Errors and Error Tolerance in Irreversible Multistep ... - ACS Publications

ARTICLE pubs.acs.org/JPCC

Errors and Error Tolerance in Irreversible Multistep Growth of Nanostructures Sagi Eppel* and Eran Rabani School of Chemistry, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv 69978, Israel

bS Supporting Information ABSTRACT: Error correction is one of the main features allowing selfassembly and other reversible growth processes to create giant, ordered molecular structures at the nanometer scale and above. On the other hand, in irreversible growth processes, such as chemical synthesis, errors (defects) are usually not removable and, as a result, growth is limited to a small number of synthetic steps and formation of relatively small structures. In this work, we develop a general model and theory to describe the behavior of errors in multistep, irreversible growth processes. Simulations of the model show that despite the large number of ways in which errors may occur in the growth, the average effect of errors seems to obey a universal pattern controlled by only a few parameters, regardless of the exact type of errors that occur. Furthermore, we show that errors that disturb growth cause damage that increases exponentially with the number of growth steps, thus leading to randomly disordered structures after a relatively small number of growth steps. If, however, growth is cooperative, the exponential increase of defects can be suppressed, suggesting that it is possible to create irreversible growth processes with error tolerance similar to that of reversible self-assembly (although without error correction). This may allow growth of ordered superstructures that exceed the limits of current systems.

1. INTRODUCTION The ability to create complex and ordered molecular structures on a nanometer size scale and above represents one of the main goals in the fields of chemistry and nanotechnology and has many potential applications in areas such as material design, catalyst chemical sensors, and molecular electronics.1-7 All of the main methods used to date to construct such structures are based on the self-assembly process, which is the main and, basically, only process in which molecules and nanoparticles arrange themselves into ordered structures.6-9 The self-assembly process is based on the second law of thermodynamics and on the tendency of systems to spontaneously arrange themselves in structures that minimize the free energy.7 The main element that enables self-assembly is the reversibility of growth. This reversibility assures that the growth will eventually lead to a structure that corresponds to the minimum free energy rather than to metastable structures of higher free energy, as is often the case in kinetic (irreversible) growth.8,10-12 Another viewpoint, often used to describe reversible selfassembly or growth, is based on error correction or healing.8,9,13-17 If correct growth is defined as growth that leads to a predictable, ordered structure, and errors are defined as growth that leads to any other structure, then in reversible growth, errors will be less stable than correct growth and will, therefore, be r 2011 American Chemical Society

“healed” by reversing the process, resulting in the detachment of the defects from the structure which will allow the system to resume the correct growth. In irreversible growth, however, once an error occurs it cannot be removed and the system diverts to an incorrect growth path leading to wrong structures.8,9 Due to its error-correction capabilities, self-assembly can be used to create giant-ordered molecular structures such as crystals, micelles, and thin layers as well as a variety of ordered nanostructures.10,11,15,18-22 On the other hand, the ability of irreversible growth processes, such as organic synthesis and polymerization, to form giant ordered structures on a nanometer size scale and above is almost nonexistent. As discussed recently, the main problem is the lack of error correction or healing in such processes.8-10,23-25 A major class of such irreversible systems is growth processes that are based on covalent bonds. This class represents the majority of organic synthesis processes, which are currently limited to relatively small ordered structures, as noted in a recent paper by Schluter: “Compared to the self-assembly’s state of the art, the alternative covalent chemistry approach to functional objects of the size of viruses, for example, is still in its infancy. There is not even one controlled three-dimensional Received: July 17, 2010 Revised: January 19, 2011 Published: March 08, 2011 5181

dx.doi.org/10.1021/jp106657a | J. Phys. Chem. C 2011, 115, 5181–5198

The Journal of Physical Chemistry C covalent growth process, either to laterally confined products (molecular nano-objects) or to extended macroscopic materials. An obvious intrinsic difficulty with the use of covalent chemistry is the lack of a healing option. Once a mistake occurs, a permanent structural or, even worse, a functional defect is created.”9 The main solution used so far in chemical syntheses was to separate the defective structures from the solution at each step and continue to the next step of the growth with only the errorless structures.4,26 The drawback of this strategy is that removing the fraction of defective structures at each step results in an exponential decline in the yield of the final products relative to the number of steps in the growth process. As a direct consequence, syntheses with a large number of steps are very limited and result in very low yields.8,9,24 An important question, that so far has not been explored, concerns the following: Even if there is an irreversible growth process in which errors can occur and cannot be fixed, and structures with errors are not separated and discarded but instead are allowed to continue to participate in the growth process, what will be the effect of these errors on the resulting structure? Or more accurately: What will be the differences and similarities between structures that result from growth with errors and the target structure that results from errorless growth? This question is important since there are several examples of macromolecules and nanostructures that can function even with a large number of defects. Such systems include DNA and proteins27 that can tolerate a wide range of mutations, as well as nanoelectronic circuits28 that can function with a large number of defective components. These systems suggest that structures with some fraction of irremovable defects can still have useful applications if the fraction of defects is kept below a certain level. The first goal of this work is, therefore, to develop a theoretical model in order to gain a general understanding of the possible effects of errors in irreversible growth systems. The goal was not to analyze errors and defects in a specific system or structure but rather to gain a general understanding of the possible effects of errors on multistep irreversible growth processes, such as the synthesis of macromolecules and various other bottom-up assembly processes.4,8,29-37 In the second part of this work we used our model to examine the ability of cooperativity to generate an error tolerance in the growth of such systems.38 Although the concept of error tolerance and correction has been addressed in previous studies,14-17 these were concerned with single step reversible growth systems and used the reversibility of the growth as a tool for removing defects from the structure. On the other hand, dealing with errors in irreversible growth systems requires a different paradigm, since the focus in this case is not on removing the defects, which is by definition impossible, but rather on minimizing the effects of errors that occur on the growth process.

2. MODEL DESCRIPTION While a large number of models have been offered to describe single-step (or one-pot) growth processes such as crystallization, polymerization, and self-assembly,16,39-48 we are unaware of any simple model that generally describes irreversible stepwise growth of discrete structures. The model we propose does not purport to accurately describe the molecular behavior of a specific system but rather aims to capture the general features of irreversible stepwise growth and synthesis processes, similar to the way in which some of the self-assembly, crystallization, and polymerization models capture the basic features of such growth

ARTICLE

Figure 1. The growth model. (A) The set of monomer types used in the growth. (B) The growth sequence is the sequence in which the different types of monomers are added to the structure. The numbers in the sequence refer to the type of monomer. (C) In the model, every monomer added to the structure is affixed to the structure in such a way that complementary binding sites on the monomer and on the structure are adjacent (representing bond formation).

processes.16,39-44 The model focuses on the growth of a single structure in each simulation, where growth is performed by binding monomers to specific sites on the face of a structure (Figure 1, panel C). In each growth step a different type of monomers is added and bound to the structure. Pairs of sites that can form bonds with each other are described in the model as complementary binding sites and are represented in the figures as faces with complementary patterns (Figure 2, panel B and all figures throughout the text). In each growth step, one type of monomer is added and bound to the structure such that complementary binding sites on the face of the monomer and the structure are adjacent (Figure 1, panel C), which describe bond formation between the structure and the monomers. Complementary binding sites can bind only in one orientation to each other (Figure 1, panel C). Binding sites in the model are meant to imitate the ability of specific functional groups to selectively bind to each other in solution. Two examples of this are the ability of electrophilic and nucleophilic sites to react and form covalent bonds and the ability of hydrogen donors and acceptors to form hydrogen bonds with each other (Figure 2, panel A).8,21,29,30,37,49 The monomers themselves are described as cubes containing six faces, where each face can be empty or contain one binding site. The model describes step-by-step growth, which refers to systems in which the growth occurs in several discrete steps, where in each step different growth processes occur. In the simplest case this refers to conditions in which different types of monomers are added to the structure in each step (Figure 1, panel C). Experimentally, this kind of growth is usually carried out by separating the structure from the solution and inserting it into a different solution with different monomers and conditions.24,29,31,50,51 The stepwise growth is described by using the final structure (product) of the current growth step as the initial structure (reactant) for the next step of the growth (Figure 1), where in each step different types of monomers are added to the structure. The sequence in which the growth steps are performed, which in this case is simply the sequence in which different monomers are added to the structure, is referred to as the “growth sequence” (Figure 1, panel B). In this case the nth 5182

dx.doi.org/10.1021/jp106657a |J. Phys. Chem. C 2011, 115, 5181–5198

The Journal of Physical Chemistry C

ARTICLE

Figure 2. Bond formation between complementary binding sites in the model and their analogous real systems. (A) Examples of bond formation between complementary binding sites in real systems. (B) Sites with functional groups that can form bonds with each other, described in the model as complementary binding sites, which are portrayed in the figure as faces with complementary patterns. (C) Examples of cases in which bonds cannot be formed since two faces have binding sites that are not complementary.

member in the growth sequence is simply the type of monomers that are added to the structure in the nth step of the growth (Figure 1, panel C). The growth sequence is constant and cannot be changed during the growth process. 2.1. Main Assumptions of the Model. A major assumption of the proposed model concerns the rigidity and structure, namely, that the structure is not flexible and cannot go through conformational changes. The majority of nanostructures and macromolecules have some flexibility and some degree of conformational freedom, which in some cases, such as proteins and foldamers, can completely determine the final structure of the molecule.52 The problem in such cases is that this flexibility usually does not allow the formation of a predictable structure. In a case where predictable structures are formed, they are usually the result of folding processes based upon weak, reversible, bonds, which are essentially self-assembly processes. Such systems have already been explored by other models53 and are not the subject of the present work. In cases where order-predictable structures may result from the growth and not from folding, there is a clear preference toward using rigid building blocks and forming rigid structures. Such systems will be the focus of this work.8,29,35,52,53 Another assumption underlying our model is that different structures cannot interact or bind with each other, namely, that each structure in the solution grows independently from the others. This is a realistic constraint which holds for the majority of such growth systems, mainly since both the concentration and diffusion rates of the structures in the solution are much smaller than that of the monomers. Furthermore, in multistep growth of nanostructures and macromolecules, it is common to affix the structure to a solid support,4,31,50,51,54 which also prevents interactions between different structures.

2.1.1. Simulation of the Model. The simulations of the model described above were executed on a 3D cubic cellular lattice space. Each cell can be empty or contain one monomer, and monomers could only be added to empty cells. Monomers can have any orientation relative to the structure. If a monomer could be added (and bind) to a cell in more than one orientation, a random orientation was chosen. The monomers themselves are described as cubes with six faces, each face could be either empty or contain one binding site; the rules of the growth are as described above in section 2 (see Supporting Information for more details on the software). 2.2. Examples of Growth of Structures in the Model. Several examples of the model are presented below and will be used in section 4 to examine the effect of errors on irreversible growth processes. These examples were chosen mainly because they demonstrate different aspects of stepwise growth. Real life analogues can be found for cases 1 and 2,4,32,55,56 while cases 3 and 4 were added since they represent interesting and possible methods for growing materials (at least possible under the assumption that errors will not destroy the growth). The number of growth steps in all of the examples is much larger than in most of the current experimental situations, which tend to be limited largely due to effects of errors on the growth. Case 1, Sheet Structure (Figure 3). The first example involves the growth of a square sheet structure, as depicted in Figure 3. The growth is based on two types of monomers, each with four identical binding sites. The binding sites on monomer 1 are complementary to those on monomer 2. Since there are only two monomers that can bind only to each other, the only possible growth sequence is 1,2,1,2,... and the only possible structure is a square sheet. Case 2, Rod Structure (Figure 4). The second example involves the growth of a rod structure. Again, the growth is based on two types of monomers, where the binding sites on monomer 5183



ARTICLE

Figure 3. Growth of a square sheet structure (case 1). (A) The types of monomers used in the growth. (B) The growth sequence. The numbers in the sequence refer to the types of monomers and their sequence is the order in which they are added to the structure. (C) The growth appears as structures separated by arrows; each arrow represents one growth step. The number that appears below the arrow denotes the monomer added to the structure in that step. Multiple subsequent arrows represent multiple subsequent growth steps. Note, that although the growth is illustrated in 2D, it contains the full information of the 3D model, the faces that do not appear in the 2D representation (front and back) are empty (contain no binding sites).

Figure 4. Growth of a rod structure (case 2).

Figure 5. Growth of a brushlike structure using two types of orthogonal interactions (case 3).

1 are complementary to binding sites on monomer 2. This time, however, each monomer has only two binding sites located on opposite faces, leading to the growth of a one-dimensional rod structure. Case 3, Brushlike Structure (Figure 5). This example involves two pairs of complementary binding sites, i.e., two types of bonds that support the formation of two types of distinguished growths (“brush body” and “brush hairs”). The horizontal growth is done by monomers 1 and 2 via the first type of interaction (marked by a single triangle pattern in Figure 5), while growth in the vertical directions (“brush hairs”) is done by monomers 3 and 4 via the second type of bond (jigsaw pattern in Figure 5). Case 4, Two-Phase Shaped Structure (Figure 6). Similar to the first example, each monomer has four identical binding sites, but this time there are two pairs of complementary binding sites and four different types of monomers. Binding sites on monomers 1

and 2 (red) are complementary to each other, and binding sites on monomers 3 and 4 (green) are also complementary to each other, i.e., monomers can only bind with monomers of a similar color. As a result, adding red monomers 1 and 2 will result in growth of the red parts of the structure, while using green monomers 3 and 4 will lead to growth only of the green parts of the structure. Since the growth of each phase (red or green) blocks the growth of the other phase, it is possible to control the shape of the structure simply by using monomers 1 and 2 to grow the red parts (growth sequence 1,2,1,2,...) or monomers 3 and 4 to grow the green parts (growth sequence 3,4,3,4,...). Note that unlike previous cases this growth was simulated in a 2D square lattice.

3. ERRORS IN GROWTH Errors in growth of real systems can occur in a large number of ways. For the sake of generality, we will consider errors as 5184



ARTICLE

Figure 6. Growth of a two-phase shaped structure using several different monomers and a nonrepetitive growth sequence (case 4).

referring to any possible change in the structure that is not part of the correct growth, i.e., any change that is not according to the rules of the growth (given in section 2). Only two limitations are imposed: First, errors are local, meaning that a single error is restricted to a size of one monomer or one site. Second, the occurrence of errors at each site is independent from all other sites and there is no correlation between the occurrences of errors at different sites. The justification for these assumptions is that errors in real systems typically involve undesirable side reactions that lead to either wrong (or missing) binding of monomers to the structure, binding of defective monomers, or other side reactions at some reactive site or functional group.4,16,26,50,57-59 In either case, the effect of the error is limited to the size of one site or one monomer. 3.1. Location of Errors. Errors are added to the growth model by making a set of random changes to the structure at the end of each growth step. In general, there are two options regarding the location in which errors may occur in the growth: Option a. Errors in each step can occur at every site in the structure. In this case, the average number of errors that occur in each growth step is approximately proportional to the size of the structure and, therefore, will increase with the number of growth steps. For cases 1-4 it was found (Supporting Information, section S.1.2) that the ratio between the average number of errors

that occur in a step (kerr) and the rate of correct growth (kcor, i.e., the number of new monomers added to the structure in the growth step) is given by kerr ¼ Pn=A kcor where n is the number of the growth steps, P is the probability for an error (per site per step), and A is a constant (2 e A e 4 for cases 1-4). Thus for structures that result from a large number of growth steps (n > A/P) the rate of error occurrence will be greater than the rate of correct growth, which means that even if we assume that each error adds only one defect, the growth will still be composed mainly of defects. Option b. Errors in each growth step can occur only in parts of the structure that can participate in the growth in that step. In this case, the average number of errors that occur in each step of the growth (kerr) is approximately proportional to the growth rate of the structure (kcor) and the probability for error (P) per site per step. This means that the ratio between the average number of errors and the correct growth rate is approximately constant and proportional to the error probability (kerr/kcor = P), so that if each error adds only one defect, than the average fraction of defects remains constant with the number of growth steps. 5185



ARTICLE

Figure 7. The fraction of defects for growth of a rod (case 1), as a function of the number of growth steps: option a, errors in each step can occur at every site on the structure; option b, errors can occur only at sites that participate in that growth step. Errors in both cases were taken not to affect the growth but the sites in which the errors occurred were considered as defects. The error probability P was taken as 10% (per site per step). Results shown are based on an average of 500 runs.

Figure 7 shows the effect of the two options of error locations on the growth of case 1, where each error adds a single defect (without further affecting the growth). In real systems, errors usually occur either as part of the growth or at least in specific, exposed functional groups,26,57 described in the model as binding sites. Errors in the backbone of the structure are relatively rare. Thus, in this work we adopt the second scenario (option b) and assume that errors can occur only in parts that participate in the growth (in this step) or at exposed binding sites. 3.2. Methods for Creating Errors in the Model. Although errors were defined as any possible change in the structure that does not lead to the target structure, there are actually many such realizations and it is impossible to explore them all. Thus, in the present work, three arbitrary sets of random changes were chosen in order to create errors. Each of the growth simulations was run with each type of errors independently to give a separate set of results. This was done to ensure that the results are general and not restricted to a specific type of error. The three error types, illustrated schematically in Figure 8, are given below: Error Type I (random addition or removal of binding sites to/ from the structure). Binding sites were randomly added with probability P1 (per site per step) to sites on the surface of the structure and existing binding sites were removed from the face of the structure with probability P2 for each binding site in the structure (limited to monomers with exposed binding sites). Error Type II (random attachment of monomers to the structure). Random monomers with random orientation (chosen from the complete set of monomers participating in the growth) were attached randomly to the face of the structure with probability P3 (per site per step), limited to sites on monomers that participated in the last growth step. In addition, exposed binding sites on the face of the structure were removed with probability P4 (per site per step). Error Type III (random replacement of monomers in the structure). Monomers that were added to the structure during the last growth step were randomly replaced with probability P5 (per monomer per step) by another monomer chosen randomly from the set of all monomers participating in the growth. The new monomer was placed in a random orientation. Hence, a monomer may be replaced by the same type of monomer but with a different orientation. 3.3. Evaluating the Effect of Errors on Growth (Defects and Vacancies). Two properties were used to quantitatively evaluate the effects of errors on the growth. The first was the fraction of defects, where the term “defects” refers to monomers

Figure 8. Illustration of the three methods used to create errors in the growth: (I) random addition or removal of binding sites to/from the structure; (II) random attaching of monomers to the structure; (III) random substitution/replacement of monomers in the structure.

that appear in the structure that resulted from growth with errors but not in the target structure that resulted from errorless growth. Note that the term “error” refers to a mistake in the growth process, whereas the term “defect” refers to a mistake in the structure. The fraction of defects (panel D in Figures 9, 10, 11, and 12, discussed below in section 4) is calculated by dividing the average number of defects by the average number of monomers in the structure. The second property is the fraction of vacancies, where the term “vacancies” refers to any monomer that appears in the target structure that resulted from the errorless growth, but does not appear in the structure that resulted from growth with errors. The fraction of vacancies (panel C in Figures 9-12, discussed below in section 4) was calculated by dividing the average number of vacancies by the number monomers in the errorless (target) structure.

4. EFFECTS OF ERRORS ON IRREVERSIBLE GROWTH: RESULTS AND DISCUSSION Figures 9-12 present results of the simulations of growth with errors for cases 1-4 (section 2.2). As can be seen, after 20-30 growth steps, the structures resulting from the growth with errors (Figures 9-12, panel B) become completely disordered and show vanishing overlap with the corresponding target structures (Figures 9-12, panel A). This happens despite the fact that the fraction of sites at which errors occur during the growth (marked in black in Figures 9-12, panel B) was relatively low, i.e., Perr < 10% for all cases shown. This implies that even a small ratio of errors can divert the growth to a completely disordered structure after several growth steps. The results also imply that the number of defects increases exponentially with the number of growth steps (Figures 9-12, panel D); this exponential growth is observed for all three error types (section 3.2). The growth in case 4 is an exception to this rule, as seen in Figure 12, panel D. In this case, the fraction of defects with respect to the number of steps increases to about 50% and then levels off. This, however, does not mean that the damage due to errors in this case is less significant. As can be seen in Figure 12, panel B, the distribution of monomers in the final structure is almost completely random. However, since there are only two 5186



ARTICLE

Figure 9. The effect of errors on the growth of the sheet structure (case 1, section 2.2): (A) the target structure that results from errorless growth, after 20 growth steps; (B) example of a structure resulting from a growth with errors, sites at which errors occur are marked in black; (C) plot of the fraction of vacancies vs the number of growth steps; (D) plot of the fraction of defects vs the number of growth steps. Each curve represents the average of 12000 simulations with one of the error types described in section 3.2. Note that some curves are concealed because of overlap. The standard errors60,61 (%) for the curves in panels C and D, respectively, are as follows: I, 2.8 10-3 and 0.14; II, 6.6 10-4 and 0.21; and III, 10-4 and 0.19.

Figure 10. The effect of errors on growth of the rod structure (case 1): (A) the structure that results from errorless growth after 30 growth steps; (B) two examples of structures that results from the same growth with errors, sites at which errors occur are marked in black; (C and D) Plot of the fraction of vacancies and defects, respectively. Each curve represents an average of 30000 simulations with one error type. The standard errors60,61 (%) for the plots in graphs C and D respectively, are as follows: I, 0.15 and 0.52; II, 0.15 and 0.52; III, 0.14 and 0.33. 5187



ARTICLE

Figure 11. The effect of errors on growth in case 3: (A) the structure that results from errorless growth, after 30 growth steps; (B) example of a structure that results from the same growth only with errors, sites at which errors occur are marked in black; (C and D) plot of the fraction of vacancies and defects, respectively. The wavy curves in panels C and D stem from the existence of two distinct growth types (i.e., “brush body” and “brush hairs”), which occur at different steps (Figure 5). Note that some curves are concealed because of overlap. Each curve represents the average of 30000 simulations with one error type. The standard errors60,61 (%) for the curves in graphs C and D respectively, are as follows: I, 0.06 and 0.18; II, 0.06 and 0.22; III, 0.05 and 0.21.

colors of monomers (red and green), even when a completely random distribution of monomers is used, there is still a finite probability that the color of the monomer at a specific site will be correct. Therefore, even in this case, the growth becomes completely random after less than 30 steps. 4.1. Possible Effects of Errors on the Growth. To understand why a small number of errors in the growth can have such a destructive influence, it is necessary to examine the possible effects of errors on the growth. In general, there are three possible ways in which errors may influence the growth (illustrated in Figure 13): (1) Errors may change the structure (add defects) with no further influence on the next steps of the growth (Figure 13, panel C). These errors are referred to as “neutral”. (2) Errors may block or prevent part of the correct growth from occurring (Figure 13, panel B). These errors are referred to as “blocking”. (3) Errors may affect the growth by starting a new path of growth that is not part of the correct growth (Figure 13, panel D). These are referred to as “branching”. In general, the effect of each error in the growth can be described by a combination of these three categories of influences. 4.1.1. Neutral Errors. Since errors of this type do not affect the subsequent steps of growth, each defect that is created must be a direct result of an error, and therefore the number of defects added to the structure equals the number of errors that occur. As a result, the fraction of defects is proportional to the probability of occurrence of this type of error, Pn (per site per step) and, thus, remains approximately constant (assuming Pn is constant) regardless of the number of growth steps (Figure 7 option b; see Supporting Information for a detailed mathematical

discussion). This means that if the probability for neutral errors is low, the fraction of resulting defects will also be low, regardless of the number of steps in the growth. Examples of errors with similar effects can be found in stepwise peptide synthesis50,51 (wrong amino acids) and in polymer growth (change of tacticity).58 4.1.2. Blocking Errors. Since blocking errors terminate part of the growth, they do not add defects but rather increase the number of vacancies. In branching and linear structures, such as cases 2 and 3 (rod and brush), a single blocking error can permanently terminate the growth of part of the structure (or all of it). In such cases, the probability of growth for any part of the structure drops exponentially with the number of growth steps (k). More accurately, the average rate of growth with blocking errors, ΔNkbl, which equals the average number of monomers attached to the structure the kth growth step, decreases exponentially with the number of steps according to ΔNk bl ¼ ð1 - Pbl Þk ΔNk where ΔNk is the number of monomers added to the structure at the kth step of the errorless growth (rate of errorless growth) and Pbl is the probability of a blocking error (per site per step). It should be noted that errors with this type of influence can affect only the growth of structures in which growth at each site is independent from that occurring at all other sites. In such cases, preventing the growth at a single site necessarily results in permanently stopping the growth of part of the structure or all of it (Figure 14). Such systems include cases 2 and 3. By contrast, the growth of cases 1 and 4 is latticelike and cannot be terminated by any single error, and therefore, blocking errors have only a minor effect on these cases, as 5188



ARTICLE

Figure 12. The effect of errors on growth in case 4: (A) the target structure, after 80 growth steps; (B) example of a structure with errors, sites at which errors occur are marked in black; (C and D) plots of the fraction of vacancies and defects, respectively. The shapes of the curves stem from the existence of two distinct phases of growth (red and green), which occur at different steps, and the fact that the growth sequence in this case is nonrepetitive (Figure 6). Each curve represents the average of 30000 growth simulations with one of the error types. The standard errors60 (%) for the plots in graphs C and D respectively, are as follows: I, 0.01 and 0.02; II, 0.02 and 0.05; III, 0.03 and 0.05.

Figure 13. Three basic types of effects errors may have on growth, illustrated on the growth of a rod structure: (A) correct growth with no errors; (B) blocking error; (C) neutral error; (D) branching error. Sites at which errors occur are marked in black.

shown in Figure 14. Examples of errors with similar effects appear in dendrimers,26 oligomers,4 and polymers,57,62 and usually involve termination of the growth site by a defective monomer,4,26 a side reaction,57,62 or blocking of a branch due to steric hindrance.24

4.1.3. Branching Errors. Branching errors that initiate wrong growth paths lead to an exponential increase in the number of defects with respect to the number of growth steps. This is seen from both the simulation results and the mathematical approximation for cases 1-3 (Figures 9-11, panel D). In the rod structure (case 2), 5189



ARTICLE

Figure 14. Different effects of errors (marked in black) that block the growth at one site, illustrated for the growth of cases 1 and 2.

the average growth of defects can be approximated by Nk def ¼

ð1 - Pbl Þk ePbr k - 1 1 - ð1 - Pbl Þk þ logð1 - Pbl Þ þ Pbr logð1 - pbl Þ

where Nkdef is the average number of defects in the structure in the kth step of the growth, Pbr is the probability of a branching error (per site per step), and Pbl is the probability of a blocking error (see section S.1 of the Supporting Information for the exact derivation of this equation). Although the growth of defects in other structures cannot be expressed by such a simple formula, it can be shown that all structures in which such errors occur show an approximateexponential growth of defects. This exponential increase is the result of two factors: First, once a wrong growth path starts, it will result in the addition of defects in all subsequent growth steps even if no further errors occur (Figure 15). Second, each new wrong growth path can, by itself, lead to errors that will initiate other wrong growth paths and so on (as illustrated in Figure 15). These results imply that even if the probability of occurrence of such errors (Pbr) is low, defects will overtake the growth after a large enough number of growth steps, resulting in a completely disordered structure. We note in passing that any exponential growth cannot last long in 3D since it will eventually result in overcrowding and will become a space-filling growth.24 However, as can be seen from the fit between the mathematical approximation and the simulation results (Figures 9 and 10, panel D), the exponential growth in the number of defects represents a good approximation of the first few dozen steps of the growth, at least for discrete, low-density structures such as those in cases 1 and 2. This clearly does not apply to the space filling growth in case 4 (Figure 12, for further discussion of this point see the Supporting Information). Examples of errors that produce similar effects can be found in polymers57,63 and dendrimers59 and can involve chain transfer reactions,57,63 the activation of a wrong functional group,50 or the failed termination of a functional group.59 4.2. Equivalence in the Effect of Different Types of Errors on the Growth. Another interesting result is the fact that the average effect of errors on the growth, as expressed by the fraction of defects and vacancies, seems to be almost identical for all error types examined, even though the errors created in each case were completely different (section 3.2). Note that each error-creating method has only one or two controllable parameters (i.e., the error probabilities P1-P5 (section 3.2)). The qualitative similarity between the average effects of different types of errors, as seen in panels C and D of Figures 9-12, is clearly a result of more than just an adjustment of one or two

parameters. Rather, it suggests that the average effect of any type of error on the growth is controlled by only a few parameters. Presumably, these are the probabilities of branching, blocking, and neutral errors (Pbr, Pbl, Pn). This hypothesis is further supported by the fact that a reasonable approximation (Supporting Information) can be obtained for the average effect of errors in terms of these probabilities alone (Figures 9 and 10, panel D). This might imply that there is a universal pattern to the average effect of errors on growth that is common to all types of errors.

5. COOPERATIVITY AND ERROR TOLERANCE The question arises from previous sections is whether it is possible to create an irreversible growth process that will also be error tolerant? By definition, error tolerance cannot, in this case, refer to conditions under which defects are removed or “healed” as in the case of reversible processes. However, it is possible to develop scenarios in which all errors in the growth are neutralized, namely, that an error might add a defect but will not further alter subsequent steps of the growth. This can create a growth process in which the fraction of defects will remain approximately constant for a large number of steps (section 4.1.1). If possible, such growth processes would not be limited by the number of growth steps and, subsequently, the size of the structures created. One possible way of creating error tolerance is through the use of cooperativity, which in this context refers to conditions under which growth at several different sites is codependent.38 Since errors are presumably restricted to one site, one might expect that they will not be able to affect such growth. In the following section we will present three cases for cooperative growth. Note that, as in previous sections, we specify general rules for the growth but not the physical mechanism creating them. Since the focus here is not on exploring a specific physical mechanism, we have left the detailed discussion regarding possible mechanisms for creating cooperative growth to the Appendix. 5.1. Cooperative Growth Processes. Three cases of cooperative growth are schematically illustrated in Figures 16 and 17. The growth of all structures is made in layers, where in each step a new layer is added. The cooperativity of the growth is expressed by having the growth at each site depend on the growth at neighboring sites. More accurately, growth will occur only if at least nc parallel adjacent binding sites are present in the same layer (where nc is the level of cooperativity of the growth). If the group of parallel binding sites has less than nc adjacent sites, then growth will not occur at any of the sites (Figure 16, case 2). By contrast, if there is a 5190



ARTICLE

Figure 15. Branching errors that initiate a wrong growth path, resulting in a proximate exponential increase in the number of defects (illustrated for case 1).

Figure 16. Schematic description of cooperative growth processes of sheet structure cases A and B (panels A and B, respectively). In order for growth to occur, a minimal number of nc adjacent, parallel binding sites must exist in the same layer: (1) When the number of parallel binding sites is larger than nc, growth will occur at all sites (nc = 4). (2) When the number of parallel binding sites is smaller than nc, growth will not occur at any site.

group of nc or more adjacent binding sites in a layer, growth will occur at all sites (Figure 16, case 1). For physical mechanisms that create growth based on such rules see the Appendix. 5.1.1. Examples of Cooperative Growth. Three cases of cooperative growth appear in Figures 16 and 17. These cases were chosen because of their simplicity as well as the fact that it is possible to conceive (and simulate) a simple mechanism that create them (Appendixes I.1 and I.2). Cases A and B (Figure 16, panels A and B, respectively). Both cases represent cooperative growth of 2D sheet structures. The growth is made in layers where in each step a new layer is added by cooperative growth and also in each step the layer width is expanded by two monomers. Note that the shapes of the growth in cases A and B are very similar; the difference between them is in the mechanism of the growth is discussed is in Appendixes I.1 and I.2. Case C (Figure 17). Cooperative growth of rod structure. The growth is similar to that of the rod structure in case 2 (Figure 4), except that here the growth is based on four parallel binding sites instead of one, which enables the use of cooperativity. 5.2. The Effect of Cooperativity. The reason cooperativity generates error resilience in a large number of systems, molecular and other, is that while errors are usually local and appear at one point or one site, cooperative processes depend on groups of sites and can therefore remain unaffected by the occurrence of a single error, or more.17,27,28,64,65 As described in section 5.1 cooperative growth can occur only when a minimal number of parallel adjacent binding sites are present. And since the assumption is that a single error is restricted to the size of one site or one monomer, occurrence of one error will not lead to wrong growth (branching). In order for errors to initiate a wrong growth path, it is necessary for at least nc parallel wrong binding sites to exist,

Figure 17. Cooperative growth of a rod structure (case C) based on four parallel binding sites. The minimal number of binding sites necessary for cooperative growth to occur is nc = 3. (A) Growth with no errors. (B) A single error that damages one binding site (marked in black) will occur with probability Pbl and leave three parallel binding sites active and thus will not stop the growth. (C) Two errors will occur simultaneously with probability (Pbl)2 and will leave only two parallel binding sites active, which will not be enough for growth to occur.

which means at least nc parallel errors occur simultaneously. Since errors are (presumably) noncorrelated, the probability for this is proportional to (Pbr)nc, where Pbr is the probability for one error (that creates a wrong binding site) to occur. Although cooperativity does not prevent the exponential growth of defects as a result of branching errors, it is a useful tool to reduce the probability for such errors in the growth (proportional to (Pbr)nc), by adjusting the level of cooperativity. 5.2.1. Cooperativity and Error Tolerance of Blocking Errors. As mentioned in section 4.1, rather than starting wrong growth paths, errors may influence the growth by terminating the correct growth (blocking errors). The cooperative growth of rod (case C) in Figure 17 illustrates how cooperativity may prevent the effect of such errors. Growth is based on four parallel binding sites. If the minimum cooperativity for growth to occur is nc = 3, then any error that damages one site will leave three unharmed binding sites, which will still allow correct growth to occur (Figure 17, panel B). Thus, to terminate growth, at least two parallel errors must occur simultaneously (Figure 17, panel C). Assuming that errors are noncorrelated, the probability for this is proportional to (Pbl)2, where Pbl is the probability for an error that damages one specific binding site. 5.3. Effects of Errors on Cooperative Growth: Results and Discussion. Figures 18, 19, and 20 present simulation results for the cooperative growth of the structures in cases A-C. It is evident that cooperative growth shows a significant tolerance to errors with a relatively small fraction of defects (Figures 18-20, panel D) and a high similarity between the target structure 5191



ARTICLE

Figure 18. The effect of errors on the cooperative growth of a sheet structure (case A): (A) the target structure after 62 growth steps; (B) example of a structure resulting from the same growth with errors, sites at which errors occur are marked in black; (C and D) Plots of the fraction of vacancies and defects, respectively. Each curve represents the average of 500 growth simulations with one of the error types described in section 3.2. The error rate for each of the error types is identical to that used in the growth of the noncooperative sheet (case 1) in Figure 9. The standard errors60 (%) for the curves in panels C and D, respectively, are as follows: I, 0.25 and 0.36; II, 0.014 and 0.6; III, 0.0364 and 0.054.

Figure 19. The effect of errors on the cooperative growth of a rod structure (case C): (A) the structure that results from errorless growth after 82 growth steps; (B) example of the structure that results from the same growth with errors (type I), sites in which errors occur are marked in black; (C and D) plot of the fraction of vacancies and defects, respectively. Each plot represents the average of 2000 simulations. The error rates for each of the error types are identical to those used in the growth of the noncooperative rod (case 2) in Figure 10. The standard errors60 (%) for the plots in graphs C and D, respectively, are as follows: I, 0.02 and 0.33; II, 0.11 and 0.44; III, 0.05 and 0.28.

produced by errorless growth and the corresponding structures that result from growth with errors (Figures 18-20, panels A and B, respectively). This observation holds even after a relatively large number of growth steps (g100), for all structures and for all types of errors. These results can be compared to the noncooperative growth of the structures in section 4, where for the same error rate and much fewer growth steps, growth either completely stopped or became completely disordered (Figures 9-12). Note that the fraction of sites in which errors occur in the cooperative growth (sites marked in black in Figures 18-20, panel B) was chosen to be equal to that of the noncooperative growth presented in section 4 (Figures 9-12). Thus, the difference does not result from the rate or type of errors but only from the way in which they affect the growth.

Panels C and D of Figures 18-20 also show that the error tolerance of this growth process was effective for all three error types examined. This supports the proposition that the addition of cooperativity to the growth provides a general tolerance for all types of errors that may occur, at least within the assumptions of the model (rigidity and locality). We note that this time error probabilities were not adjusted to create an overlap between the average effect (defects and vacancies) of the different error types (as in section 4). Instead, for all three error types we used identical error rates as those used in the noncooperative growths of cases 1-4 described in section 4. This was done in order to allow for a more objective comparison between the results of the noncooperative growth of section 4 and the results of the cooperative growth in this section. 5192



ARTICLE

Figure 20. The effect of errors on the cooperative growth of a sheet structure (case B): (A) the structure produced by errorless growth after 60 growth steps; (B) example of a structure produced by the same growth with errors (type I), sites at which errors occur are marked in black; (C and D) plot of the fraction of vacancies and defects, respectively. Each curve represents an average of 2000 simulations. The error rates for each of the error types are identical to those used in the growth of case 1 in Figure 9. The standard errors60 (%) for the curves in graphs C and D, respectively, are as follows: I, 0.13 and 0.06; II, 0.17 and 0.07; III, 0.12 and 0.04.

Figure 21. Fraction of defects for cooperative growth with different levels of cooperativity (nc): (A and B) cooperative growth of sheets structures (cases A and B respectively); (C) cooperative growth of rod structure (case C). All cases refer to growth with error type I.

5.3.1. Level of Cooperativity (nc) and Its Influence on the Fraction of Defects. Examination of the effect of the degree of cooperativity (nc) on the fraction of defects (Figure 21) reveals that for growth with a high level of cooperativity (nc g 3,4), the fraction of defects remains low and almost does not increase with the number of growth steps. This means that the probability that a combination of errors in a growth with a high level of cooperativity will initiate a wrong growth path is negligibly small. However, some defects are still added to the structure as a direct result of the errors. For the same growth with lower cooperativity (nc g 2,3, Figure 21), the fraction of defects remains low but increases slowly after a large number of steps. This means that the probability that a combination of errors will initiate a wrong growth path (branching error) is low, but not completely negligible.

Therefore, such errors will affect the growth after a large number of growth steps, but at a much smaller rate than in noncooperative growth. For the same growth but with no cooperativity (nc = 1, Figure 21), the number of defects rises exponentially with the number of steps in the growth and the target structure is overtaken by the growth of defects after a small number of steps. 5.3.2. Effect of the Level of Cooperativity nc on the Fraction of Vacancies. The results presented above seem to suggest that the higher the cooperativity of nc the higher the precision with which growth follows the target structure. However, since cooperativity was defined as a state in which the growth of nc neighboring sites is codependent, it implies that errors that affect growth at one site can, in some cases, stop the growth at (nc - 1) neighboring sites (Figure 22). This explains why the 5193



ARTICLE

Figure 22. An error at a certain growth step (marked in black) leaving only two binding sites (denoted by red arrows) to its right, which is below the threshold (nc = 4) for cooperative growth. As a result of the cooperativity, this single error stops the growth at several neighboring sites.

Figure 23. Fraction of vacancies for cooperative growth with different levels of cooperativity: (A and B) Cooperative growth of sheet structures (cases A and B, respectively); (C) cooperative growth of rod structure (case C). All the cases were simulated with error type I.

fraction of vacancies increases with the level of cooperativity for all cases examined, as can be seen from the graphs in Figure 23. For sheet structures (cases A and B) such errors will not completely terminate the growth (Figure 22). Therefore, after the first few growth steps the fraction of vacancies stops increasing and remains nearly constant with respect to the number of steps (Figure 23, panels A and B). This means that in these cases the fraction of vacancies depends on the level of cooperativity but not (for the long-term) on the number of growth steps. In the case of the rod in case C (Figure 17), growth is limited to four binding sites, and as discussed in section 5.2.1, increasing the cooperativity also increases the probability for a combination of errors that will permanently terminate the growth. In this case, the fraction of vacancies continues to increase even after a large number of steps (Figure 23, panel C); however this increase is still much smaller than that of the noncooperative rod (case 2, Figure 10), in which a single error suffices to completely terminate growth.

6. CONCLUSION We have developed a general theoretical approach to describe the behavior of errors in irreversible multistep growth processes. The approach taken was based on the development of a simple model that captures the basic features of multistep irreversible growth and synthesis processes.8,30,35 The model was used to explore the effect of different types of errors on various multistep

growth scenarios. Several major conclusions can be drawn from our work: (1) Errors that alter the next steps of the growth (by either starting a wrong growth path or terminating a correct growth path) will result in damage that grows exponentially with the number of growth steps (exponential increase in the number of defects or exponential decrease in the rate of correct growth). (2) The average effect of errors on the growth, as expressed by the fraction of defects and vacancies, seems to be similar for all error types. This suggests that the effect of errors on stepwise irreversible growth is controlled by only a few parameters, regardless of the exact type of error. Therefore, errors of different kinds can be treated in a similar manner. (3) Our examination of the ability of cooperativity to create an error tolerance in such growth systems showed that even if the growth is irreversible and defects cannot be removed, cooperativity can still reduce the effect of errors by creating conditions in which errors are prevented from further affecting the growth (errors might add defects but will not lead to wrong growth or terminate the correct growth). Therefore, for such systems, the effect of errors does not increase exponentially with the number of growth steps but rather remains approximately constant for a large number of steps and approximately proportional to the errors rate in the growth. 5194



ARTICLE

Figure 24. Allosteric cooperative growth (case A): (A) first stage, growth of layer of N monomers creates 2N - 1 bonds (marked by black line); (B) second stage, layer extension. At the end of each stage all monomers in the structure are irreversibly cross-linked.

Although our results are based on a theoretical model and are therefore limited to the assumptions of this model (mainly the rigidity and locality of errors), they still suggest that it is possible to create irreversible growth processes that will exhibit error tolerance similar to that displayed in reversible self-assembly growth (but without error correction). Such systems may allow growth with a very large number of steps without loss of precision, which might lead to the creation of ordered structures of size and complexity that exceed the capability of current systems.

’ APPENDIX I: PHYSICAL MECHANISM FOR COOPERATIVE IRREVERSIBLE GROWTH PROCESS In this Appendix, we provide the details of two specific cooperative irreversible growth processes. Each step in the processes presented here is based on two intermediate stages: The first cooperative and reversible and the second noncooperative and irreversible. Since a sequence of two processes, the first reversible and the second irreversible, is always irreversible and since the sequence of two stages, one cooperative and the second noncooperative, is always cooperative, the total sequence of two intermediate stages is irreversible and cooperative. In general, two basic types of cooperative processes exist in molecular systems: allosteric and chelate, both of which exist only in reversible systems.38 To see how such processes might be used to create the irreversible growth described in the text, we suggest two possible processes, which are discussed briefly in the following sections. These growth processes were simulated to yield the results discussed in section 5. I.1. Processes Based on Allosteric Cooperativity (Case A). An allosteric type of cooperativity refers to a condition in which a group of monomers binds to the structure with higher stability than that resulting from the binding of each monomer by itself.38 In this case, cooperativity of nc is achieved by choosing conditions in which the binding of nc adjacent monomers reduces the system’s free energy, and will therefore occur, while at the same time the binding of nc - 1 monomers increase the system free energy and therefore will not occur. For example, consider a situation in which each monomer can form only one bond with the structure (bond I, Figure 24), but in addition, can form two bonds with other monomers at adjacent sites (bond II, Figure 24). For simplicity, we assume that the free energy of a bond breaking F is the same for both bonds (I and II).

Figure 25. Growth based on chelate cooperativity (case B): (A) stage 1, monomers bind reversibly to the structure (bond I); (B) stage 2, monomers in the same layer are linked by irreversible bonds (bond II); (C) stage 3, selectively dissociating parts that bind at less than nc sites.

If we take the free energy barrier for attaching one monomer to the structure (S) to be between F and (3/2)F cooperativity of two (nc = 2) will result, since binding of one monomer will create one bond and will therefore increase the system free energy while binding of two adjacent monomers will create three bonds and reduce the free energy. In general, the degree of cooperativity is determined by the balance between the gain in energy (from bonds formation) and loss in entropy (from monomers affixing). Mathematically, the conditions for creating cooperativity of nc are 2ðnc - 1Þ - 1 S 2nc - 1 < < nc - 1 F nc where 2nc - 1 is the number of bonds formed in a complex of nc adjacent monomers (bonds marked as black lines in Figure 24 panel A). Since such cooperativity can be created only under reversible conditions, a second irreversible stage is required. This can be achieved simply by cross-linking the monomers in the structure using irreversible bonds, as demonstrated in several recent publications.66-71 Figure 24, illustrates the growth of the structure in case A, based on the mechanism described here. The first stage of cooperative layer growth (Figure 24 panel A), is based on the conditions described above. The second stage of layer extension (Figure 24 panel B) is based on noncooperative growth (condition of S < F). It was necessary to add this stage in order to compensate for the shrinking of the layers as a result of errors described in section 5.3.2. I.2. Chelate-Type Cooperativity (Case B). Chelate-type cooperativity is achieved when a single molecule can bind to the structure through more than one bond,38,62,72-74 where to a reasonable approximation the dissociation constant of the molecule decreases exponentially with the number of bonds.38,72 A three-stage approach (illustrated in Figure 25) based on this principle was simulated to give the growth of case B. In the first stage (Figure 25 panel A), monomers bind to the structure reversibly (using reversible bond I, Figure 25). In the second stage (Figure 25 panel B), adjacent monomers in the same layer are linked irreversibly to each other, forming essentially one inseparable segment. In this model, this was achieved by using 5195



ARTICLE

Figure 26. Illustration of the Monte Carlo methods used to simulate the growth of the reversible parts.

Figure 27. Cooperative growth of case C, based on mechanism in Appendix I.1. Bond I links between the monomer and the structure while bond II forms between adjacent monomers in a layer. The growth based on one stage. After each stage all monomers in the structure are irreversibly cross-linked.

linker monomers (using irreversible bond II, Figure 25), although in real systems this is often done directly.67,69,73,75-77 The resulting segment is bound by a large number of reversible bonds and therefore chelate cooperativity is achieved. In the third stage (Figure 25 panel C), segments that are based on a small number of monomers and are bound to the structure at less than nc sites selectively dissociate using the chelate cooperativity, leaving only segments that are bound to the structure with nc sites or more. Assuming the dissociation constant of the segment is given by72 koff = A exp(-NbF) where F is the normalized free energy for breaking a single bond (F ≈ Ea/kBT), Nb is the number of bonds of the segment, and A is a constant, then the conditions to create a cooperativity level of nc are given by A-1 expððnc - 1ÞFÞ , tm , A-1 expðnc FÞ where tm is the time scale associated with the third stage. In this case, growth will not occur for less than nc sites (will occur in the first stage but will dissociate in the third stage), while growth of nc sites or more will occur and will be practically irreversible. Finally note that the processes described in panels A-C of Figure 25 leave exposed binding sites in the edge of the new layer, which are terminated in the fourth stage (Figure 25 panel D) and the process is repeated to give the growth of case B.78 Simulation of the Cooperative Stages. The processes presented in previous sections (I.1 and I.2) contain reversible stages that cannot be simulated directly using the model described in section 2. To describe the reversible cooperative stages, a simulation

based on Monte Carlo and rate equation must be used. Our approach is somewhat similar to Winfree’s KTAM model.16 As in the model described in section 2, sites on the monomer surface can form bonds with complementary binding sites on the structure (Figure 26, panel A). Again, bonds form when the complementary binding sites on the monomer and the structure are adjacent (section 2). In this case, each bond has a constant free energy, F (normalized by kBT), and the binding free energy of the monomer to the structure is taken as the sum of the energy of all of its bonds72 (Nb). In every step of the simulation the rate constants for all possible association and dissociation reactions in the structure are calculated (Figure 26, panel B). The dissociation constant for each monomer in the structure is calculated as koff = A exp(-NbF). The association constant, kon = A exp(-S), is assumed to be constant during the simulation (assuming constant concentration of monomers in the solution). The next reaction (association or dissociation of monomers) is then chosen randomly, with probability (Pn), which is equal to the rate constant of this reaction, (kn), divided by the general rate constant (kall): Pn = kn/kall (Figure 26, panel C). The general rate constant is equal to the sum of rate constants of all possible reactions: kall = ∑n=1kn. The time of the system (t) is then increased by -(log(ξ)/kall), where ξ is a random number between 0 and 1. If the time (t) exceeds the predetermined time limits tm, the simulation (of the growth step) ends; otherwise the chosen reaction is performed and the procedure is repeated. When finished, the final structure resulting from the simulation of a growth stage is used as the initial structure for the simulation of the next growth stage. Monomers that bond irreversibly to the structure (Appendix I.1) will not participate in the dissociation reactions. Also, groups of monomers that bind irreversibly to each other but bind reversibly to the structures (Appendix I.2) are treated as a single entity that binds to the structure with energy that is equal to the sum of all of its monomer bonding energies. Finally, the simulation was done with 3D cubic cellular lattice and monomers identical to those described at the end of section 2.1.1, allowing structures resulting from each model (in section 2 and here) to be used by the other (in different stages of growth). I.4. Technical Details and Parameters for Simulation for Cases A-C. The growth simulation for cases A and B (based on the mechanism explained in Appendixes I.1 and I.2) are illustrated in Figures 24 and 25, respectively. Case C is illustrated in Figure 27; this case is also based on the mechanism of Appendix I.1 (case A) only with rod instead of sheet structure (Figure 27). Each step of the growth of cases A-C was divided into several 5196


The Journal of Physical Chemistry C stages. Each stage was simulated by the approach described in Appendix . The parameters for each growth stage for cases A-C appear below (note that the stage letters refer to the panels of the figures). (A = 1 for all cases and stages) Case A (Figure 24): Stage A: S = 20, nc S 2 ðn - 1ÞS 1 þ , F ¼ 2nc - 1 3 2ðnc - 1Þ - 1 3 tm = 6 exp(2S - F) Stage B: S = 1, F = 20, tm = 10 exp(-S) Case B (Figure 25): Stage A: S = 1, F = 20, tm = 10 exp(-S) Stage B: Based on irreversible bonds (II) and where simulated based on the model in section 2. Stage C: (kon = 0, F = 20 (for bond I), tm = exp(F(nc - 0.5)). Case C (Figure 27): S = 20, 4S 2 4S 1 þ , F ¼ 4 þ nc 3 3 þ nc 3 tm = 6 exp(2S - F) where S = log(kon), nc is the cooperativity level, A,tm, F, and kon are defined in Appendix . Note, that the growth stages described above are counted as single steps in the graphs in section 5.

’ ASSOCIATED CONTENT

bS

Supporting Information. Derivation of the approximations of the numerical results and further discussion of the results and methods. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the Israel Science Foundation (Grant No. 283/07). We thank Professor Moshe Portnoy and Professor Uri Sivan for their useful and insightful discussions and Professor Mark Ratner for valuable remarks on the manuscript. E.R. thanks the Miller Institute for Basic Research in Science at UC Berkeley for partial financial support via a Visiting Miller Professorship. ’ REFERENCES (1) Goddard, W. A., III; Brenner, Donald W.; Lyshevski, Sergey E.; Lafrate, Gerald J. Handbook of Nanoscience Engineering, and Technology , 2nd ed.; CRC Press: Boca Raton, FL, 2007. (2) Gilardi, G.; Meharenna, Y. T.; Tsotsou, G. E.; Sadeghi, S. J.; Fairhead, M.; Giannini, S. Biosens. Bioelectron. 2002, 17, 133. (3) Tour, J. M. Molecular Electronics: Commercial Insights, Chemistry, Devices, Architecture and Programming; World Scientific Pub Co., Inc.: Singapore, 2003. (4) Tour, J. M. Chem. Rev. 1996, 96, 537. (5) Chen, J.; Wang, W.; Klemic, J.; Reed, M. A.; Axelrod, B. W.; Kaschak, D. M.; Rawlett, A. M.; Price, D. W.; Dirk, S. M.; Tour, J. M.; Grubisha, D. S.; Bennett, D. W. Molecular wires, switches, and memories. In Molecular Electronics II; Aviram, A., Ratner, M., Mujica, V., Eds.; New York Academy of Sciences: New York, 2002; Vol. 960; pp 69.

ARTICLE

(6) Lehn, J. M. Rep. Prog. Phys. 2004, 67, 249. (7) Lehn, J. M. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 4763. (8) Allis, D. G. S.; James, T. Nanostructural architectures from molecular building blocks. In Handbook of Nanoscience, Engineering, and Technology, 2nd ed.; Goddard, W. A., III, Brenner, Donald W., Lyshevski, Sergey E., Lafrate, Gerald J., Eds.; CRC: Boca Raton, FL, 2007; pp 18/1. (9) Schluter, A. D. A covalent chemistry approach to giant macromolecules with cylindrical shape and an engineerable interior and surface. In Functional Molecular Nanostructures; Topics in Current Chemistry, Vol. 245; Springer: Berlin and New York, 2005; pp 151. (10) Cote, A. P.; Benin, A. I.; Ockwig, N. W.; O’Keeffe, M.; Matzger, A. J.; Yaghi, O. M. Science 2005, 310, 1166. (11) Lund, K.; Liu, Y.; Lindsay, S.; Yan, H. J. Am. Chem. Soc. 2005, 127, 17606. (12) Langner, A.; Tait, S. L.; Lin, N.; Rajadurai, C.; Ruben, M.; Kern, K. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 17927. (13) Hopfield, J. J. Proc. Natl. Acad. Sci. U.S.A. 1974, 71, 4135. (14) Rowan, S. J.; Cantrill, S. J.; Cousins, G. R. L.; Sanders, J. K. M.; Stoddart, J. F. Angew. Chem., Int. Ed. 2002, 41, 898. (15) Fujibayashi, K.; Hariadi, R.; Park, S. H.; Winfree, E.; Murata, S. Nano Lett. 2008, 8, 1791. (16) Winfree, E. Simulation of Computing by Self-Assembly, PhD Thesis, Caltech, 1998. (17) Winfree, E.; Bekbolatov, R. Proofreading tile sets: Error correction for algorithmic self-assembly; 9th International Workshop on DNA Based Computers (DNA9), Madison, WI, 2003. (18) Sun, J. Q.; Wu, T.; Sun, Y. P.; Wang, Z. Q.; Zhang, X.; Shen, J. C.; Cao, W. X. Chem. Commun. 1998, 1853. (19) Chworos, A.; Severcan, I.; Koyfman, A. Y.; Weinkam, P.; Oroudjev, E.; Hansma, H. G.; Jaeger, L. Science 2004, 306, 2068. (20) Winfree, E.; Liu, F. R.; Wenzler, L. A.; Seeman, N. C. Nature 1998, 394, 539. (21) Frattarelli, D.; Schiavo, M.; Facchetti, A.; Ratner, M. A.; Marks, T. J. J. Am. Chem. Soc. 2009, 131, 12595. (22) Li, D. P.; Zhou, X. H.; Liang, X. Q.; Li, C. H.; Chen, C.; Liu, J.; You, X. Z. Cryst. Growth Des. 2010, 10, 2136. (23) El-Kaderi, H. M.; Hunt, J. R.; Mendoza-Cortes, J. L.; Cote, A. P.; Taylor, R. E.; O’Keeffe, M.; Yaghi, O. M. Science 2007, 316, 268. (24) Tomalia, D. A.; Mardel, K.; Henderson,Scott A.; Holan, G.; Esfand, R. Dendrimers—An Enabling Synthetic Science to Controlled Organic Nanostructures. In Handbook of Nanoscience, Engineering, and Technology, 2nd ed.; Goddard, W. A., III, Brenner, Donald W., Lyshevski, Sergey E., Lafrate, Gerald J., Eds.; CRC Press: Boca Raton, FL, 2007; p 542. (25) Liu, J. W.; Wernette, D. P.; Lu, Y. Angew. Chem., Int. Ed. 2005, 44, 7290. (26) Tang, M. X.; Redemann, C. T.; Szoka, F. C. Bioconjugate Chem. 1996, 7, 703. (27) Gupta, M. K. IEEE Eng. Med. Biol. Mag. 2006, 25, 46. (28) Heath, J. R.; Kuekes, P. J.; Snider, G. S.; Williams, R. S. Science 1998, 280, 1716. (29) Schafmeister, C. E. Sci. Am. 2007, 296, 76. (30) Grill, L.; Dyer, M.; Lafferentz, L.; Persson, M.; Peters, M. V.; Hecht, S. Nat. Nanotechnol. 2007, 2, 687. (31) Gupta, S.; Macala, M.; Schafmeister, C. E. J. Org. Chem. 2006, 71, 8691. (32) Levins, C. G.; Schafmeister, C. E. J. Org. Chem. 2005, 70, 9002. (33) Mathias, J. P.; Stoddart, J. F. Chem. Soc. Rev. 1992, 21, 215. (34) Park, S. H.; Finkelstein, G.; LaBean, T. H. J. Am. Chem. Soc. 2008, 130, 40. (35) Schafmeister, C. E.; Brown, Z. Z.; Gupta, S. Acc. Chem. Res. 2008, 41, 1387. (36) Stoddart, J. F. Chem. Br. 1988, 24, 1203. (37) Gothelf, K. V.; Brown, R. S. Chem.—Eur. J. 2005, 11, 1062. (38) Hunter, C. A.; Anderson, H. L. Angew. Chem., Int. Ed. 2009, 48, 7488. (39) (a) Rabani, E.; Reichman, D. R.; Geissler, P. L.; Brus, L. E. Nature 2003, 426, 271–274. (b) Sztrum, C. G.; Rabani, E. Adv. Mater. 5197


The Journal of Physical Chemistry C 2006, 18, 565–571. (c) Yosef, G; Rabani, E. J. Phys. Chem. B 2006, 110, 20965–20972. (d) Kletenik-Edelman, O.; Ploshnik, E.; Salant, A.; Shenhar, R.; Banin, U.; Rabani, E. J. Phys. Chem. C 2008, 112, 4498– 4506. (e) Kletenik-Edelman, O.; Sztrum-Vartash, C. G.; Rabani, E. J. Mater. Chem. 2009, 19, 2872–2876. (40) Neves, U. P. C.; Botelho, A. L.; Onody, R. N. Phys. A (Amsterdam, Neth.) 2003, 324, 455. (41) Vogt, M.; Hernandez, R. J. Chem. Phys. 2002, 116, 10485. (42) Majid, I.; Jan, N.; Coniglio, A.; Stanley, H. E. Phys. Rev. Lett. 1984, 52, 1257. (43) Levi, A. C.; Kotrla, M. J. Phys.: Condens. Matter 1997, 9, 299. (44) Meirovitch, H. J. Phys. A: Math. Gen. 1987, 20, 6059. (45) Troisi, A.; Wong, V.; Ratner, M. A. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 255. (46) Fortuna, S.; Troisi, A. J. Phys. Chem. B 2009, 113, 9877. (47) Chen, T.; Lamm, M. H.; Glotzer, S. C. J. Chem. Phys. 2004, 121, 3919. (48) Jankowski, E.; Glotzer, S. C. J. Chem. Phys. 2009, 131, 8. (49) Tancini, F.; Genovese, D.; Montalti, M.; Cristofolini, L.; Nas, L.; Prodi, L.; Dalcanale, E. J. Am. Chem. Soc. 2010, 132, 4781. (50) Meienhofer, J.; Waki, M.; Heimer, E. P.; Lambros, T. J.; Makofske, R. C.; Chang, C. D. Int. J. Pept. Protein Res. 1979, 13, 35. (51) Houghten, R. A. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 5131. (52) Hill, D. J.; Mio, M. J.; Prince, R. B.; Hughes, T. S.; Moore, J. S. Chem. Rev. 2001, 101, 3893. (53) Thirumalai, D.; O’Brien, E. P.; Morrison, G.; Hyeon, C. Annu Rev Biophys 2010, 39, 159. (54) Dahan, A.; Portnoy, M. J. Polym. Sci., Part A: Polym. Chem. 2005, 43, 235. (55) Sakamoto, J.; van Heijst, J.; Lukin, O.; Schluter, A. D. Angew. Chem., Int. Ed. 2009, 48, 1030. (56) Levins, C. G.; Schafmeister, C. E. J. Am. Chem. Soc. 2003, 125, 4702. (57) Satoh, K.; Kamigaito, M. Chem. Rev. 2009, 109, 5120. (58) McCord, E. F.; McLain, S. J.; Nelson, L. T. J.; Arthur, S. D.; Coughlin, E. B.; Ittel, S. D.; Johnson, L. K.; Tempel, D.; Killian, C. M.; Brookhart, M. Macromolecules 2001, 34, 362. (59) Leininger, S.; Stang, P. J.; Huang, S. P. Organometallics 1998, 17, 3981. (60) Refer √ to the average standard error of the mean for each plot, given by S/ N where S is the standard deviation and N number of runs. (61) Shirley, D.; Stanley, W.; Daniel, C. Statistics for Research, 3rd ed.; Wiley Interscience: New York, 2004. (62) Odian, G. In Principles of Polymerization, 4th ed.; Wiley-Interscience; New York, 2004; Chapter 2, p 50. (63) Baughman, T. W.; Wagener, K. B. Recent advances in ADMET polymerization. In Metathesis Polymerization; Springer-Verlag: Berlin, 2005; Vol. 176; p 1. (64) Isokawa, T.; Abo, F.; Peper, F.; Adachi, S.; Lee, J.; Matsui, N.; Mashiko, S. Int. J. Mod. Phys. C 2004, 15, 893. (65) Whitty, A. Nat. Chem. Biol. 2008, 4, 435. (66) Calderone, C. T.; Puckett, J. W.; Gartner, Z. J.; Liu, D. R. Angew. Chem., Int. Ed. 2002, 41, 4104. (67) Laughrey, Z. R.; Gibb, B. C. Templates in Chemistry II; Topics in Current Chemistry, Vol. 249Springer: Berlin and New York, 2005; p 67 (68) Li, X. Y.; Liu, D. R. Angew. Chem., Int. Ed. 2004, 43, 4848. (69) Ziegler, A.; Mamdouh, W.; Heyen, A. V.; Surin, M.; Uji, H.; Abdel-Mottaleb, M. M. S.; De Schryver, F. C.; De Feyter, S.; Lazzaroni, R.; Hoger, S. Chem. Mater. 2005, 17, 5670. (70) Sun, J.; Tang, H. D.; Jiang, J. Q.; Xie, P.; Zhang, R. B.; Fu, P. F.; Wu, Q. Polymer 2003, 44, 2867. (71) Gothelf, K. V.; Brown, R. S. Chem.—Eur. J. 2005, 11, 1062. (72) Hunter, C. A.; Ihekwaba, N.; Misuraca, M. C.; Segarra-Maset, M. D.; Turega, S. M. Chem. Commun. 2009, 3964. (73) Hoffmann, M.; Wilson, C. J.; Odell, B.; Anderson, H. L. Angew. Chem., Int. Ed. 2007, 46, 3122. (74) Badjic, J. D.; Nelson, A.; Cantrill, S. J.; Turnbull, W. B.; Stoddart, J. F. Acc. Chem. Res. 2005, 38, 723.

ARTICLE

(75) Lin, J. B.; Xu, X. N.; Hang, X. K.; Li, Z. T. J. Org. Chem. 2008, 73, 9403. (76) Feldman, K. S.; Lee, Y. B. J. Am. Chem. Soc. 1987, 109, 5850. (77) Cao, Y. D.; Wang, L. Y.; Bolte, M.; Vysotsky, M. O.; Bohmer, V. Chem. Commun. 2005, 3132. (78) Work on a simplified version of this mechanism is currently in progress.

5198


Errors and Error Tolerance in Irreversible Multistep ... - ACS Publications

Recommend Documents