_REF][START_REF] Armstrong | Reactions of complex metalloproteins studied by protein-film voltammetry[END_REF]. The main attributes and the most important theoretical and experimental achievements of the SWV have been thoroughly reviewed in a very recent monograph [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF]. Quantifying the strengths of interactions between biologically important compounds, and revealing the mechanistic pathways of many proteins and enzymes attached to the surface of the working electrodes are just some of the new exciting fields in which square-wave voltammetry has been explored as a simple and powerful tool [START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF][START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Armstrong | Bioelectrochemistry of Biomacromolecules[END_REF][START_REF] Armstrong | Reactions of complex metalloproteins studied by protein-film voltammetry[END_REF][START_REF] Leger | Enzyme electrokinetics: using protein film voltammetry to investigate redox enzymes and their mechanisms[END_REF][START_REF] Obrien | Low-Temperature electrochemistry of Chromium porphyrins -characterization of transient species[END_REF].

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The SWV technique allows accurate quantification of various kinetic and thermodynamic parameters connected to electron transfer reactions of many redox active compounds in very elegant ways [1-3, 12,13]. Considering the electrode reaction of a single compound, the main physical parameters that describe the electron transfer reaction between the electrode and the investigated compound are: the standard redox potential E o (which is tightly connected to the magnitude of the energetic barrier of electron transfer), the number of exchanged electrons involved in the elementary act of electrochemical transformation n, the electron transfer coefficient α (i.e. the symmetry factor of the energetic barrier [START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF]), and the standard rate constant of electron transfer k s . The recent theoretical considerations of various redox systems analyzed by SWV permitted valuable information about the kinetic and thermodynamic parameters of electron transfer steps by various redox systems [START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF][START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Garay | Quasi-reversible EC reactions at spherical microelectrodes analysed by square-wave voltammetry[END_REF][START_REF] Garay | Square-wave voltammetry of quasi-reversible electrode processes with coupled homogeneous chemical reactions[END_REF][START_REF] Miles | The theory of square wave voltammetry at uniformly accessible hydrodynamic electrodes[END_REF][START_REF] Miles | Simulation of square-wave voltammetry at a channel electrode: E, EC and ECE processes[END_REF][START_REF] Brookes | Simulation of square wave voltammetry: quasireversible electrode processes[END_REF][START_REF] Mirceski | Surface catalytic mechanism in square-wave voltammetry[END_REF][START_REF] Mirceski | Adsorption effects in square-wave voltammetry of an EC mechanism[END_REF][START_REF] Gulaboski | Square-wave voltammetry of cathodic stripping reactions. diagnostic criteria, redox kinetic measurements, and analytical applications[END_REF][START_REF] Mirceski | EC mechanism of an adsorbed redox couple. Volume vs surface chemical reaction[END_REF][START_REF] Mirceski | Charge transfer kinetics in thin-film voltammetry. Theoretical study under conditions of square-wave voltammetry[END_REF][START_REF] Mirceski | Square-wave voltammetry of an EC reaction of a partly adsorbed redox couple[END_REF]. To date, the kinetics of electron transfer reactions have been most commonly characterized by cyclic voltammetry [START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF][START_REF] Laviron | Adsorption, autoinhibition and autocatalysis in polarography and in linear potential sweep voltammetry[END_REF][START_REF] Laviron | Use of linear potential sweep voltammetry and of AC voltammetry for the study of the surface electrochemical reaction of strongly adsorbed systems and of redox modified electrodes[END_REF]. The kinetic characterization with the square-wave voltammetry has become particularly interesting due to the simplicity and accuracy of the developed methods [START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF][START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Garay | Quasi-reversible EC reactions at spherical microelectrodes analysed by square-wave voltammetry[END_REF][START_REF] Garay | Square-wave voltammetry of quasi-reversible electrode processes with coupled homogeneous chemical reactions[END_REF][START_REF] Miles | The theory of square wave voltammetry at uniformly accessible hydrodynamic electrodes[END_REF][START_REF] Miles | Simulation of square-wave voltammetry at a channel electrode: E, EC and ECE processes[END_REF][START_REF] Brookes | Simulation of square wave voltammetry: quasireversible electrode processes[END_REF][START_REF] Mirceski | Surface catalytic mechanism in square-wave voltammetry[END_REF][START_REF] Mirceski | Adsorption effects in square-wave voltammetry of an EC mechanism[END_REF][START_REF] Gulaboski | Square-wave voltammetry of cathodic stripping reactions. diagnostic criteria, redox kinetic measurements, and analytical applications[END_REF][START_REF] Mirceski | EC mechanism of an adsorbed redox couple. Volume vs surface chemical reaction[END_REF][START_REF] Mirceski | Charge transfer kinetics in thin-film voltammetry. Theoretical study under conditions of square-wave voltammetry[END_REF][START_REF] Mirceski | Square-wave voltammetry of an EC reaction of a partly adsorbed redox couple[END_REF]. While the peak potentials (and peak-to-peak separation) as a function of the SW frequency have commonly been used to determine the kinetics of redox reactions from the dissolved state [START_REF] Osteryoung | Square-Wave Voltammetry[END_REF][START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF][START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Odea | Theory of square-wave voltammetry for kinetic systems[END_REF], the phenomenon of "quasireversible maximum" (based on the peak-current measurements) has emerged as a simple and viable method to determine kinetic parameters of surface-confined redox reactions [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Odea | Characterization of quasi-reversible surface processes by square-wave voltammetry[END_REF][START_REF] Mirceski | Surface catalytic mechanism in square-wave voltammetry[END_REF][START_REF] Mirceski | Adsorption effects in square-wave voltammetry of an EC mechanism[END_REF][START_REF] Gulaboski | Square-wave voltammetry of cathodic stripping reactions. diagnostic criteria, redox kinetic measurements, and analytical applications[END_REF][START_REF] Mirceski | EC mechanism of an adsorbed redox couple. Volume vs surface chemical reaction[END_REF][START_REF] Mirceski | Charge transfer kinetics in thin-film voltammetry. Theoretical study under conditions of square-wave voltammetry[END_REF][START_REF] Mirceski | Square-wave voltammetry of an EC reaction of a partly adsorbed redox couple[END_REF]. In the present paper, we present theoretical results that are related to the measuring of kinetic parameters of the electron transfer reactions, in which the half-peak width of the net square-wave voltammograms is the main studied feature. To the best of our knowledge, there is no

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 5
paper in the literature about the kinetic measurements extracted from the half-peak widths of the square-wave voltammograms. We develop a new theoretical method to determine the kinetics of electrode reactions from the half-peak width of the square-wave voltammograms. Redox reactions in a dissolved state and surface redox reactions of strongly adsorbed redox couple (which can be considered as a diffusionless redox system) were analyzed. The new method is a simple and viable alternative for quick and accurate assessment of the physical parameters related to the electron transfer reactions of redox systems.

Results and Discussions

Theoretical net SW voltammograms are bell-shaped curves characterized by peak potential ∆E p , peak current Ψ p , and half-peak width ∆E p/2 (figure 1). The half-peak width is an item defined as the width (measured in millivolts) at the half-height of the net SW voltammetric peak. All these parameters of the voltammetric curves are a function of the potential modulation parameters (frequency f, amplitude E sw , and potential increment ∆E), the number of exchanged electrons n, the electron transfer coefficient α, and the temperature T, as well as of the dimensionless electrode kinetic parameter K. For redox reactions taking place from a dissolved state, the dimensionless kinetic parameter K is defined as K = k s (fD) -0.5 , while for the surface redox reaction K is defined as K = k s /f. By k s we assign the standard rate constant of electron transfer (k s is given in cm s -1 for reaction from the dissolved state and in s -1 for surface redox reactions), while D (cm 2 s -1 )

is the common diffusion coefficient of both species of the dissolved redox couple.

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

6 Detailed studies of the features of simple surface redox reaction and redox reaction from the dissolved state related to the influence of the kinetic parameter K, E sw , α, and n under conditions of square-wave voltammetry can be found elsewhere [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Odea | Characterization of quasi-reversible surface processes by square-wave voltammetry[END_REF][START_REF] Odea | Theory of square-wave voltammetry for kinetic systems[END_REF]. In this work, we focus on the influence of the kinetic parameters of the electron transfer reaction on the half-peak widths of theoretical square-wave voltammograms.

A. Analysis of the half-peak width of the SW voltammograms in the case of an electrode reaction taking place from dissolved state

In this case, we consider simple redox reaction taking place from dissolved state that can be described by the following reaction scheme:

A (dissolved) + ne -⇔ B (dissolved) (I)
The half-peak width of the SW voltammograms of a redox reaction from dissolved state (at constant temperature) is influenced by the normalized square-wave amplitude nE sw , the electron transfer coefficient α, and the dimensionless kinetic parameter K. The sensitivity of the half-peak width to the number of exchanged electrons n and the electron transfer coefficient α has been briefly mentioned in the work of O'Dea and Osteryoungs' [START_REF] Odea | Characterization of quasi-reversible surface processes by square-wave voltammetry[END_REF][START_REF] Odea | Theory of square-wave voltammetry for kinetic systems[END_REF] In the next set of figures, we show in more details how the SW voltammetric responses are affected by n and α. In figure 2A, the net square-wave voltammograms calculated for 3 different values of n are shown. Increasing the number of electrons involved in the direct act of the electrochemical transformation is followed by a decrease of the half-peak width of the simulated SW voltammograms. This feature is a direct consequence of the sensitivity of the shape of forward and backward components of the corresponding

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 7
square-wave voltammetric responses to n. Both, forward and backward component are getting much narrower as n increases (see figure 2B). Since the net current in SWV is defined as a difference between the forward and the backward current components, the shape of the net SWV peak always reflects changes of the forward and/or backward SW components. Besides the number of electrons, the half-peak widths of theoretical squarewave voltammograms are also sensitive to the charge transfer coefficient α (see figures 3A and 3B). Increasing of the value of electron transfer coefficient α decreases the peakto-peak separations between the forward and backward current components (figure 3A), which again results in a decrease of the net half-peak widths (figure 3B).

Determination of charge-transfer coefficient alpha (α α α α) from the net square-wave voltammograms

Most commonly, for a given number of electrons exchanged n, the half-peak width of the net voltammograms of an irreversible electrode reaction depends mainly on the electron transfer coefficient α [START_REF] Osteryoung | Square-Wave Voltammetry[END_REF][START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF][START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Odea | Theory of square-wave voltammetry for kinetic systems[END_REF]. In the region of a very sluggish (irreversible) electron transfer, the half-peak width ∆E p/2 is a linear function of the temperature. The magnitude of the half-peak widths decreases linearly with the temperature, with a slope being inversely proportional of α (see figure 4). The dependence ∆E p/2 vs. T in the irreversible region is defined as

T nF R z E ) ( p/2 α = ∆
, where the factor z is a constant depending on E sw .

Since the half-peak width is independent on the electrode kinetic parameter K (K = k s (Df) - 0.5 ) in the irreversible kinetic region (ln(K) < -3) [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Odea | Theory of square-wave voltammetry for kinetic systems[END_REF], the electron transfer coefficient can be estimated from the analysis of ∆E p/2 vs. T (see figure 4). At this stage, it is worth

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8
to stress that the temperature dependence should be done in some reasonable temperature interval (i.e. not very drastic changes in T) in order to avoid the strong changes of the value of the diffusion coefficient (and strong changes in the values of K, correspondingly), which is, indeed, temperature sensitive parameter. It is worth mentioning, that in the real experiment, if a quasireversible electrode reaction is being considered, the transfer of the electrode system into the irreversible kinetic region can be achieved by increasing of the frequency of the potential modulation.

Effect of the kinetic parameter K to the half-peak width of square-wave voltammograms

The calculated curves representing the effect of the dimensionless kinetic parameter K on the half-peak widths for several values of the electron transfer coefficient α are shown in figures 5A and 5B (for n =1 and n = 2, respectively). In all cases, the dependences between ∆E p/2 and ln(K) are sigmoidal. In the region of very sluggish electron transfer (roughly for ln(K) < -3) as well as in the region of very fast electron transfer (ln(K) > 0), the half-peak width is insensitive to the kinetic parameter K. However, in the region of modest electron transfer (the so-called "quasireversible region"), significant changes in the half-peak widths occur. Increasing of the value of K in the quasireversible region is followed by a decrease of the magnitude of the half-peak width. The origin of such behaviour is revealed by analysing the properties of the forward and backward components of the SW voltammograms as a function of K (figure 6A). As the value of K increases, the peak-to-peak separation between the forward and the backward current components decrease. This results in a decrease of the half-peak width of the net SW voltammograms in the quasireversible kinetic region (see figure 6B). In the

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 9
quasireversible region, the magnitude of the half-peak width is a linear function of ln(K) (see figure 5 A, B), with a slope being inversely proportional to the electron transfer coefficient α. The equations corresponding to the theoretical linear dependences between the half-peak widths ∆E p/2 and ln(K) in the quasireversible region of electron transfer are given in Table 1 (for n = 1 and n = 2 and several square-wave amplitudes and α values).

The equations given in Table 1 can be routinely used for quick determination of the standard rate constant of electron transfer k s from the simple half-peak widths measurements of the experimental square-wave voltammograms. This is a very simple approach for precise determination of the standard rate constant of the electrode reaction.

In the real experiment, one needs to measure the half-peak width only for a given set of parameters of the potential modulation. With a help of equations listed in Table 1, one can calculate the value of the electrode kinetic parameter K, provided that n, α, and D are already known. From the definition of the electrode kinetic parameter, it follows that the standard rate constant of the electrode reaction can be calculated from the following simple relation:

f D K k = s
. This procedure has been applied for a series of experimental systems presented in the literature listed in Table 3.1. The standard rate constants measured with other SW voltammetric methods (third column) are compared to the values estimated with the approach presented here (fourth column).

A C C E P T E D M A N U S C R I P T

B. Sensitivity of the half-peak widths of SW voltammograms by the surface redox reactions

In this part we elaborate the half-peak width features of theoretical SW voltammograms of a simple surface electrode reaction that can be described by following reaction scheme:

A (adsorbed) + ne -⇔ B (adsorbed) (II)
Both, the reactant A and the product B are strongly adsorbed to the surface of electrode surface, so that the mass transport can be completely neglected. All general principles of part A can also be applied to the surface electrode reaction. Again, for a constant temperature, the magnitudes of the half-peak widths of the simulated square-wave voltammograms depend in the same way on n, α, and the kinetic parameter K (in this situation K is defined as K = k s /f). Similar to the case of a dissolved redox couple, the dependence ∆E p/2 vs. T in the irreversible kinetic region can be used for the determination of electron transfer coefficient α (see figure 8). Theoretical curves of the half-peak widths ∆E p/2 as function of the kinetic parameter ln(K) are calculated for several values of α and n =1 and n = 2 and are presented in figures 7A and 7B, respectively. For the case of a very slow electron transfer reaction (ln(K) < -2), and for fast electron transfers (ln(K) > -0.2) the magnitude of the theoretical half-peak widths are insensitive to K (it is important to stress that in the regions of very fast electron transfers (ln(K) > 0.2), the net SW peak splits into two peaks, a phenomenon discussed elsewhere) [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Odea | Characterization of quasi-reversible surface processes by square-wave voltammetry[END_REF]27]. Within these two kinetic regions i.e. -2 < ln(K) < -0.2, the half-peak widths of theoretical net square-wave voltammograms decreases considerably by increasing of the value of ln(K).

The effect of the K to the forward, backward and net components of the theoretical SW voltammograms in the quasireversible region of electron transfer can be easily obtained from the voltammograms presented in figure 9A and 9B, respectively. The slopes of the linear parts of the theoretical curves presented in figure 7A and 7B are again inversely proportional to the values of α. The equations corresponding to the region of the linear dependence between ∆E p/2 and ln(K) are given in Table 2. As for the case of redox reactions taking place from dissolved state, the equations given in Table 2 can be used for a very simple and precise determination of the value of the standard rate constant of electron transfer k s and even from the single measurements of the half-peak widths of the experimental voltammograms, providing that the values of n and α are known. This method can be regarded as a viable alternative for the methods of "quasireversible maximum" and "splitting SW peaks" [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF] for accessing the kinetic parameters of the simple surface redox reactions featuring moderate electron transfers.

A C C E P T E D M A N U S C R I P T

Conclusions

The demands for rapid and precise determinations of the kinetic parameters of electron (charge) transfer reactions by many biologically important redox systems require development of simple and reliable methods. To date, the majority of the physical parameters characterizing the electron transfer reactions by various biologically relevant redox systems have been commonly accessed from the cyclo-voltammetric experiments [START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF]. The recent developments of several theoretical works analysing different redox systems studied under conditions of square-wave voltammetry provided a large number of elegant methods for assessing a variety of thermodynamic and kinetics parameters [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF]27]. Generally, the magnitudes of the SW peak potentials and the SW peak currents, analysed as a function of the time (frequency), have been explored as main items for the determination of the kinetic parameters of the electron transfer reactions [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF]. In this work, we present a theoretical set of results, which shows that the half-peak width of the net square-wave voltammograms can be used as a method for quick and reliable determination of the physical parameters characterizing the electron transfer reactions of a dissolved state and on the electrode surface reactions. With the help of the theoretical equations given in the Tables 1 and2, we have estimated the standard rate constant k s of numerous biologically relevant redox systems, by using this new method based on the half-peak width measurements. The estimated values of k s are given in Table 3.1. and methodologies. This shows that the half-peak width measuring method of the SW voltammograms can be successfully explored as a routine tool for rapid and precise determination of the kinetic parameters of electron transfer reactions featuring moderate electron transfers. Moreover, it can be also seen as a simple and viable alternative for the other voltammetric methods developed for kinetics characterizations of the electron transfer rates, as well as for assessing the ion-transfer kinetics by the four-electrode experiments of ion transfer across the liquid-liquid interface [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Scholz | Electrochemistry of Immobilized Particles and Droplets[END_REF]. In particular, the method can be of a significant importance for the kinetic characterizations of the electron transfer rates by surface-confined metalo-protein reactions studied with protein-film voltammetry [START_REF] Armstrong | Bioelectrochemistry of Biomacromolecules[END_REF][START_REF] Armstrong | Reactions of complex metalloproteins studied by protein-film voltammetry[END_REF][START_REF] Leger | Enzyme electrokinetics: using protein film voltammetry to investigate redox enzymes and their mechanisms[END_REF][START_REF] Jeuken | Insights into gated electron-transfer kinetics at the electrode-protein interface: A square wave voltammetry study of the blue copper protein azurin[END_REF][START_REF] Hirst | Fast-scan cyclic voltammetry of protein films on pyrolytic graphite edge electrodes: Characteristics of electron exchange[END_REF][START_REF] Mcevoy | Protein film cryovoltammetry: demonstrations with a 7Fe ([3Fe-4S]+[4Fe-4S]) ferredoxin[END_REF]. The main advantage of this method, against the familiar cylco-voltammetric method based on the dependence of the peak-to-peak separation as a function of the scan rate [START_REF] Bard | Electrochemical Methods, Fundamentals and Applications[END_REF], is the avoidance of the IR drop effect, which is inevitably present when using large scan-rates in cyclo-voltammetric measurements. By using the corresponding equations given in Tables 1 and2, we give a simple and easy method to determine the kinetic parameters of various charge-transfer reactions without making comprehensive frequency analysis, thus avoiding the undesired IR drop effects to the charge transfer kinetics [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF].

At the end, it is worth to mention once again that the estimation of the standard rate constants of electron transfer with the help of theoretical equations given in Tables 1 and2

A C C E P T E D M A N U S C R I P T

Appendix:

Mathematical Model and Simulation Details

Redox reaction from dissolved state

A quasireversible electrode reaction of two chemically stable species dissolved in the electrolyte solution is described by:

A (dissolved) + ne -⇔ B (dissolved) (I)
The electrode mechanism (I) can be described mathematically by the following set of equations:

        ∂ ∂ = ∂ ∂ 2 2 ) ( ) ( x A c D t A c (1)         ∂ ∂ = ∂ ∂ 2 2 ) B ( ) B ( x c D t c (2) t = 0, x→ ∝ : c(A) = c(A)*; c(B) = 0 (3) t > 0, x→ ∝ : c(A) + c(B) = c(A)* ∞ → (A) c , 0 (B) → c t > 0, x = 0: nFS I x c D x c D x x =       ∂ ∂ - =       ∂ ∂ = = 0 0 ) B ( ) A ( (4)
- - = = t x t d D nFS I A c A c 0 0 ) ( ) ( 1 * ) ( ) ( τ τ τ π (6) ∫ - = = t x t d D nFS I B c 0 0 ) ( ) ( 1 ) ( τ τ τ π (7).
By substituting equations ( 6) and ( 7) into the (5) one gets the integral equation ( 8) that represents mathematically the quasireversible electrode reaction of a dissolved redox couple under voltammetric conditions:

        - - - - - = ∫ ∫ τ τ π φ τ τ π αφ d t D nFS I d t D nFS I A c k nFS I t t s 0 0 ) ( 1 1 ) exp( ) ( 1 1 * ) ( ) exp( (8)
The solution of equation ( 8) under conditions of square-wave voltammetry was obtained with the help of numerical method proposed by Nicholson and Olmstead [START_REF] Nicholson | Olmsetad Electrochemistry: Calculations, Simulation and Instrumentation[END_REF]. Both, the time variable and dimensionless current were discretized. To each time period t = jd, where d is the time increment, a certain current Ψ j can be ascribed. The numerical solution for the equation ( 8) reads:

A C C E P T E D M A N U S C R I P T [ ] ) exp( 1 ( 50 ) exp( 2 1 )) exp( 1 ( 50 2 1 ) exp( 1 1 1 j j j i i j i j j j K S K φ π αφ φ π αφ + - +       Ψ + - - = Ψ ∑ - = + - (9), where 1 - - = j j S j
, and K is the dimensionless kinetic parameter defined as

Df k K s =
. For the calculations, the time increment d = 1/(50f) was used, which means that each SW half-period was divided into 25 increments. The dimensionless current Ψ is normalized as:

D f A nFSc I Ψ * ) ( = .
A comprehensive study of this redox mechanism can be found elsewhere [START_REF] Mirceski | Square-Wave Voltammetry, Theory and Application[END_REF][START_REF] Odea | Theory of square-wave voltammetry for kinetic systems[END_REF].

Surface electrode reaction

In the second case, we consider surface confined (diffusionless) electrode reaction of strongly adsorbed redox couple that can be represented by following reaction scheme:

A (adsorbed) + ne -⇔ B (adsorbed) (II)
It is assumed that all participants in the electrode mechanism (II) are strongly adsorbed on the electrode surface. During the voltammetric experiment the mass transport of all species is neglected. The electrode mechanism (II) is mathematically represented by the following set of equations:

dΓ(A)/dt = -I/(nFS) (10)
dΓ(B)/dt = I/(nFS) (11)

t = 0; Γ(A) = Γ(A)*; Γ(B) = 0 (12) t > 0; Γ(A) + Γ(B) = Γ(A)* (13)
.

A C C E P T E D M A N U S C R I P T

Γ is a symbol of the surface concentration, which is a function of time t, whereas Γ(A)*

is the total surface concentration of all species. The solutions are:

∫ - = t nFS I Γ Γ 0 d ) ( * (A) (A) τ τ (14) ∫ = t nFS I Γ 0 d ) ( (B) τ τ (15).
Considering again the Buttler-Volmer formalism, at the electrode surface the following condition applies:

[ ] ) B ( ) exp( ) A ( ) exp( Γ Γ k nFS I s φ αφ - - = (16),
where k s (s -1 ) is the heterogeneous electron exchange rate constant corresponding to the 13) and ( 14) into the equation (15) yields:

      - - - = ∫ ∫ t t s nFS I nFS I Γ k nFS I 0 0 d ) ( ) exp( d ) ( * (A) ) exp( τ τ φ τ τ αφ (17)
Integral equation ( 16) is a general mathematical solution of the simple surface electrode mechanism. Numerical solution according to the method of Nicholson and Olmstead [START_REF] Nicholson | Olmsetad Electrochemistry: Calculations, Simulation and Instrumentation[END_REF] is:

[ ] ) exp( 1 50 ) exp( 1 50 ) exp( 1 1 ) exp( 1 1 m m m j j m m Ψ K Ψ φ φ φ αφ + - +       + - - = ∑ - = (18). A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 19
In this case, the dimensionless redox kinetic parameter K is defined as K = k s /f, while the dimensionless currentΨ is normalized as

f nFSΓ I Ψ * (A) = .
All the simulations have been performed with help of the MATHCAD software. *for the theoretical calculations, we used the experimentally determined values for electron transfer coefficient α provided in the corresponding references.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

  experimentally determined and the theoretically calculated data of k s in Tables 3.1 and 3.2 fall within the range of statistical errors of the applied experimental

  relies to the previously determined values of the electron transfer coefficient α (by both considered redox mechanisms), and the diffusion coefficient D (by the redox A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 14 reaction from dissolved state). For the determination of electron transfer coefficient α one can use the theoretical equations and approaches described at the figure 4 and figure 8, which are also explained in more details in our recent publication [27b]. For the estimation of the values of the diffusion coefficient D one is advised to use the methodologies proposed in references [2,3].

  . c(A) and c(B) are the concentrations of the species A and B, while c(A)* is the bulk concentration of A. I is symbol of the current, S is the electrode surface area (cm 2 ), and D is the diffusion coefficient (cm 2 s -1 ) which is assumed to be equal for both A and B. In the case of the charge transfer controlled redox reaction, at the electrode surface the following condition holds:[START_REF] Scholz | Electrochemistry of Immobilized Particles and Droplets[END_REF] with k s (cm s -1 ) is assigned the heterogeneous electron exchange rate constant corresponding to the standard redox potential o A/B E of the electrode reaction (I), α is the cathodic electron transfer coefficient, and )relative electrode potential. The symbols for n, F, R and T are the number of exchanged electrons, the Faraday constant, the universal gas constant and the temperature, respectively. The solutions for c(A) and c(B) at the electrode surface have been obtainedwith help of Laplace transformations and they read:

∫

Figure 1 .

 1 Figure 1. Theoretical net square-wave voltammogram of a simple redox reaction taking place from dissolved state. The simulation conditions were: n = 1, E sw = 50 mV; ∆E = 4 mV, α = 0.4; T = 298 K, D = 5x10 -6 cm 2 s -1 . The value of the kinetic parameter was ln(K) = -1.21.

Figure 2 .

 2 Figure 2. Redox reaction from dissolved state: ln(K) = -0.068, while all the other conditions

Figure 3 .

 3 Figure 3. Redox reaction from dissolved state: ln(K) = -0.83, while all the other conditions

Figure 4 .

 4 Figure 4. Redox reaction from dissolved state: Temperature dependence of the net SW halfpeak widths ∆E p/2 simulated for various electron transfer coefficients in the region of very

Figure 5 .

 5 Figure 5. Redox reaction from dissolved state: Theoretical dependence of the half-peak

Figure 6 .

 6 Figure 6. Redox reaction from dissolved state: A) Effect of the kinetic parameter K to the

Figure 7 .

 7 Figure 7. Surface redox reaction: Theoretical dependence of the half-peak widths of the net

Figure 8 .

 8 Figure 8. Surface redox reaction: Temperature dependence of the net SW half-peak widths ∆E p/2 simulated for various electron transfer coefficients in the region of very slow electron

Figure 9 .

 9 Figure 9. Surface redox reaction: A) Effect of the kinetic parameter K to the shape of the

  theoretical calculations, we used the experimentally determined values for electron transfer coefficient α and diffusion coefficients D provided in the corresponding references.A C C E P T E DM A N U S C R I P T

  Figure 2

Table 3 .

 3 2., together with the k s values from the square-wave voltammetric experiments determined by other authors. Obviously, the estimated k s values are consistent with those

derived from other square-wave voltammetric methods. The small discrepancies that

Table 1 . 1

 11 Simple redox reaction from dissolved state: Equations of the linear dependences between the SW half-peak width ∆E p/2 and ln(K) in quasireversible region.SW voltammograms were simulated for T = 298 K, ∆E = 4mV, n = 1;

	nE sw / mV	α	Equation of the dependence ∆E p/2 -ln(K)
		0.3	∆E p/2 /mV = -43.73 ln(K) + 105
		0.4	∆E p/2 /mV = -37.94 ln(K) + 105
	30	0.5	∆E p/2 /mV = -27.85 ln(K) + 105
		0.6	∆E p/2 /mV = -15.87 ln(K) + 105
		0.7	∆E p/2 /mV = -10.90 ln(K) + 105
		0.3	∆E p/2 /mV = -49.50 ln(K) + 115
		0.4	∆E p/2 /mV = -32.10 ln(K) + 115
	40	0.5	∆E p/2 /mV = -16.05 ln(K) + 115
		0.6	∆E p/2 /mV = -14.62 ln(K) + 115
		0.7	∆E p/2 /mV = -05.40 ln(K) + 115
		0.3	∆E p/2 /mV = -49.23 ln(K) + 120
		0.4	∆E p/2 /mV = -32.06 ln(K) + 120
	50	0.5	∆E p/2 /mV = -16.70 ln(K) + 120
		0.6	∆E p/2 /mV = -15.78 ln(K) + 120
		0.7	∆E p/2 /mV = -10.60 ln(K) + 120

Table 1 . 2 .

 12 Simple redox reaction from dissolved state: n = 2; All the other conditions are same as those by Table1.1.

	nE sw / mV	α	Equation of the dependence ∆E p/2 -ln(K)
		0.3	∆E p/2 /mV = -32.60 ln(K) + 70
		0.4	∆E p/2 /mV = -16.30 ln(K) + 70
	30	0.5	∆E p/2 /mV = -10.80 ln(K) + 70
		0.6	∆E p/2 /mV = -08.70 ln(K) + 70
		0.7	∆E p/2 /mV = -05.40 ln(K) + 70
		0.3	∆E p/2 /mV = -32.60 ln(K) + 75
		0.4	∆E p/2 /mV = -19.00 ln(K) + 75
	40	0.5	∆E p/2 /mV = -10.85 ln(K) + 75
		0.6	∆E p/2 /mV = -08.70 ln(K) + 75
		0.7	∆E p/2 /mV = -05.40 ln(K) + 75
		0.3	∆E p/2 /mV = -32.56 ln(K) + 80
		0.4	∆E p/2 /mV = -21.80 ln(K) + 80
	50	0.5	∆E p/2 /mV = -10.85 ln(K) + 80
		0.6	∆E p/2 /mV = -07.05 ln(K) + 80
		0.7	∆E p/2 /mV = -05.60 ln(K) + 80

Table 2 .

 2 [START_REF] Osteryoung | Square-Wave Voltammetry[END_REF]. Simple surface redox reaction: Equations of the linear dependences between the SW half-peak width ∆E p/2 and ln(K) in quasireversible region. SW voltammograms were simulated for T = 298 K, ∆E = 2 mV, n = 1

	T / K	α	nE sw / mV	Equation of the dependence ∆E p/2 -ln(K)
			30	∆E p/2 /mV = -18.40 ln(K) + 85.00
		0.3	40	∆E p/2 /mV = -18.80 ln(K) + 100.00
			50	∆E p/2 /mV = -17.40 ln(K) + 105.00
			30	∆E p/2 /mV = -16.90 ln(K) + 85.00
		0.4	40	∆E p/2 /mV = -16.60 ln(K) + 100.00
			50	∆E p/2 /mV = -13.70 ln(K) + 105.00
			30	∆E p/2 /mV = -11.20 ln(K) + 85.00
	298.15	0.5	40	∆E p/2 /mV = -08.80 ln(K) + 95.00
			50	∆E p/2 /mV = -08.10 ln(K) + 100.00
			30	∆E p/2 /mV = -12.40 ln(K) + 85.00
		0.6	40	∆E p/2 /mV = -10.90 ln(K) + 95.00
			50	∆E p/2 /mV = -10.30 ln(K) + 100.00
			30	∆E p/2 /mV = -7.40 ln(K) + 85.00
		0.7	40	∆E p/2 /mV = -8.90 ln(K) + 95.00
			50	∆E p/2 /mV = -7.10 ln(K) + 95.00

Table 2 . 2 .

 22 Simple surface redox reaction: n = 2. All the other conditions are same as those by Table2.1.

	T / K	α	nE sw / mV	Equation of the dependence ∆E p/2 -ln(K)
			30	∆E p/2 /mV = -50.00 ln(K) + 60.00
		0.3	40	∆E p/2 /mV = -54.00 ln(K) + 62.00
			50	∆E p/2 /mV = -48.50 ln(K) + 62.00
			30	∆E p/2 /mV = -27.50 ln(K) + 54.00
		0.4	40	∆E p/2 /mV = -27.50 ln(K) + 60.00
			50	∆E p/2 /mV = -25.00 ln(K) + 60.00
			30	∆E p/2 /mV = -15.00 ln(K) + 52.00
	298.15	0.5	40	∆E p/2 /mV = -16.50 ln(K) + 56.00
			50	∆E p/2 /mV = -12.50 ln(K) + 56.00
			30	∆E p/2 /mV = -07.50 ln(K) + 50.00
		0.6	40	∆E p/2 /mV = -10.00 ln(K) + 52.00
			50	∆E p/2 /mV = -07.50 ln(K) + 52.00
			30	∆E p/2 /mV = -05.00 ln(K) + 50.00
		0.7	40	∆E p/2 /mV = -07.00 ln(K) + 52.00
			50	∆E p/2 /mV = -05.00 ln(K) + 50.00

Table 3 . 1 .

 31 Experimental and theoretical values of the standard rate constants of electron transfer of some redox systems (from dissolved state). The theoretical values were estimated from the half-peak width of square-wave violtammograms by using corresponding equations given in Table1.

	Compound	Experimental conditions	Experimental k s /cm s -1	Theoretical k s /cm s -1	Reference
	Eu 3+	Mercury electrode, 0.01 M HClO 4	0.006±0.001	0.005±0.001	[32]
		Glassy carbon
	cercosporin	electrode,	0.019±0.005	0.025±0.004	[33]
		1 M HClO 4 + ACN
		Mercury electrode,
	ZnCl +	pH = 2 (NaClO 4 +	0.0045±0.0005	0.0035±0.00045	[34]
		HClO 4 )
		Tungsten Electrode,
	La 3+	Molten-chlorides	0.004±0.001	0.006±0.0005	[35]
		mixtures
		Tungsten Electrode,
	Y 3+	Molten-chlorides	0.0037±0.001	0.0045±0.0005	[35]
		mixtures
	Re 6 Se 8 (CH 3 CN) 6 ] 2+	Pt electrode, Propylene carbonate	0.0035±0.0015	0.00315±0.0035	[36]
	Zn 2+	Mercury electrode pH = 7	0.00026±0.0001 6	0.00020±0.0001	[37]
	2-[[2-hydroxyethyl)amino]eth naphtoquinone yl]amino]-1-4-	Mercury electrode dimethylformamide N,N-	0.041±0.014	0.025±0.01	[38]
	2-[[2-dimethyl)amino]ethyl]a naphtoquinone mino]-1-4-	Mercury electrode dimethylformamide N,N-	0.041±0.013	0.025±0.01	[38]

Table 3 . 2 .

 32 Experimental and theoretical values of the standard rate constants of electron transfer of some redox systems (surface reactions). The theoretical values were estimated from the half-peak width of square-wave voltammograms by using corresponding equations given in Table2.

	Compound	Experimental conditions	Experimental k s /s -1	Theoretical k s /s -1	Reference
	Cynolline	Mercury electrode, pH = 7.0	455±110	360±40	[39]
	Azobenzene	Mercury electrode, pH = 4.65	28±10	35±5	[40]
	Mo(VI)-fulvic acid complex in presence of phenantroline	Mercury electrode, pH = 2.0	8±2	18±4	[41]
		Plane pyrolitic graphite
		electrode modified with
	Myoglobin	film of	48±2	33±5	[42]
		didodecylmethylammonium
		bromide
	Cytochrome C	ITO electrode,	8.4±2	4.80±1.5	[43]
		Pyrolitic graphite electrode
	Hemoglobin-AQ55	modified with polyester	60±10	39.70±6	[44]
		sulfonic acid
	2-Hydroxy-5-[(4-sulfophenyl)azo]benzoic acid	Mercury electrode, pH = 4.00	24±6	21.80±4	[45]
	adriamycin	Mercury electrode pH = 4.65	49±12	42.80±8	[46]
	Cd(II)-Ferron complex	Mercury electrode pH = 10.80	0.17±0.01	1.00±0.5	[47]
	4-(dimethylamino)-azobenzene	Mercury electrode pH = 6.00	102±2	119±10	[48]
	Cd(II)-8-OH-quinoline	Mercury electrode pH = 6.70	7±1	8.80±2	[49]
	Cu(II)-8-OH-quinoline	Mercury electrode pH = 6.70	1.5±0.1	2.2±1.5	[49]

Acknowledgments:

Rubin Gulaboski thanks Alexander von Humboldt Stiftung for providing a postdoctoral fellowship.

This project was funded by two competitive research grants from the Saarland University (HOMFOR both to I. Bogeski)