extfieldsofstudy: Chemistry Physics

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Dynamics of polydisperse irreversible adsorption: a pharmacological example

Many drug delivery systems suffer from undesirable interactions with the host immune system. It has been experimentally established that covalent attachment (irreversible adsorption) of suitable macromolecules to the surface of the drug carrier can reduce such undesirable interactions. A fundamental understanding of the adsorption process is still lacking. In this paper, the classical random irreversible adsorption model is generalized to capture certain essential processes involved in pharmacological applications, allowing for macromolecules of different sizes, partial overlapping of the tails of macromolecules, and the influence of reactions with the solvent on the adsorption process. Working in one dimension, an integro-differential evolution equation for the adsorption process is derived and the asymptotic behaviour of the surface area covered and the number of molecules attached to the surface is studied. Finally, equation-free dynamic renormalization tools are applied to study the asymptotically self-similar behaviour of the adsorption statistics.

Introduction
Random sequential adsorption (RSA) is a classical model for various physical, chemical or biological problems [5]. In the simplest form, RSA processes can be formulated as sequential addition to a structure of objects that cannot overlap, and once inserted, cannot move or leave the structure [1]. In this paper, we present a pharmacological example in which application of the RSA model can provide meaningful qualitative insights. Motivated by pharmacological applications, we present a slight generalization of the classical RSA model to enable us to study the effects of polydispersity and partial overlap of adsorbing macromolecules on the surface of a virus. We also study the dependence of the adsorption process on interactions (reactions) of the adsorbing macromolecules with the solvent.
The paper is organized as follows. In Section 2, we will introduce the motivating pharmacological example and our questions of interest. In Section 3, we will introduce the generalized random sequential adsorption (gRSA) model suitable for capturing essential features of the pharmacological problem from Section 2, which we formulate in one dimension. The analytical results for this model are presented in Section 4. We derive the governing integro-differential equation for the evolution of gaps between polymers, and compute the asymptotical properties of the quantities of interest, namely the number of macromolecules adsorbed and the total area (in one dimension, length) they cover. In Section 5, we apply equation-free methods to the computational study of the system. The main idea that underlies this equation-free computer-assisted analysis is the design and execution of appropriately-initialized short bursts of stochastic simulations; the results of these are processed to estimate coarse-grained quantities of interest -in this case the self-similarly evolving shape of the gap statistics in the problem. Finally, in Section 6, we discuss the higher dimensional case and summarize the connections between the theory and the experimental data.

Pharmacological background
Many medical conditions such as cancer, heart disease and heritable disorders (hemophilia, cystic fibrosis, muscular dystrophy etc) have faulty, mutant genes as an underlying cause. Healthy, normal genes can be readily synthesized in the laboratory but introducing them into diseased cells remains a challenge. Many research groups are studying viruses, such as adenovirus, as a means to introduce normal genes into diseased cells. For therapeutic use, the virus' own DNA is usually partially or completely replaced by the gene of interest. The most common adenovirus strain used for this purpose is adenovirus type 5 (Ad5), because it is easy to manipulate and is non-pathogenic in humans [14]. Ad5 has been used with great success to treat diseases in laboratory animals but the results have not been replicated in humans. One of the greatest problems with using Ad5 in humans is the presence of neutralizing antibodies. In addition, the viruses often infect non-target cells, particularly the liver, causing unwanted toxicity. Our laboratory is exploring the use of hydrophilic polymers such as poly[N-(2-hydroxypropyl)methacrylamide] (pHPMA) to coat virus particles and protect them from neutralising antibodies by steric shielding. This technique is very effective at protecting the virus and permitting it to be retargeted to specific cell types [6,7,12]. The polymer has multiple esters along its length that are used to bind to the amino side chain of lysine residues on the virus surface. In a coating reaction the polymers bind randomly to the virus surface until (a) all of the lysine residues are occupied or (b) lysine residues are rendered inaccessible (obscured) by polymer chains. We know little about the orientation of polymers on the virus surface or how to optimise the coating reaction because there are no techniques to visualise the orientation of polymers on the virus surface.
Representative questions one would like to answer are: How many polymer molecules will become attached to the viral surface by a given time? How large is the surface area covered by the polymer coat at that time? In this paper, the theoretical approach is chosen to address these questions for simplified models of the adsorption process. Since the adsorption process is driven by the diffusion of molecules to the surface of the virus, and since the adsorption is effectively irreversible, a suitable modification of the classical RSA might be applied to model the process. It is important to take into account polydispersity in the polymer size. Even if we prepare the polymer molecules with a specified target molecular weight, some relatively small molecules of the polymer will always be present, and they will diffuse faster than the larger molecules. A smaller molecule can reach the surface of the virus at a higher rate. We will consider the adsorption of polymers whose diameters are distributed according to a probability distribution function P (z). The solution is assumed to be well-mixed. We also define p(z) as the probability distribution function of the particles which can reach the surface in a single time step. An important modelling issue lies in a good choice of P (z) (e.g. it might be the Gamma distribution) and in a realistic relation between P (z) and p(z). In Section 3, we simply specify p(z) (avoiding the above questions). The long term dynamics of the polydisperse adsorption depends on the behaviour of p(z) close to zero. We will use different distributions p(z) given by (3.1).
Finally, the reactive groups of the polymer molecules can also react with the solvent before reaching the surface. This is the case for commonly used biocompatible pHPMA reactive polymers [15]. If all reactive groups of a polymer are already hydrolyzed then the polymer cannot covalently bind to the surface of the virus. Hence, we have to consider that only a fraction r(t) of the polymers is still reactive at time t. Depending on the form of r(t), different polymer coats may be created. This issue will be discussed in more detail in Section 4.4.
Polymers are long flexible molecules [4]. The pHPMA polymer molecule has (one or more) reactive group(s) which can react with the primary amino groups on the viral surface. As a result, a polymer molecule becomes covalently (irreversibly) attached to the surface at a point. The rest of the polymer is not attached (unless another covalent bond is created) and it freely "wiggles" in the space above the viral surface. Having in mind that the "wiggling tail" does not perfectly shield the underlying surface, we generalize the classical RSA model to allow partial overlap of the adsorbing macromolecules, i.e. we allow some squeezing of the polymers.
Let N(t) be the number of polymers attached to the surface at time t. Let A(t) be the total area of the surface covered by adsorbed polymers at time t. Since we cover the surface by polymers of different sizes, there is no obvious relation between A(t) and N(t). However, both variables A(t) and N(t) are of practical interest as discussed below.
We cover the surface of the virus by polymers to protect the surface from unwanted interactions. Hence, the number A(t) gives us the simple characterization of the area of the surface which is protected by the polymer coat. The unwanted interactions are not the only problem which one has to overcome in order to use viruses as a drug delivery system. Another important task is to retarget the virus to infect the cells of interest (e.g. cancer cells) via new receptors. Assuming that we put one "targeting" group per polymer molecule, the number of targeting molecules will be equal to N(t).
If we considered the adsorption of the same-size nonoverlapping objects of the area a, then we would have A(t) = aN(t). In our case, the adsorbing molecules have different sizes. There is no obvious relation between A(t) and N(t) and both quantities are of interest. In the following sections, we will present theoretical approaches to compute the time evolution of A(t) and N(t).

A simple generalization of random sequential adsorption
Random sequential adsorption has been extensively studied during the last several decades [5]. The theoretical work is more mature in one dimension with information in higher dimensions mostly coming from numerical simulations [10]. If we consider fixed size objects, then RSA usually starts from an empty surface and continues until the time when no further object can be placed, the so-called "jamming limit". If the objects to be covered have spherical symmetry [13] then the coverage approaches the jamming limit as t −1/d where t is time and d is the dimension. The asymptotic behaviour can be more complicated for objects of different shape [5]. As argued in Section 2, polydispersity is often present in real systems. If we allow adsorbing particles (in one spatial dimension) of arbitrarily small length, then the coverage approaches the full coverage as t → ∞. Relatively less is known for polydisperse adsorption. One-dimensional analytical results can be found in [10], where it is assumed that the attached polymers prevent binding of other polymer molecules that would overlap with them. In reality, the polymer molecules are stretching during the adsorption process, creating a polymer brush (for semitelechelic polymers) after sufficiently long time [11,3]. Thus each molecule "covers" a smaller surface area at later times. Consequently, it is possible to adsorb more molecules onto the surface. Here, we take this fact into account and we modify the random sequential adsorption algorithm accordingly. We state our generalized random sequential algorithm (gRSA) in one dimension as follows.
gRSA algorithm: We consider adsorption of small intervals of different sizes onto the interval [0, 1], the adsorbing domain. At each time step, the size of a small interval is chosen randomly according to the probability distribution function p(z). We select randomly the position of the center w inside the adsorbing domain [0, 1] and we make an attempt to place the small interval of length z there. If the center w of the segment to be adsorbed lies inside a segment already placed, the adsorption is rejected. If the position of the center w is chosen in the gap (x 1 , x 2 ) between attached polymers, then the new polymer segment is adsorbed with probability ξ(z, w − x 1 , x 2 − w) where w − x 1 and x 2 − w are distances of the center w of the polymer from the endpoints of the gap (x 1 , x 2 ).
The parameters of gRSA which have to be specified include the probability distribution function p(z) and the probability ξ(z, w − x 1 , x 2 − w). In what follows, we assume that the lengths of polymers are distributed according to the formula In the latter case, the maximum probability of adsorption is achieved for w = x 1 +x 2 2 , for which ξ(w) = x z . The formulas (3.2) and (3.3) give the same probability density function ξ(·) for z = x as is desirable. The plot of ξ as a function of w is given in Figure  1. Formula (3.2) is shown in Figure 1(a) where the gap size x = x 2 − x 1 is greater than the length of the new polymer segment z. Formula (3.3) is shown in Figure 1(b) where the gap size x = x 2 − x 1 is less than the length of the new polymer z.
To explain the motivation behind formula (3.2), three possible cases of the relative position of the new (red) interval of the length z ≤ x and the gap (x 1 , x 2 ) are shown in Figure 2. In Figure 2(a), the red interval is rejected because its middle point w lies inside a polymer segment which is already adsorbed to the surface. Hence, the probability of adsorption is 0, the same probability as in the classical RSA model. In Figure 2(c), the red segment of the length z does not overlap with neighbouring polymers, and we (a) Figure 1: The probability ξ as a function of w for gRSA model allow it to be adsorbed with probability ξ = 1. Cases in Figure 2(a) and Figure 2(c) are treated as in the classical RSA model. In Figure 2(b), the center of the red polymer is inside the gap but the red polymer overlaps with neighboring polymers segments. This polymer would be rejected by the classical RSA model. We believe it is more realistic to consider that such a polymer will be adsorbed with some nonzero probability which continuously interpolates between the cases shown in Figures 2(a) and 2(c), i.e. between zero for w − x 1 = 0 and 1 for w − x 1 = z. Formula (3.2) takes this fact into account, using simple linear interpolation. Formula (3.3) naturally extends the formula (3.2) for polymer segment lengths greater than the gaps (see also Figure 1).
Having explained the new rules for adsorbing the polymer, we must also specify what part of the surface is actually covered. We will assume that the new polymer covers only the intersection of the intervals This guarantees that a possibly long, newly adsorbed polymer will not "spill over" and cover any part of the neighboring gaps.

Analysis of gRSA
Let G(x, t) be the concentration of gaps (holes) of length x at time t and let C(x, t) be the corresponding cumulative probability distribution function; that is, The total length of the surface that is covered by polymers at time t, A(t), is directly related to G(x, t) by The number of polymers attached to the surface at time t, N(t), can be also related to G(x, t), as we will see in Section 4.1. Thus, the starting point of the analysis of the system is the derivation of the evolution equation for the distribution function of gaps G(x, t).
A gap of length x can be created from a larger gap (of length y > x) by adsorbing a suitable interval to the system. Thus the evolution of the concentration of gaps G(x, t) is given by the equation

Using (3.2) and (3.3), equation (4.3) can be rewritten in the following form
Hence, We assume that the lengths of polymers are distributed according to formula (3.1) for α > −1 and small ε ≪ 1. Moreover, we assume that there are already no holes of the length greater than ε/2 in the system, i.e. G(x, t) = 0 for x > ε/2. Then (using (3.1)), equation (4.5) can be rewritten (for x < ε/2 and α = 0) as Equation (4.6) (or (4.7)) is the desired integro-differential equation for G(x, t). If we solve (4.6), we can compute the evolution of A(t) by (4.2). The equation for the evolution of N(t) is given in the next section.

Evolution of N (t)
At each time step, an interval of length between (z, z + dz) is chosen with probability p(z)dz. This interval can be placed in any gap of size x with probability gives the probability that the randomly chosen position of the polymer of length z will be accepted. Thus the probability of attaching a polymer of any length at one time step is equal to Using a continuous approximation for N(t), we find that N(t) satisfies the following ordinary differential equation Taking p(z) to be given by (3.1) and ξ(z, u, x − u) to be given by (3.2) -(3.3), and considering the regime where all gaps are already less than ε (i.e. G(x, t) = 0 for x > ε), we obtain Before analyzing (4.6) and (4.10) further, we summarize some results from the literature on classical RSA.

Some results for the classical RSA
If we consider particles of the same length ε so that p(z) = δ(z − ε), and if we choose then our gRSA algorithm reduces to the classical RSA algorithm. The evolution equation (4.3) can be used to verify known one-dimensional results about fixed segment size, nonoverlapping random sequential adsorption [10], namely one can show that the jamming limit is asymptotically approached as t −1 [13]. Random sequential adsorption with a probability distribution p(z) given by (3.1) and probability ξ(z, w − x 1 , x 2 − w) given by (4.11) has been studied in [10]. Then equation (4.3) for x < ε reads as follows (assuming that initially there exist no holes of length greater than ε in the system, i.e. G(x, t) = 0 for x > ε) (4.12) The scaling ansatz [10] for the concentration G(x, t) can be written as Defining the moments and using (4.13), we obtain Moreover, multiplying equation (4.12) by x γ and integrating over x, one can derive the equation for moments, where B(·, ·) is Beta function. We define the function γ(α) implicitly by the equation F ( γ, α) = 0. If γ is equal to γ(α), then the moment M γ is independent of time. Hence, using (4.15), we obtain the relation a = b + b γ(α) between the coefficients of the scaling ansatz (4.13) and the parameter α of the model. Finally, substituting the scaling ansatz (4.13) in (4.12), we find that b = (α + 2) −1 . Thus, the scaling of moments (4.15) can be rewritten in the form (4.17) Using (4.2) and (4.17), we obtain The graph of the function ω(α) is given in Figure 3(a). The equation (4.9) for p(z) given by (3.1) and probability ξ(z, w − x 1 , x 2 − w) given by (4.11) reads as follows: Using (4.17), we obtain (for σ(α) > 0) (4.20) The graph of the function σ(α) is given in Figure 3

Evolution of gRSA
The temporal evolution of the gRSA model is more complex than the cases discussed in Section 4.2. To see this, we use the moments M γ (t) defined by (4.14). Multiplying equation (4.6) by x γ and integrating over x, we can derive the equation for moments (for α = 0), where B(·, ·) is Beta function and H(γ, α) is defined as .
First, consider the case α < 0; the dominant term on the right-hand side of (4.21) is the term ε −α−1 H(γ, α)M γ+α+2 . The zeroth order moment, gives the total number of gaps at time t. At leading order, we have There is a constant α . = −0.415 such that H(0, α) is positive for α < α and negative for α > α. We immediately see that To illustrate the result (4.23), we will execute two stochastic simulations with the gRSA algorithm. We will use ( Next, we will study the behaviour of the system for α < α. Here, we will assume the scaling ansatz (4.13). Differentiating (4.22) with respect of γ, we obtain where ψ 0 is the polygamma function. For each α < α, the equation defines implicitly the function γ(α). If γ is equal to γ(α), then the moment M γ is independent of time. Hence, using (4.15), we obtain the relation between the coefficients of the scaling ansatz (4.13) and the parameter α of the model. Finally, substituting the scaling ansatz (4.13) into (4.21), one can find that b = (α+2) −1 .
Hence, the scaling of moments (4.15) can be rewritten in the form   Using (4.2) and (4.26), we obtain The graph of the function ω(α) is given in Figure 5(a). We also plot ω(α) given by (4.18) for comparison. Using (4.10) and (4.26), we also obtain (4.28) The graph of the function σ(α) is given in Figure 5(b); we also plot σ(α) given by (4.20) for comparison. To illustrate the results (4.27) and (4.28), we will execute two gRSA stochastic simulations. We will use (3.1), (3.2) and (3.3) where α = −0.5 or α = −2/3. We choose ε = 10 −3 . The results are given in Figure 6 where the time evolution of the number of gaps and the number of adsorbed polymers are shown. The time is scaled according to In Figure 6, we also plot the cumulative distribution function C(x, t) for different times (i.e. for different numbers of polymers attached). Using the suitable rescaling C(x, t) = C(kx, t), the curves collapse to a single curve as shown in Figure 7.  Finally, one can easily show that formula (4.27) works also for the case α ≤ α < 0. To illustrate it, we plot the time evolution of the quantity [1 − A(t)] −1 for α = −0.1 and α = −0.3 in Figure 4 (bottom panel) using the scaling (4.27). On the other hand, formula (4.28) is no longer correct because integrating of (4.10) implies that N(t) is approaching a constant value. More precisely, N(t) ∼ C + t σ(α) for σ(α) < 0. If α is positive than (4.23) is valid, i.e. we have (4.23) for any α > α.

Time dependent concentration of reactive polymers
As discussed before, the reactive groups on our polymers are capable of reacting with the solvent before reaching the surface [15]. It may therefore be more realistic to consider that only a fraction of polymers r(t) is still reactive at time t. Here, r(t) ∈ [0, 1], r(0) = 1 and r(t) is a decreasing function of time.
The random sequential adsorption algorithm has to be modified as follows: at each time step, we generate the random number uniformly distributed in the interval (0, 1). If this number is greater than r(t), then the selected polymer has lost its binding site through reaction with the solvent (it cannot be adsorbed), and we continue with the next step. Otherwise, we choose randomly a position on the interval and we attempt to place the polymer there.
Depending on the form of function r(t), different dynamics can be observed. First, let us suppose that In this case, we can find a relation between the modified random sequential algorithm and the previous results. At each time t, we can compute the average waiting time ∆t before a reactive polymer hits the surface as the solution of the equation Solving (4.31), we find (t + ∆t) 1−λ = t 1−λ + 1 − λ. Hence, Consequently, we can make use of the formulas (4.18) and (4.20), or formulas (4.27) and (4.28), in the case (4.30). For example, using (4.18) and (4.20), we obtain that the quantities A(t) and N(t) satisfy the following asymptotic behaviour To illustrate the formula (4.32), we stochastically simulate the gRSA model with the probability distribution p(z) given by (3.1) and the probability ξ(z, w − x 1 , x 2 − w) given by (4.11), where the fraction of the reactive polymers in the system decreases with time according to (4.30). We select ε = 10 −3 and verify the asymptotic behaviour (4.32) for α = −0.5 and λ = 0.5. Then (4.32) implies The time evolution of A(t) and N(t) is given in Figure 8 (top panels). We also present results for α = 0 and λ = 0.33 in Figure 8 (bottom panels). Again, we scale the time according to (4.32). The reactive group is lost by chemical reaction with the solvent. It might be more natural to consider (instead of (4.30)) that the fraction of reactive polymers is exponentially decreasing, i.e. r(t) = e −λt for λ > 0. (4.33) The formula (4.33) gives rise to qualitatively different dynamics for the system, as opposed to the dynamics associated with (4.30). For simplicity, let us assume that every polymer with a functional reactive group can be adsorbed (which will give a bound on N(t) from above). Then the average number of adsorbed polymers N(t) is which implies that the number of adsorbed polymers does not approach infinity as in the previous case.

Equation-free analysis of gRSA
In the previous theory, we assumed the scaling ansatz (4.13) (see also [10]) for G(x, t) and we computed the time dependence of the quantities of interest A(t) and N(t). A related interesting question is whether we can also compute the profile Φ from (4.13). One possibility is to substitute (4.13) in equation (4.6) and solve it numerically for Φ but we will not proceed this way. Instead, we demonstrate the computation of self-similar profile Φ using only short-time appropriately initialized simulations of the stochastic gRSA model. In this equation-free context [9,2], it is easier to work with the cumulative distribution function C(x, t), which can be obtained from G(x, t) through (4.1); C(x, t) is less noisy than G(x, t) (e.g. [8]). Using (4.1) and (4.13), we obtain Figure 9: Schematic of the equation-free mapping Ψ.
Hence, we see that the cumulative density function C(x, t) scales as To compute the profile C, we can use an equation-free iterative fixed point algorithm which is shown schematically in Figure 9. Starting with the initial guess C 0 , we compute the sequence of profiles C K , K = 1, 2, 3, . . . , where C K+1 = Ψ(C K ), for K = 0, 1, 2, 3, . . . , (5.2) and where the mapping Ψ is obtained as the composition of the following four steps: (a) Given the cumulative density profile C K , create one or more microscopic realizations of gaps in the unit interval such that the initial cumulative density function is C(·, 0) = C K . (b) Use the microscopic simulator (i.e. use the gRSA algorithm) for a short time ∆t.
One possible way to rescale C(·, ∆t) is to compute the average gap size a 0 from C(·, 0) and the average gap size a ∆t from C(·, ∆t). Then the C K+1 can be computed by We now present illustrative results obtained by this fixed point computation (5.2) using the gRSA algorithm. We will use (3.1), (3.2) and (3.3) where α = −0.5 or α = −2/3. We choose ǫ = 10 −3 . The results of long term simulations for these parameter values were already shown in Figure 6. Our goal is to use the iterative formula (5.2) to compute the scaled cumulative distribution function profile which was shown in Figure 7. This algorithm allows us to find the self-similar shape by performing simulations while simulating at a scale (at relatively larger average gap sizes) where the evolution is relatively fast, compared to the long-term dynamics close to jamming. The initial guess is given as which means that initially all our gaps have the same size 1.5 × 10 −4 . At each iteration step (see Figure 9), we place 1000 gaps according to the cumulative distribution function C K to the interval [0, 1]. We evolve the simulation until 100 new polymers are placed. We then rescale the new cumulative distribution according to (5.3) and we compute C K+1 . Several first iterations are shown in Figure 10 (top panels). We see that after 20 iterations, we have effectively reached the steady state (the stationary shape of the self-similarly evolving gap distribution). More precisely, the error between iterations is small and it is not further systematically decreasing. The comparison of the equationfree 20th iteration with the results obtained by the long-time simulations are also shown in Figure 10 (bottom panels). Finally, we note that many other algorithmic possibilities for the computation of the profile C exist. The equation (5.2) seeks a fixed point of the mapping Ψ. Instead of successive substitution, other fixed point algorithms implemented in a matrix-free fashion through short simulation bursts can be used to find stationary solutions -for example, Newton-GMRES iterations [2]; these would be able to converge on even dynamically unstable self-similarly evolving distributions.

Discussion
In this paper, motivated by a pharmacological example involving polymer coating of a virus surface, we studied certain aspects of polydisperse adsorption of macromolecules in one spatial dimension. We presented an extension of the classical random sequential adsorption algorithm to capture better certain essential properties of the pharmacological model system currently used in drug development research. We introduced partial overlapping of adsorbing macromolecules, i.e. we considered that the polymers are not rigid objects but they can be deformed while attaching to the surface. We found two distinct asymptotic regimes. Depending on the parameters of the processes involved, we can observe that either (a) the number of gaps between polymers asymptotically approaches zero, or that (b) the number of gaps asymptotes to infinity and the gap distribution acquires an asymptotically self-similar profile.
We also briefly discussed the impact of a possible reaction of the polymers with the solvent on gRSA dynamics. Again, two possibilities exist. If the decay of the reactive groups is relatively weak, then the dynamics of the system remains qualitatively  unchanged and the system only evolves on a slower time scale. On the other hand, if the reactive groups decay exponentially, this decay ultimately wins over the polynomial time asymptotics of gRSA. From the applications point of view, it therefore becomes crucial to know the corresponding rate constants in order to reliably predict what type of behaviour one might expect over the time scales of interest. Typically, the coating process is performed overnight in the laboratory and different reactive groups have different half lives; measuring these rates becomes an important task.
In this paper we worked in one spatial dimension and provided analytical results about the long time behaviour of gRSA models. The analytical approach was based on two important facts: we knew what the good macroscopic observables for describing the system behaviour were and we were able to write down analytically tractable equations for these observables. The good observable for our system was a distribution of gaps G(x, t) between adsorbed polymers. If we know the initial distribution of gaps G(0, t) one could easily predict G(x, t) at future times.
On the other hand, if we know (hope) that the gap distribution G(x, t) is a good observable for the system of interest but we do not know the evolution equation for G(x, t), then it is still possible to use the equation-free methods [9]. The main idea of the equation-free methods is to use the short bursts of appropriately initialized microscopic/stochastic computations to estimate macroscopic quantities of interest on demand. Hence, if one does not have an explicit coarse-grained evolution equation for the system statistics, one can in principle avoid long, brute-force simulations. This might be the case for one-dimensional adsorption problems with more complicated microscopic evolution rules.
The situation becomes significantly more difficult in the higher dimensional case. Here, the analytical theory is far behind in development, and the literature contains mostly computational results. The first question for higher dimensional adsorption is the nature of the "good" coarse-grained observables for the system. Good observables (the variables in terms of which the unavailable effective model would be written) are necessary for developing a useful analytical theory. Knowing appropriate coarsegrained observables is also an important feature of equation-free algorithms. Having one-dimensional analogues in mind, we see that one needs an effective way to describe the statistics of "gaps" (free space) in higher dimensions. If we cannot estimate (by intuition or by suitable algorithms for the detection of low-dimensionality in high-dimensional data) effectively good observables for the system, then the direct, brute-force computationally intensive simulations might be the only modelling option. In this paper we showed cases where we could do more than brute-force simulation and provided analytical results giving insights into the dynamics of gRSA.
The problems studied in this paper were motivated by the pharmacological example mentioned above, and realistic predictive modelling of the problem clearly requires extensive model parameter information that must come from experimental data. As we showed, we can expect different dynamics of the problem depending on the values of the parameters of the polymer and the virus which are used. Obtaining reliably such parameters and bounds on their uncertainty for our particular model problem is non-trivial, and we are not yet ready to report about it.


L-Proline sensor based on layer-by-layer immobilization of thermostable dye-linked L-proline dehydrogenase and polymerized mediator

Abstract L-Proline sensor has been fabricated through multilayer assembly of recombinant dye-linked L-proline dehydrogenase (L-proDH) originally from hyperthermophilic archaeon (Therococcus profundus) and polymerized mediator, poly(allylamine) ferrocene (PAA-Fc), on a gold electrode by electrostatic layer-by-layer alternate adsorption method. The characteristic redox peaks of ferrocene groups were obvious exhibited in cyclic voltammograms, and both anodic and cathodic peaks increased with PAA-Fc/L-proDH bilayer number which confirmed the formation of a multilayer structure. Further experiments showed that an electrode fabricated in this way catalyzed the oxidation of L-proline. It suggested that polymerized mediator could transfer electrons between electrode surface and immobilized L-proDH. Ampeormetric experiments showed that the current response to L-proline increased with PAA-Fc/L-proDH bilayer number. The stability of the sensor was measured, and it kept a 40% relative response after 30 days storage in buffer under 4 °C.

Introduction
As an essential component, L-proline is important for proper functioning of joints and tendons. It not only implicates some disorders in the human body, but is also a practical subject in foods and pharmaceutical industries [1,2]. Several methods for L-proline determination have been developed and reported. They were based on fluorescence [3], chromatography [4][5][6] and electrochemiluminescence [7]. Although, these methods can determine the substrate in a relatively sensitive scale, chromatography and fluorescence methods always require a long time for analysis and pre-treatment of sample. The disadvantages of chemiluminescence include expensive and toxic reagents and interferences from impurities in real samples.
As a simple and sensitive method, electrochemical detection has been developed for the determination of amino acids. For example, biosensors incorporating broad-spectrum amino acid oxidase (AAO) with electrochemical transducer were utilized in the construction of amperometric biosensors for the determination of various kinds of substrates with high sensitivity and rapid analytical speed [8,9]. Furthermore, biosensors fabricated by immobilizing specific amino acid enzyme have also been reported [10].
Previously, dye-linked L-proline dehydrogenase (L-proDH) originally from hyperthermophilic archaeon (Thermococcus profundus) has been expressed as a recombinant protein [11,12]. The enzyme showed high thermal stability, and a dye compound such as 2,6-dichlorophenolindophenol (DCPIP) could be used as an artificial electron acceptor for enzyme reaction. Our research aim is to develop an electrochemical L-proline sensing system based on this enzyme.
Design and preparation, especially immobilization of this enzyme is a crucial procedure in the construction of enzymatic biosensors. This is because enzymes should be stabilized and easily contacted with substrates, and the immobilization procedure should be beneficial to electron transfer between enzyme and electrode surface by redox mediator [13,14]. As a novel method proposed by Decher and co-workers, layer-bylayer (LBL) adsorption method has been widely used in the assembly of nanoscale films composed of polyelectrolytes, such as proteins, DNA and, etc. It has been proved that LBL method is a promising way especially for the immobilization of biomolecules [15][16][17][18][19]. LBL method, which is based on electrostatic being forced between oppositely charged polyelectrolytes has been utilized on the immobilization of enzymes and construction of a biosensing system [20,21]. Various kinds of enzymes have been immobilized by LBL, and enzymes encapsulated in these integrated multilayer films showed superior chemical stability in long storage period and relative thermostability [22,23]. On the other hand, a biosensing system based on the alternated assembly of enzyme and polymerized mediator were developed, which made it possible to construct a reagentless sensor system [24][25][26][27][28]. In our previous work, LBL method has been used in the construction of a bienzyme biosensor [29].
In the present study, dye-linked L-proDH and polymerized mediator were fabricated on a gold electrode by LBL method. The electrochemical characteristics of the modified electrode and its application on the construction of L-proline biosensing system were investigated.

Synthesis of polymerized mediator
The polymerized mediator was synthesized from poly(allylamine) and ferrocene carboxaldehyde according to the procedure reported by Hodak et al. [24]. Eight milligrams ferrocene carboxaldehyde was dissolved in 5 ml methanol and added dropwise within 1 h to 30 ml of methanolic solution containing 40 mg poly(allylamine) and 0.26 ml triethylamine. The mixture was stirred for 1 h at room temperature, then sodium borohydride was carefully added in portions at 0 8C, and stirred continually for 90 min. Finally, the mixture was dried in vacuo condition and the residue was extracted with distilled water. The aqueous solution was purified by membrane dialysis against water. The polymer obtained was referred to as PAA-Fc and used for further experiments.

Purification of L-proDH
Recombinant dye-linked L-proDH originally from hyperthermophilic archaeon was prepared and purified according to previous procedures [11]. The activity of the enzyme was routinely determined spectrophotometrically by measuring the reduction rate of DCPIP at 595 nm according to the previous procedures described [11,12]. The reaction mixture was composed of 200 mM Tris-HCl buffer (pH 7.5), 0.1 mM DCPIP, 100 mM L-proline and enzyme solution with a total volume of 1.0 ml. One unit of activity was defined as the amount of enzyme that reduced 1 mmol of DCPIP per min at 50 8C. The molar absorption coefficient of 2.15! 10 4 M K1 cm K1 was used for calculating the reduction of DCPIP. The activity of the purified enzyme was measured at 6.0 units ml K1 (with 3.0 mg protein).

Assembly of multilayer films
The gold electrode was finely polished by 1 mm diamond and 0.05 mm aluminum slurry and scanned over the potential range from C0.0 to C1.7 V vs. Ag/AgCl in 0.05 mM H 2 SO 4 solution for about 30 min until the same cyclic voltammogram was obtained. The cleaned electrode was immersed in 10 mM 3-MPA ethanolic solution for 12 h, followed by rinsing with ethanol and water. A negative charged surface was achieved in this way.
Multilayer assembly was fabricated on 3-MPA modified electrode by alternative adsorption in PAA-Fc and L-proDH solution. The electrode was firstly immersed in 2 mg ml K1 PAA-Fc solution for 20 min. Then it was rinsed with Mill-Q water and immersed in 3 mg-protein ml K1 L-proDH (6.0 units ml K1 ) in 10 mM phosphate buffer (pH 7.2) for another 20 min to immobilize the enzyme. A multilayer film could be obtained by repeating the above steps.

Electrochemical experiments
Electrochemical measurements were carried out on Hokuto Denko potentiostat HA-502 and function generator (HB-104) with an X-Y recorder, Model U-335 from Pantos Co. (Japan). A typical three electrode system was used, with Pt wire as the counter electrode, Ag/AgCl as the reference electrode and modified Au electrode as the working electrode. In all electrochemical experiments, 10 mM phosphate solution (pH 7.2) was used as the buffer, and the buffer solution was bubbling by using high purity nitrogen for more than 5 min before measurements to eliminate the influence of oxygen. A potential of C0.45 V vs. Ag/AgCl was set in amperometric experiments.

Results and discussion
The general properties of this thermostable dye-linked L-proDH were thoroughly characterized by Sakuraba et al. [11,12]. The enzyme (molecular weight 160 kDa) consists of four subunits (abgd), and b subunit catalyzes the dye-linked L-proline dehydrogenase reaction. As a dye-linked enzyme, L-proDH can use some organic compounds (such as DCPIP) as electron acceptors. For the construction of L-proline biosensing system, it is important to find a suitable mediator for L-proDH.
Because ferrocene and its derivatives are the most common mediators for various kinds of oxidoreductase, ferrocene carboxylic acid was used for preliminary experiments. Both L-proDH and ferrocene carboxylic acid were free in the buffer, and cyclic voltammograms were measured in the absence and presence of L-proline. Only specific redox peaks of ferrocene carboxylic acid were exhibited in the absence of L-proline. But after the addition of L-proline, the anodic peak increased dramatically and cathodic peak nearly disappeared. This confirmed the mediated reaction happened and ferrocene was an efficient mediator for this enzyme.
In order to develop reagentless biosensing system, a polymerized mediator (PAA-Fc) containing ferrocene groups as redox species was synthesized. Total iron content in the PAA polymer was estimated to be approxymately 20% spectrophotometrically at 440 nm using a calibration curve forferrocene carboxylic acid [24]. Both PAA-Fc and L-proDH were immobilized on Au electrode by LBL method. The reaction scheme is shown in Fig. 1.

Formation of PAA-Fc/L-proDH multilayer film on Au electrode and electrochemical behavior of the modified electrode
The adhesion layer was obtained after adsorption of 3-MPA (pK a 5.2) on Au electrode via its thiol end groups. At the buffer (pH 7.2), the electrode carries negative charges due to partially deprotonation of the carboxyl acid groups [30]. Because PAA-Fc carries positive charges and L-proDH carries negative charges under present experimental conditions, the multilayer film could be formed by an electrostatic force between oppositely charged polyelectrolyte and enzyme (as shown in Fig. 2). Characteristics of the electrode modified with PAA-Fc/ L-proDH multilayer film were investigated by cyclic voltammetry. Fig. 3 shows the cyclic voltammograms of Au electrode modified with one to six PAA-Fc/L-proDH bilayers. A pair of oxidation-reduction peaks were obviously apparent (with midpoint at C0.37 V vs. Ag/AgCl). Both anodic and cathodic peaks increased with the number of PAA-Fc/L-proDH bilayers, which suggested that a multilayer structure had formed on the electrode surface. The amount of ferrocene groups in the multilayer film could be estimated by integrating the area of oxidation peaks in Fig. 3. The charge density of electrochemical active ferrocene was shown in the insert figure. It was observed that the amount of ferrocene groups increased with the bilayers, from one to five, but seemed to be saturated for the sixth bilayer. In the previous paper, QCM was used for measuring and analyzing the amount of enzyme and polymer, and a linear increase of adsorption amount was found for assembly more than 10 bilayers by LBL method [19]. So, it could be evaluated that nearly same amount of Fc in each mediator layer. For multilayer assembly, Fc moieties on the inner layers are electrochemically active, but those moieties on the outer layers may not be active because electron hopping is not sufficient. The electrochemical characteristic of the multilayer film was further studied under different scan rates. Cyclic voltammetric experiments of Au electrode modified with three and six PAA-Fc/L-proDH bilayers were carried out at different scan rates (from 40 to 200 mV s K1 ), and the influence of the scan rate on anodic current peaks is shown in Fig. 4. The anodic peak currents increased linearly with the scan rate for Au electrode modified with three PAA-Fc/L-proDH bilayers, indicating a diffusion-free electron transfer between ferrocene groups and electrode. On the other hand, for Au electrode modified with six PAA-Fc/L-proDH bilayer film, the anodic peak currents did not depend linearly but were proportional to the square root of the scan rate. As illustrated by Liu et al. [31], a charge diffusion contributes more or less to electron transfer for the thicker films fabricated by LBL method. This might be the reason for the phenomena observed in these experiments.

Electrocatalytic oxidation of L-proline on electrode modified with PAA-Fc/L-proDH multilayer film
The response of the multilayer film modified electrode to L-proline was characterized by cyclic voltammetry. Cyclic voltammograms of Au electrode modified with four PAA-Fc/ L-proDH bilayers before and after addition of L-proline are shown in Fig. 5. In the absence of L-proline, cyclic voltammogram gave specific redox peaks of ferrocene only. But after the addition of L-proline, the anodic peak increased and cathodic peak decreased, which suggested that a mediated reaction occurred.
It could be inferred that after immobilization of PAA-Fc and L-proDH, some ferrocene groups near the active center of the enzyme undertook the electron transfers. The total reaction process is evaluated as following (L-proDH ox and L-proDH red represent the oxidized and reduced form of the dye-linked L-proline dehydrogenase, respectively). L-proline diffused to the surface of the multilayer film modified electrode, and was oxidized to pyrroline-5-carboxylate by immobilized L-proDH ox . The L-proDH red produced in the enzymatic reaction, reduced the mediator PAA-Fc C to PAA-Fc. Finally, the PAA-Fc produced was then electrochemically oxidized and gave anodic current response. By this way, the communication between immobilized enzyme and electrode was achieved which is essential for the construction of biosensing system, and the immobilized L-proDH was 'wired' by PAA-Fc.

Steady-state current response of the modified electrode to L-proline
The typical steady-state amperometric response of gold electrode modified with four PAA-Fc/L-proDH bilayers is shown in Fig. 6. The applied anodic potential was set at C0.45 V vs. Ag/AgCl. It was found clearly that anodic current increased with the injection of L-proline solution.
The influence of bilayer number on amperometric response was further investigated and shown in Fig. 7. The current response to substrate increased with the bilayer number two to six. It was attributed to the more immobilized enzyme and mediator with the layer number increasing. But the current response obtained was relatively low which might be due to the low activity of the enzyme solution used in the fabrication of the multilayer film. Now efforts are being made to prepare a more concentrated enzyme solution with high activity.

Long-term stability of the multilayer film
The stability of the L-proline sensor was also investigated by measuring its current response to L-proline during long-term storage. Au electrode was modified with four PAA-Fc/L-proDH bilayers, and its response to 8 mM L-proline was measured. The modified electrode was stored in the buffer under 4 8C when not in use. The relative response after long-term storage is shown in Fig. 8. No decrease in response to L-proline was observed for the first 12 days, which suggested that the enzyme kept its activity after immobilization by LBL method, and the mutilayer film was stable. The response decreased continuously after 12 days. It seemed to be due to the deactivation of L-proDH, desorption of PAA-Fc, or the enzyme from the multilayer film. About 40% of the relative response was retained after 30 days storage.

Conclusion
In the present work, LBL method has been utilized in the construction of L-proline sensor based on multilayer assembly of polymerized mediator (PAA-Fc) and dye-linked L-proDH on Au electrode. Cyclic voltammetry showed clearly redox peaks of ferrocene, and both cyclic voltammetry and amperometry experiments proved that catalyzed electrochemical oxidation of L-proline was achieved on a modified electrode, which   confirms that electrochemical communication has been achieved between immobilized enzyme and electrode by the polymerized mediator in the multilayer film. The current response to L-proline increased with PAA-Fc/L-proDH bilayer number, and the L-proline sensor with multilayer film showed relatively long-term stability.


Graphical Abstract: ChemBioChem 7/2006

The cover picture shows the alignment of the two geometric isomers (E isomer in blue, Z in turquoise) of 2-(3-benzylidene)-3-oxo-2,3-dihydrobenzo[b]thiophene-7carboxylic acid methyl ester at the active site of rabbit 12/15-lipoxygenase. In the Z configuration, the benzoyl moiety conflicts with the side chain of the C-terminal I663, which normally ligates the catalytic non-heme iron. Light irradiation induces Z-to-E isomerization, which flips the benzoyl ring over towards I593, the side chain of which limits the depth of the substrate binding pocket. There are no steric constraints for binding of the E isomer. Further details can be found in the article by H. Kuhn et al. on p. 1089 ff.

-1016
Transforming Carbonic Anhydrase into Epoxide Synthase by Metal Exchange Enantioselective epoxidation of styrene was observed in the presence of manganese-containing carbonic anhydrase as catalyst. The probable oxygentransfer reagent is peroxymonocarbonate, which has a structural similarity with the hydrogenocarbonate substrate of the natural reaction. Styrene was chosen as the enzyme possesses a small hydrophobic cavity close to the active site. Does aglycone linker conformation matter? Both E-and Z-linked a-C-galactosylceramides have been prepared by application of the Julia-Lythgoe-Kocienski synthesis to a C-formylgalactose and sulfones derived from phytosphingosine. Assays of cytokine levels induced by GalCers in mice suggest that the E geometry better resembles the preferred conformations of the flexible, saturated C-glycoside and the O-glycoside when bound to the CD1d molecule of antigen-presenting cells.

-1027
Target-Assisted Selection of Galactosyltransferase Binders from Dynamic Combinatorial Libraries. An Unexpected Solution with Restricted Amounts of the Enzyme Fix and look: Dynamic combinatorial chemistry coupled to an irreversible transformation has been used to identify enzyme binders in a library directed towards an a-1,3-galactosyltransferase. Small amounts of the protein induce an accumulation of the nonbinding amine products such as A resulting from the reduction of the corresponding imine binders. The binding properties of the imines are best mimicked by amides such as B.

-1030
Thermostable Glycosynthases and Thioglycoligases Derived from Thermotoga maritima b-Glucuronidase By using active-site mutants of the Thermotoga maritima b-glucuronidase, glucuronic acid-containing glycosides and thioglycosides have been made in high yields. These enzymes have potential use in the chemoenzymatic synthesis of a range of glycoconjugates, such as glycosaminoglycans (X = OH, NAc, H), that contain b-linked glucuronic and galacturonic acids. Substrate activity screening was used to rapidly identify a novel and potent (k inact /K i = 59 000 m À1 s À1 ) nonpeptidic chymotrypsin inhibitor with M W < 500. The inhibitor is more potent than the best reported tetrapeptidyl phosphonate chymotrypsin inhibitor and demonstrated selectivity over a panel of other serine proteases, including the closely related enzyme cathepsin G.

-1047
Efficient Sequence-Specific Purification of Listeria innocua mRNA Species by Triplex Affinity Capture with Parallel Tail-Clamps A gripping tail. The formation of stable triplexes of parallel tail-clamps with mRNA molecules is shown under acidic and also under neutral and slightly basic pH conditions (see schematic representation). As an example of the fore-seen applications of this process, efficient and specific purification of Listeria innocua iap mRNA molecules was achieved from an L. innocua total RNA solution by triplex affinity capture.

-1077
Inactivation of gilGT, Encoding a C-Glycosyltransferase, and gilOIII, Encoding a P450 Enzyme, Allows the Details of the Late Biosynthetic Pathway to Gilvocarcin V to be Delineated Road map to gilvocarcin V. Two structural features that are crucial for the biological activity of antitumor antibiotic gilvocarcin V-its C8 vinyl group and C-glycosidically bound d-fucose residue-are shaped in late biosynthetic steps by the enzymes GilOIII and GilGT, respectively (see scheme). We demonstrate this by a series of experiments that include the analysis of mutant strains in which the gilOIII and gilGT genes were deleted.

-1088
On the Sequence of Bond Formation in Loline Alkaloid Biosynthesis Who's next? The order of bond-forming steps in the biosynthetic pathway to loline alkaloids was determined by feeding deuterated compounds to Neotyphodium uncinatum. Proline and homoserine combine to form N-(3-amino-3-carboxypropyl)proline, which is decarboxylated and cyclized to form 1-aminopyrrolizidine. The addition of an oxygen atom to 1-aminopyrrolizidine completes the biosynthesis. Scyllo-or myo-. Physicochemical studies with potentiometric and NMR titrations show that, at physiological PH, the microprotonation patterns of synthetic scyllo-inositol pentakisphosphate (scyllo-InsP 5 ) resemble those of its close relative, the intracellular signalling molecule myo-Ins(1,3,4,5,6)P 5 , as shown by their conditional probabilities (%) at pH 7.5. This makes scyllo-InsP 5 a promising tool for identifying those biological effects of myo-InsP 5 that are mediated by specific binding proteins (centre), rather than by nonspecific electrostatic effects.
Supporting information on the WWW (see article for access details).
* Author to whom correspondence should be addressed. ADDENDUM Shortly before the submission of our manuscript on microarrays of lipolytic enzymes to ChemBioChem in 2005, Dr. Funeriu and his colleagues published a study that described "Enzyme family-specific and activity-based screening of chemical libraries using enzyme microarrays". [1] The authors named this type of application "Thematic Enzyme Microarrays" (TEMA). In ref.

BOOKS
[8] on p. 532 of our publication, we wanted to emphasize that the same authors had already successfully applied hydrogel NHS slides in enzyme microarray technology before our laboratory used them for the preparation of lipase and esterase chips.  Members of the GDCh must place their orders directly at the publishers (via Subscription Service above).

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text:
Reply to ``Remarks on the simulation of Cl electrosorption on Ag(100) reported in Electrochimica Acta 50 (2005) 5518"

We reply to remarks by Lang and Horanyi on the meaning of the notion of"electrosorption valency"used in I. Abou Hamad et al., Electrochim. Acta 50 (2005) 5518. It is concluded that, contrary to the assertion of Lang and Horanyi, the magnitude of the current in the external circuit upon adsorption of an ion of charge ze with partial charge transfer is indeed given by an electrosorption valency gamma such that |gamma e|<|ze|. We believe the conclusion of Lang and Horanyi to the contrary is the result of an excessively severe charge-neutrality requirement.

Introduction
In their interesting discussion article [1], Láng and Horányi (LH) question the concept of electrosorption valency [2,3] as interpreted in our recent paper on Cl electrosorption on Ag(100) [4]. As we understand their argument, its central point is the requirement of charge neutrality in the solution phase and separately in a surface layer consisting of the specifically adsorbed ions (traditionally known as the Inner Helmholtz Layer, IHL) and the adjoining part of the metal in the working electrode. These two separate conditions, when applied simultaneously, lead them to the conclusion that the charge transported through the external circuit due to the specific adsorption of an ion of charge ze must be ze, regardless of possible partial discharge of the adsorbate, unless coadsorption of ions of opposite charge takes place.
We argue that this double application of the charge-neutrality requirement overconstrains the problem and leads to an erroneous conclusion. The correct region over which charge neutrality should be applied must include the parts of the solution phase close to the IHL, traditionally known as the Outer Helmholtz layer (OHL) and the diffuse double layer (DDL). See Fig. 1 for definitions of these terms. This polarized region, representing the half-cell of the working electrode, is separated from the rest of the system by fictitious bounding surfaces inside the uniform, macroscopically uncharged bulk media (electrolyte and metal, respectively). This schematic partitioning of the system removes the need to consider the counter electrode explicitly in calculating the charge transport. A completely analogous condition of charge neutrality is independently obeyed at the counter electrode. The current through the working electrode into the external circuit induces a current of opposite sign, but with the same magnitude, through the counter electrode. For instance, if chloride adsorbs at the working electrode, a (partial) negative charge will be released into the external circuit, while a corresponding negative charge will, on average, be injected through the counter electrode. In the present case, this latter charge will most probably transform a corresponding amount of H + into 1 2 H 2 , thus maintaining electroneutrality in the entire system.
Our conclusion, for which we argue below, is that the expressions for electrosorption valency and adsorbate dipole moment used in Ref. [4] correspond to the approximations of excess supporting electrolyte, as well as to identifying the electrosorption valency with the negative of the partial charge-transfer coefficient. While not exact, these are reasonable approximations, supported a posteriori by the good agreement between the numerical results for our model system and the experimental adsorption isotherms. In the process we reconfirm Vetter and Schultze's relation between the electrosorption valency and the current in the external circuit [3].
Details of our arguments, including the approximations used in Ref. [4] (and also in Ref. [5]), are given below. In Sec. 2 we discuss the definition of the electrosorption valency; in Sec. 3 we discuss its relation to the current in the external circuit; and in Sec. 4 we obtain its relation to the surface dipole moment. Our conclusions are summarized in Sec. 5.

Electrosorption Valency
The electrosorption valency was thermodynamically formalized by Vetter and Schultze [2,3] to account for the current in the external circuit during electrosorption with partial charge transfer [6]. Abbreviated derivations have been presented later (see, e.g., Ch. 18 of Ref. [7]), but we believe the most detailed treatment is given in Refs. [2,3], as supplemented by Refs. [8,9,10]. Results of these early papers were recently summarized by Schultze and Rolle [11,12].
Vetter and Schultze consider an electrosorption reaction for an ion of valence z with the possible charge transfer of a noninteger number λ of electrons, described by the equation [2,3,8,9,10] To avoid extrathermodynamic complications arising from considering the microscopic structure of the DDL [13], we here restrict ourselves to the case of excess supporting electrolyte, so that the potential at the OHL, φ OHL , is the same as that in the bulk electrolyte, φ e , [2,3,9], which we can define equal to zero without loss of generality. See Fig. 1. This is indeed the condition corresponding to the experiments reported in Refs. [4,5]. The electrode potential is then E = φ met + cst., where φ met is the potential of the metal. Thus, in the case of excess supporting electrolyte, differentiation with respect to E is the same as with respect to (φ met − φ OHL ). The excess of supporting electrolyte also reduces the concentration of adsorbate ions S z in the DDL, while a relatively low concentration of adsorbate ions relative to the solvent reduces their concentration in the OHL relative to the IHL. In this approximation we can therefore replace the total surface excess of adsorbate ions, Γ S , by the surface coverage θ of the specifically adsorbed, partially discharged species S z+λ in the IHL [2]. The coverage is defined as the number of specifically adsorbed ions per adsorption site on the surface. In what follows, we shall use θ as an approximation for the more general Γ S .
In Ref. [2], the electrosorption valency is defined as where µ S is the chemical potential of S z in the bulk solution. From this and a general adsorption isotherm corresponding to the reaction equation (1), they obtain the relation where e is the elementary charge unit and C D is the capacitance of the compact double layer (metal vs. IHL). The subscript PZC refers to the Potential of Zero Charge. The value of γ at the PZC is Here, κ ad and κ w refer to the effects of the dipole moments of the adsorbate and water, respectively, and can usually be neglected for inorganic, aqueous electrolytes. The factor g = is the ratio of the potential difference between the adsorbate and the OHL to that between the metal and the OHL and is frequently of the order of 0.2. A simple interpretation of the schematic Fig. 1 with a near-linear potential profile would lead to the traditional interpretation of g as a purely geometric factor. For more realistic microscopic models that lead to a nonlinear potential profile, such as illustrated in Fig. 1, the connection of g to the geometry of the interface region is less clear.
Equations (3) and (4) reveal two important approximations in our Refs. [4,5]. The first is that the integral in Eq. (3) is approximated by a linear function in the coverage θ. This is reasonable since any changes in γ with E are likely to be largely due to the increased crowding on the surface.
The second approximation, which is expressed in Sec. 2.2 of our Ref. [4] (and also quoted verbatim in the fifth paragraph of LH [1]), is that we take g = 0. This leads to Lorentz' approximation γ PZC = −λ [6] and corresponds to the situation that the full potential drop happens between the bulk metal and the IHL. A somewhat better approximation is probably the minimum value for water, g min ≈ 0.16, obtained by Schultze and Koppitz [10].

Surface Charge and Potentiostatic Current
The crucial problem of the interpretation of γ in terms of the current in the external circuit is treated by Vetter and Schultze in Ref. [3]. The central point is that even a partial discharge corresponding to λ = 0 merely corresponds to a redistribution of charge in the interface region. Therefore, all currents in the circuit are capacitive, corresponding to changes in the excess charge density on the metal, q met , and the corresponding quantity on the electrolyte side, q e . (By the requirement of charge neutrality, q e = −q met .) Considering q met as a function of θ and E, one thus gets the current density From the electrocapillary equation for the compact double layer (i.e., the full differential of the surface free energy density or "surface tension" σ), they obtain where z met , Γ met , and Γ e − refer to the ions and conduction electrons of the electrode. The current density at constant potential , i pot , is thus obtained from Eq. (5) by setting dE/dt = 0. The necessary relation to γ as defined in Eq.
(2) is found by writing and using the Maxwell relation obtained from the electrocapillary equation, together with the standard equality for any three quantities related by a single equation, Thus, To appreciate this derivation it is important to realize that the current at constant potential is not a simple quantity. As pointed out by Schmickler [14], conceptually it involves two steps: first the adsorption of the ion and corresponding buildup of the image charge, which changes the potential , followed by readjustment of the charges to bring the potential back to its original value. Without a detailed, microscopic model, these semimacroscopic, thermodynamic results cannot tell us in detail how the charges are distributed in the interface region. This is a major limitation of the concept of electrosorption valency.
The applicability of the results summarized above to reversible electrodes (as opposed to perfectly polarizable ones) was questioned by Frumkin, Damaskin, and Petrii in Ref. [13]. In response, Vetter and Schultze explicitly established the validity of their results for reversible electrodes in Ref. [9].

Surface Dipole Moment
While the quantities discussed in Sec. 3 are purely thermodynamic quantities (in the case of excess supporting electrolyte), a microscopic theory of the interface structure is necessary to estimate the dipole moment associated with adsorption of an ion [14,15,16]. Generally, the dipole moment of a (onedimensional) charge distribution q(x) is defined as The result is independent of the coordinate system if the integration limits are chosen such that charge neutrality is obeyed over [x 1 , x 2 ]: x 2 x 1 q(x) dx = 0 [17]. The charge distribution produces a potential difference, φ 2 − φ 1 = p/ǫ, where ǫ is the dielectric constant of the medium. The details depend on the microscopic model.
The approximation used in Ref. [4] is the commonly used one [18,19], due to Bange et al. [20] and Schmickler [7,14], where C H is the Helmholtz capacity. This result is derived under the condition of charge neutrality (but not necessarily vanishing dipole moment) over the DDL. It can be obtained simply within the picture illustrated in Fig. 1 as follows. The dipole moment is related to the potential drop from the metal to the layer of specifically adsorbed ions as p = ǫ(φ met − φ IHL ) = ǫ(1 − g)E.
Assuming the field dependence of γ in Eq. (3) can be included in g and λ, and ignoring the polarization terms κ when solving Eq. (4) for (1 − g), we obtain By setting C H = e(z + λ)/E, we get Eq. (13), which can be rewritten in terms of (1 − g) and the partial charge-transfer coefficient λ as The prefactor zeǫ/C H can be viewed as an effective dipole distance. However, due to screening by both the liquid and the electron gas of the metal, this distance is generally much smaller than the ionic radius of the adsorbate [14,15]. Equation (15) with g = 0 was the one used in Ref. [4] for the dipole moment, and therefore for the lateral adsorbate interactions.
An alternative approach is to consider the surface dipole moment as the basic, physical quantity, and γ as a derived quantity linked to p by Eq. (13) under assumption of the geometric model described in Fig. 1.

Conclusion
In this discussion paper we have summarized arguments showing that the electrosorption valency γ as defined by Vetter and Schultze [2] gives the correct result for the current in the external circuit due to adsorption of an ion of charge ze, Eq. (11) [3], as well as the relation between γ and the chargetransfer coefficient λ, Eq. (4) [2]. These results show that there is no need for coadsorption with an oppositely charged ion to reduce the current per adsorbate particle from ze, to γe. We note that the derivation of these results requires charge neutrality over the whole interface region, as indicated in Fig. 1, not separately over the electrolyte and the compact double layer, as we understand the argument of LH to imply. We believe the reason for their conclusion that partial charge transfer cannot change the current is a result of their overly strict charge-neutrality requirement.
Beyond these general results, we have shown that the approximations used in Refs. [4,5] are excess supporting electrolyte and vanishing of the factor g. While certainly not exact, these assumptions are reasonable for the systems studied and give very good agreement between the computer simulations of the lattice-gas model and the experimental adsorption isotherms.
Finally we note that the arguments given here are based on classical thermodynamics with no specific assumptions on the microscopic structure of the interface region, beyond the charge neutrality. In order to obtain explicit results for microscopic parameters without fitting to experiments, one would need quantum-statistical-mechanical calculations that are still beyond our computational capacity. Only such future calculations have the potential to determine explicitly such quantities as the surface dipole moment and the spatial distribution of charge and dipole moments in the whole interfacial region.  Fig. 1. Schematic picture of the interface region at the working electrode (a), the electrostatic potential φ as a function of distance x perpendicular to the surface in the general case (b) and in the case of excess supporting electrolyte (c). Here, IHL stands for the inner Helmholtz layer, OHL for the outer Helmholtz layer, and DDL for the diffuse double layer. Large circles with a minus sign represent the adsorbate ions, and small circles with an arrow indicating a dipole moment represent water molecules. After Ref. [2].