## Introduction

As seen in DNA-replication, the template effect1,2 is a powerful strategy for efficient synthesis of assemblies whose building blocks are well arranged, such as macrocycles3,4,5,6,7,8,9, cages10,11, interlocked and knotted molecules12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27, and submicrometer-sized materials28,29,30,31,32,33,34. Basically, this effect makes the interconnection of the building blocks feasible by their preorganization by binding to the template molecule(s). As a consequence, the template effect is effective in the synthesis of assemblies composed of conformationally flexible building blocks. When the building blocks are connected by irreversible bonds, once an undesired bond is formed, such intermediates can never lead to the desired structure. Thus, the template molecule biases the reaction towards the correct pathway by modulating the energy landscape so as to selectively decrease the energy barriers of proper bond connections under kinetic control2. In molecular self-assembly, where building blocks are connected by reversible bonds, the reaction is rather controlled by thermodynamics. Owing to the reversible nature of the chemical bonds between the building blocks, even when the building blocks are connected in an incorrect manner, the thermodynamically most stable structure is produced under thermodynamic control by correction of the interconnections. Therefore, the thermodynamic template effect plays a major role in most molecular self-assembly systems13,15,18. As the template molecule that stabilizes the final assembly would also stabilize the transition states and the intermediates produced along the correct self-assembly pathway(s), the importance of the kinetic template effect on molecular self-assembly has been implied but is not clearly understood.

Here we report a bifurcation of self-assembly pathways by a kinetic template that determines the product, a kinetically trapped micrometer-sized sheet structure or the thermodynamically most stable 1-nm-sized coordination cage. In the absence of the template molecule (NO3), the self-assembly of ditopic ligand 1 and Pd(II) source ([PdPy*4](OTf)2, Py* indicates 3-chloropyridine) leads to a metastable micrometer-sized sheet structure as a major product under kinetic control (Fig. 1). With the template, the (NO3)@[Pd214]4+ cage, where one NO3 anion is bound to the cavity of the cage, is produced in 79% yield through the on-pathway. Quantitative and numerical analyses of the self-assembly of the cage in the presence of the template anion revealed that the production of the intermediates for the cage is significantly accelerated by the template anion, which is also confirmed by the acceleration of the formation of a cyclic model structure in the presence of NO3 and the strong binding of the cyclic structure with NO3. Besides the stabilization of the final cage by the encapsulation of NO3, a kinetic template effect accelerates the formation of primitive intermediates for the cage by the template anion and contributes strongly to the bifurcation of the self-assembly pathways.

## Results

### Micrometer-sized sheet formation on the off-pathway

The self-assembly of the helical [Pd214]4+ cage was reported by Steel and McMorran35,36. Due to the high flexibility of the ditopic ligand 1, the resulting cage can bind a variety of anionic species (PF6, BF4, ClO4, ReO4, and I) in its cavity, whose size can be changed by twisting the helix in an induced-fit manner. The order of the binding strength for anions is ClO4 > BF4 > ReO4 > PF6, which suggests that smaller anions are more strongly bound in the cage with the exception that ClO4 (62.5 Å3) is a better guest than BF4 (49.3 Å3)36. The self-assembly of the cage from 1 and [PdPy*4](OTf)2 in CD3NO2 at 298 K was carried out to investigate the intermediates produced during the self-assembly by QASAP (quantitative analysis of self-assembly process)37,38,39. In our previous research on the self-assembly process of this cage from 1 and [PdPy*4](BF4)2, a 200-nm-sized sheet structure was transiently produced during the self-assembly and this sheet structure was finally converted to the (BF4)@[Pd214]4+ cage40. When the self-assembly of the cage was carried out using OTf as a counter anion and monitored by 1H NMR spectroscopy (Fig. 2, Supplementary Fig. 1 and Supplementary Tables 16), the cage formation was significantly retarded (Fig. 3a). The yield of the (OTf)@[Pd214]4+ cage at 1 day is only 6.3%, while that of the (BF4)@[Pd214]4+ cage at 1 day is 68.3%40. The encapsulation of one OTf anion in the cage was confirmed by 19F NMR and 19F DOSY measurements (Supplementary Fig. 2 and Supplementary Table 7). This result indicates that the counter anion largely affects the self-assembly process. STEM measurements of a reaction mixture after the convergence of the self-assembly indicate that a sheet structure was also produced in the self-assembly from 1 and [PdPy*4](OTf)2 (Fig. 4a–e). However, the size of the sheet (over 100 μm) is much larger than that produced from 1 and [PdPy*4](BF4)2. Several dark areas can be seen in the STEM images, which would be formed by tangling of the sheet, suggesting that this sheet is a flexible, soft material. The same tangled sheet structures were observed by AFM measurements of a dried sample of the reaction mixture (Fig. 4f, g). Apart from the tangled parts (Fig. 4i), this sheet is relatively smooth and a step with a gap of 4.44 nm was found (Fig. 4h).

The formation of the sheet was not prevented even in the case where the self-assembly was carried out at 70 °C and the yield of the cage was not improved by heating (Supplementary Figs. 3 and 4), indicating that the sheet produced from 1 and [PdPy*4](OTf)2 is more stable than the sheet produced from 1 and [PdPy*4](BF4)240 and that most of the (OTf)@[Pd214]4+ cages produced under this condition (35%) would not be derived from the sheet. Indeed, when the self-assembly was conducted with BArF (tetrakis[3,5-bis(trifluoromethyl)phenyl]borate), which is too large to be encapsulated in the [Pd214]4+ cage, the [Pd214]4+ cage was produced in 17% yield without a template anion (Supplementary Figs. 5 and 6).

As the sheet structure is kinetically trapped, the self-assembly using OTf anion tends to promote the growth of the sheet structure under kinetic control, suggesting that OTf is not a good guest for the [Pd214]4+ cage. As expected, competition experiments indicate that smaller BF4 is a stronger guest than OTf (90.5 Å3) for the cage (Supplementary Figs. 79 and Supplementary Table 8). To check the ability of counter anions to convert the sheet into the cage, n-Bu4NX (X: ClO4, BF4, or PF6) was added after the formation of the sheet prepared from 1 and [PdPy*4](OTf)2 (Supplementary Figs. 1016 and Supplementary Tables 911). The rate of the conversion of the sheet to the cage became faster by the addition of smaller guests, indicating that the conversion ability seems to correlate to the size of the anions.

### The cage formation on the on-pathway

In the competition experiments of various anions, it was found that NO3 (42.3 Å3) is more strongly bound to the cage than ClO4 (Supplementary Fig. 17). Therefore, the addition of n-Bu4NNO3 considerably accelerated the conversion of the sheet to the cage compared with n-Bu4NClO4 or n-Bu4NBF4 (Supplementary Fig. 16 and Supplementary Table 12). We then carried out the self-assembly from 1 and [PdPy*4](OTf)2 in the presence of n-Bu4NNO3 to prevent the formation of the sheet during the self-assembly (Fig. 2 and Supplementary Tables 1317). The 1H NMR signals of the (NO3)@[Pd214]4+ cage were clearly observed even at 5 min and the self-assembly reaction almost converged in 1 day to produce the cage in 79% yield (Fig. 3b), indicating that the cage formation is dramatically accelerated by NO3. The Ha signal of the (NO3)@[Pd214]4+ cage was largely shifted downfield by 0.36 ppm as compared with that of the (OTf)@[Pd214]4+ cage (Fig. 2), which suggests that NO3 was strongly bound to the cavity of the cage.

### Crystal structure of the (NO3−)@[Pd214]4+ cage

The structure of the (NO3)@[Pd214]4+ cage was revealed by single-crystal X-ray analysis (Fig. 5, Supplementary Figs. 18 and 19, and Supplementary Tables 18 and 19). Single crystals of the (NO3)@[Pd214]4+ cage were obtained from a solution of the cage in CH3NO2 by slow diffusion of i-Pr2O at 298 K for 10 days. The cage was strongly twisted with a twist angle of 84.6° accompanied by a short Pd···Pd distance of 7.167 Å, which are similar to those of the (I)@[Pd214]4+ cage (84.8°, 7.442 Å)36. The NO3 anion in the cage was disordered between two sites with each occupancy of 50% at the Wyckoff position of 4a on the C2 axis, lying on a plane perpendicular to the helix axis of the cage. The shortest distance between the Ha hydrogen and the oxygen atoms of the NO3 is 2.33 Å, which is shorter than the sum of the van der Waals radii of hydrogen and oxygen atoms, indicating that the CH···O hydrogen bond stabilizes the inclusion complex41,42,43,44,45. Similar hydrogen bonds are found in the crystal structure of (ClO4)@[Pd214]4+36, suggesting that the stronger binding affinity of ClO4 to the cage than that of BF4, which is smaller than ClO4, is due to the stronger hydrogen bonds between ClO4 and the cage. The average distance between the oxygen atoms in the NO3 and the benzene ring of the ligand 1 is 3.72 Å. When the Pd···Pd distance decreases by increasing the twist angle of the cage, the electron-rich aromatic walls in the cage are slightly moved further from each other, reducing the electrostatic repulsion between the walls and the anion encapsulated in the cage. This would be the reason why the [Pd214]4+ cage prefers smaller anions.

### Investigation of the self-assembly process of the cage

The self-assembly process of the cage was investigated by the n–k analysis37,38,39. The n value indicates the average number of the Pd(II) ions bound to a ditopic ligand 1 in a species [Pda1bPy*c]2a+ (n = (4a – c)/b), while the k value indicates the ratio of the Pd(II) ions to the ditopic ligands (k = a/b). The experimentally obtained (n, k) values, which are indicated as (〈n〉, 〈k〉), are the (n, k) values of the average composition of all the intermediates [Pda1bPy*c]2〈a〉+, which can be determined by the quantification of all the substrates (1 and [PdPy*4]2+) and the products ((NO3)@[Pd214]4+ and Py*). The change in the (〈n〉, 〈k〉) value with time suggests the type of occurring reactions and the mainly produced intermediates during the self-assembly.

In the absence of NO3 (off-pathway), the 〈n〉 value largely and monotonically increased with time, while the 〈k〉 value fluctuated around 0.55 (Fig. 3c). As most of the substrates were consumed within 30 min, the increase in the 〈n〉 value after 30 min suggests the growth of the oligomers and/or cross-linking by intra- and/or intermolecular ligand exchanges to release Py*, which causes a decrease in the 〈c〉 value. The (〈n〉, 〈k〉) value reached (1.82, 0.52) at 2 h and stayed around (1.83, 0.52) until the end of the self-assembly. Considering that the largest n value among possible intermediates with the k value of 0.5 on the on-pathway is 1.75 for [Pd214Py*]4+, the experimental, 〈n〉 value of 1.83, which is higher than 1.75, suggests the production of intermediates larger than the cage40, which is consistent with the observation of the sheet structure.

In the self-assembly reaction carried out in the presence of NO3 (on-pathway), the (〈n〉, 〈k〉) value stayed around (1.67, 0.50) from 10 min to 12 h (Fig. 3d), while the cage was drastically yielded to up to 72%, after which the self-assembly almost reached convergence. The steady (〈n〉, 〈k〉) value during the production of a large amount of the cage indicates that dominant intermediates have (n, k) values close to the observed (〈n〉, 〈k〉)46. The (〈n〉, 〈k〉) value of (1.67, 0.50) is situated in between the (n, k) values of [Pd214Py*2]4+ (1.5, 0.5) and of [Pd214Py*]4+ (1.75, 0.5), which correspond to the final precursors of the cage on the on-pathway. This result indicates that these two partial cage structures are mainly produced as intermediates from an early stage of the self-assembly and then the intramolecular cross-linking occurs as the rate-determining step. This would be because the rigidity of [Pd214Py*2]4+ and [Pd214Py*]4+ hinders the formation of the intermediates and transition states with a five-coordinate Pd(II) center that should be formed during intramolecular ligand exchanges on Pd(II) centers47,48. As a consequence, the template NO3 anion much contributes to the early stage of the self-assembly of the (NO3)@[Pd214]4+ cage; the NO3 anion kinetically affects the self-assembly to a great extent.

To demonstrate the strong binding between primitive intermediates and the NO3 anion on the on-pathway, the complexation of a model [Pd212]4+ macrocycle (Pd indicates Pd(N,N,N’,N’-tetramethylethylenediamine)) with the NO3 anion was tested (Fig. 6). Upon the addition of n-Bu4NNO3 in a CD3NO2 solution of the [Pd212]4+ macrocycle, the Ha signal was as largely shifted downfield (0.36 ppm) as was observed in the (NO3)@[Pd214]4+ cage (Supplementary Fig. 20). This result indicates that NO3 can be strongly bound to the cyclic structure, which suggests that the intramolecular cyclization of (NO3)@[Pd212Py*5]4+ to the (NO3)@[Pd212Py*4]4+ macrocycle should be promoted by the kinetic template effect of the NO3 anion. Indeed, the formation of the [Pd212]4+ macrocycle completed within 5 min in the presence of NO3, while the yield of the macrocycle without NO3 at 5 min is negligible (Supplementary Fig. 20). The competition experiment for NO3 between the [Pd214]4+ cage and the [Pd212]4+ macrocycle indicates that the cage is a stronger host for NO3 than the macrocycle (Supplementary Fig. 21), which is because the cage is structurally more organized than the macrocycle. Therefore, NO3 anion has a bilateral template effect on the on-pathway: (i) the kinetic template effect that accelerates intramolecular cyclization of the intermediates on the on-pathway, and (ii) the thermodynamic template effect that stabilizes the final cage.

### Numerical analysis of the self-assembly process of the cage

To investigate the self-assembly process of the (NO3)@[Pd214]4+ cage on the on-pathway in more detail, numerical analysis (NASAP: numerical analysis of self-assembly process49,50,51) was carried out based on the experimental data. The numerical analysis enables us to learn about the intermediates that cannot be deduced by QASAP. As the behavior of the self-assembly of the (NO3)@[Pd214]4+ cage is similar to that of the cage self-assembly from the rigid ditopic ligand, NASAP for the present cage was carried out in a similar way to the rigid cage system50. A minimum reaction network considering substrates, products, and 25 intermediates composed of no more components than in the cage ([Pda1bPy*c]2a+ (a ≤ 2, b ≤ 4)) was prepared. All the reactions in the network (68 in total) are classified into the five types shown in Fig. 7a. As a result of NASAP, we found several solutions with different sets of rate constants reproducing well the experimental results of time-dependent existence ratios of the substrates and the products in addition to the (〈n〉, 〈k〉) values (Fig. 7b–d). However, all the solutions have similar features as follows:

1. (1)

Very similar k1 = 102.9 min–1 M–1, k2 = 103.4 min–1 M–1, k5 = 10–1.5 min–1, and k–5 = 10–0.4 min–1 M–1 values were obtained.

2. (2)

As the rates of backward reactions except in the final step (k–5) are significantly smaller than those of the corresponding forward reactions, these rate constants (k–1, k–2, k–3, and k–4) do not much affect the self-assembly, which is confirmed by the result that the number of backward reactions is negligibly small in all the reaction steps except for the final stage. This result indicates that even though the Pd(II)–N bond formation is reversible, the self-assembly proceeds in an irreversible manner, which would be because the template anion navigates the self-assembly along the correct pathways.

According to the experimental result showing the considerably quick formation of (NO3)@[Pd212]4+ (Fig. 6), the large k3 value can be estimated. Based on this, a suitable set of rate constants in the cyclization (k3 and k–3) and cross-linking (k4 and k–4) steps were chosen from the data sets that reproduce the experimental data. One of the typical results is shown in Fig. 7b–e. k3 (101.9 min–1) and k4 (101.8 min–1) are much larger than k5 (10–1.5 min–1), which indicates that the intramolecular cross-linking in the early stage of the self-assembly is promoted by the kinetic template effect of NO3. In the early stage of the self-assembly, [Pd21Py*6]4+ is mainly produced (Fig. 7e). Then, [Pd212Py*4]4+, [Pd213Py*3]4+, and [Pd214Py*2]4+ are produced. Finally, [Pd214Py*]4+ becomes the dominant species, which is due to a small k5.

### Kinetic template effect in the cage assembly

According to the results mentioned above, the energy landscape of the self-assembly of the cage is schematically shown in Fig. 8. Whether the self-assembly proceeds towards the formation of the cage (on-pathway) or towards the sheet structure (off-pathway) is kinetically controlled by the difference between the energy barriers of the intra- and intermolecular reactions in the primitive dinuclear intermediates such as [Pd212Py*5]4+. Without NO3, the energy barrier of the formation of the sheet (ΔGoff) is smaller than that of the (OTf)@[Pd214]4+ cage (ΔGon), which is mainly due to the preference of open structures with S-shaped ditopic ligands, so the metastable sheet structure is kinetically trapped through the off-pathway. In the presence of NO3, this good template anion induces the C-shaped conformation in the dinuclear complexes, which decreases the energy barrier of the formation of the cage by preorganization to open the on-pathway (ΔGoff > ΔGon).

## Discussion

In the self-assembly from ditopic ligand 1 and [PdPy*4]2+, the reaction pathway was largely affected by the counter anions. For the self-assembly with OTf anions (off-pathway), the very weak template effect of OTf caused the production of the kinetically trapped sheet structure larger than 100 μm. In contrast, the presence of NO3 in the self-assembly afforded the thermodynamically most stable (NO3)@[Pd214]4+ cage, whose production was significantly accelerated by the kinetic template effect (on-pathway). QASAP for the cage formation elucidated that the NO3 anion promotes the reactions in the early stage of the self-assembly to a greater extent than those in the final step. This result is supported by the high binding affinity of a cyclic [Pd212]4+ model complex for NO3 and the acceleration of the [Pd212]4+ macrocycle formation in the presence of NO3. NASAP also supports the strong kinetic template effect on the on-pathway, showing that the rate constants of the intramolecular cyclization in the early stage of the self-assembly (k3 and k4) are much larger than that in the final step (k5). NO3, the best template anion, does not only efficiently convert the kinetically trapped sheet structure to the cage but is also able to alter the self-assembly pathway itself by the strong kinetic template effect on the early stage of the self-assembly to take the on-pathway.

## Methods

### General

1H and 19F NMR spectra were recorded using a Bruker AV-500 (500 MHz) spectrometer. All 1H spectra were referenced using a residual solvent peak, CD3NO2 (δ 4.33).

### Materials

Unless otherwise noted, all solvents and reagents were obtained from commercial suppliers (TCI Co., Ltd., WAKO Pure Chemical Industries Ltd., KANTO Chemical Co., Inc., and Sigma–Aldrich Co.) and were used as received. CD3NO2 was purchased from Acros Organics and used after dehydration with Molecular Sieves 4 Å. Ditopic ligand 1 was prepared according to the literature35.

### Monitoring of the self-assembly of the (OTf−)@[Pd214]4+ cage

A 2.4 mM solution of [2.2]paracyclophane in CHCl3 (125 μL), which was used as an internal standard, was added to two NMR tubes (tubes I and II) and the solvent was removed in vacuo. A solution of [PdPy*4](OTf)2 (36 mM) in CD3NO2 was prepared (solution A). Solution A (50 μL) and CD3NO2 (500 μL) were added to tube I. The exact concentration of [PdPy*4](OTf)2 in solution A was determined through the comparison of the signal intensity of [2.2]paracyclophane by 1H NMR. A solution of ditopic ligand 1 (36 mM) in CHCl3 (100 μL) was added to tube II and the solvent was removed in vacuo. Then CD3NO2 (550 μL) was added to tube II and the exact amount of 1 in tube II was determined through the comparison of the signal intensity with [2.2]paracyclophane by 1H NMR. 0.50 eq. (against the amount of ligand 1 in tube II) of solution A (ca. 50 μL; the exact amount was determined based on the exact concentration of solution A and of ligand in tube II) were added to tube II at 263 K. The self-assembly of the (OTf)@[Pd214]4+ cage was monitored at 298 K by 1H NMR spectroscopy. An example of the 1H NMR spectra is shown in Fig. 2 and Supplementary Fig. 1. The exact ratio of 1 and [PdPy*4](OTf)2 was unambiguously determined by the comparison of the integral value of each 1H NMR signal with those of [2.2]paracyclophane. The amounts of 1, [PdPy*4]2+, [Pd214]4+, and Py* were quantified by the integral values of each 1H NMR signal against the signal of the internal standard ([2.2]paracyclophane). In order to confirm the reproducibility, the same experiment was carried out four times (runs 1–4) in total. These data, the average values of the existence ratios, and the (〈n〉, 〈k〉) values are listed in Supplementary Tables 16, 911, and 13–17.

### Determination of the existence ratio of each species

The relative integral value of each 1H NMR signal against the internal standard [2.2]paracyclophane is used as the integral value in this description. We define the integral values of the signal for the substrates, the products, and the kinetically trapped species at each time t as follows; IL(t): 1/4 of the integral value of the e proton in free ligand 1; IM(t): the integral value of the f proton of Py* in [PdPy*4]2+; ICage(t): 1/4 of the integral value of the e proton in (X)@[Pd214]4+; IPy*(t): the integral value of the f proton of free Py*.

IM(0) was determined based on the exact concentration of solution A determined by 1H NMR and the exact volume of solution A added into tube II.

IL(0) was determined by 1H NMR measurement before the addition of solution A into tube II.

Existence ratio of [PdPy*4]2+: As the total amount of [PdPy*4]2+ corresponds to IM(0), the existence ratio of [PdPy*4]2+ at t is expressed by IM(t)/IM(0).

Existence ratio of 1: As the total amount of free ligand 1 corresponds to IL(0), the existence ratio of 1 at t is expressed by IL(t)/IL(0).

Existence ratio of Py*: As the total amount of Py* corresponds to IM(0), the existence ratio of Py* at t is expressed by IPy*(t)/IM(0).

Existence ratio of (X)@[Pd214]4+: As (X)@[Pd214]4+ is quantified based on 1, the existence ratio of the cage at t is expressed by ICage(t)/IL(0).

Existence ratio of the total intermediates not observed by 1H NMR: The existence ratio of the total intermediates not observed by 1H NMR is determined based on the amount of ligand 1 in the intermediates. Thus the existence ratio of the total intermediate not observed by 1H NMR is calculated by subtracting the other species containing 1 (free 1 and (X)@[Pd214]4+) from the total amount of 1 (IL(0)). The existence ratio of the total intermediates at t is expressed by (IL(0) – IL(t) – ICage(t)).

a〉: The total amount of Pd(II) ions corresponds to IM(0)/4. The amount of Pd(II) ions in [PdPy*4]2+ at t corresponds to IM(t)/4. The amount of Pd2+ ions in (X)@[Pd214]4+ at t corresponds to ICage(t)/2. The amount of Pd(II) ions in the intermediates at t is thus expressed by IM(0)/4 – IM(t)/4 – ICage(t)/2.

b〉: The total amount of ligand 1 corresponds to IL(0). The amount of free ligand 1 at t corresponds to IL(t). The amounts of ligand 1 in (X)@[Pd214]4+ at t corresponds to ICage(t). The amount of ligand 1 in the intermediates at t is thus expressed by IL(0) – IL(t) – ICage(t).

c〉: The total amount of Py* corresponds to IM(0). The amount of free Py* at t corresponds to IPy*(t). The amount of Py* in [PdPy*4]2+ at t corresponds to IM(t). The amount of Py* in the intermediates at t is thus expressed by IM(0) – IPy*(t) – IM(t).

The 〈n〉 and 〈k〉 values are determined with these 〈a〉, 〈b〉, and 〈c〉 values by Eqs. (1) and (2).

$$\left\langle n \right\rangle = \frac{{4\left\langle a \right\rangle -\left\langle c \right\rangle }}{{\left\langle b \right\rangle}}$$
(1)
$$\left\langle k \right\rangle = \frac{{\left\langle a \right\rangle }}{{\left\langle b \right\rangle }}$$
(2)

### Quantitative analysis of the self-assembly process

In QASAP, all the substrates (1 and [PdPy*4](OTf)2), and the products (the (X)@[Pd214]4+ cage and Py*) were quantified by 1H NMR spectroscopy during the self-assembly and then the amount of the intermediates not observed by 1H NMR (Int) and the average composition of the unobservable intermediates, Pda1bPy*c, were obtained. The existence ratios of the substrates and the products and 〈a〉, 〈b〉, 〈c〉, 〈n〉, and 〈k〉 with time are listed in Supplementary Tables 16 and 917 and plotted in Fig. 3.

### STEM measurement

[PdPy*4](OTf)2 (0.595 μmol), 1 (1.19 μmol), and n-Bu4NOTf (1.19 μmol) were quantified and mixed in CD3NO2 (ca. 650 μL; [1]0 = 1.82 mM). The solution was left for 4 weeks at 298 K and then diluted 170 times with CH3NO2. Two microliter of the diluted solution was poured onto the substrate (TEM microgrids, Okenshoji Co. Ltd.). Then the substrate was freeze-dried and measurements were carried out. The STEM image is shown in Fig. 4b–e.

### AFM measurement

[PdPy*4](OTf)2 (0.595 μmol), 1 (1.19 μmol), and n-Bu4NOTf (1.19 μmol) were quantified and mixed in CD3NO2 (ca. 650 μL; [1]0 = 1.82 mM). The solution was left for 4 weeks at 298 K and then diluted 170 times with CH3NO2. Ten microliter of the diluted solution was poured onto the substrate (graphene). Then the substrate was freeze-dried and vacuumed overnight. The AFM image is shown in Fig. 4f–i.

### Anion competition experiments

19F DOSY NMR measurements confirm the encapsulation of OTf in the cavity of the [Pd214]4+ cage. Anion competition experiments have been carried out by adding 4 eq. to cage (1 eq. to the anion already present in solution) of tetrabutylammonium (n-Bu4N) salts of various polyatomic anions. Displacement of the encapsulated anions has been observed through 1H and 19F NMR measurements (Supplementary Figs. 79).

One hundred microliter of a 3 mM CD3NO2 solution of the cage (0.3 μmol) was added to an NMR tube. Four hundred microliter of CD3NO2 were then added. At this stage, the cage is encapsulating OTf, and the addition of a different anion will determine the relative binding strengths to the cage of the two anions. Fifty microliter of a 24 mM solution of the tetrabutylammonium salts of the anion of interest (1.2 μmol) was added to the NMR tube and the encapsulation process was monitored by 19F NMR spectroscopy. Next, 50 μL of a 24 mM solution of a different anion were added to the solution. The solutions of the cage and anions were prepared beforehand based on the following molecular weights: cage 1865.5 g mol–1; n-Bu4NPF6 387.43 g mol–1; n-Bu4NBF4 329.27 g mol–1; n-Bu4NClO4 341.91 g mol–1.

### Determination of the relative binding affinities of anions

In order to quantitatively determine the relative affinities of the different anions to the [Pd214]4+ cage, anion competition experiments starting from the cage encapsulating the strongest bound anion (NO3) were performed (Eq. (3)).

$${\mathrm{NO}}_{3}^{-} @ {\mathrm{cage}} + {\mathrm{X}}^{-} \rightleftarrows ({\mathrm{X}}^{-}) @ {\mathrm{cage}} + {\mathrm{NO}}_{3}^{-}$$
(3)

A 2.4 mM solution of [2.2]paracyclophane in CHCl3 (125 μL), which was used as an internal standard, was added to an NMR tube and the solvent was removed in vacuo. Two hundred microliter of a 0.5 mM solution of (NO3)@[Pd214]4+ in CD3NO2 were added to the NMR tube, followed by 300 μL of CD3NO2. The exact concentration of (NO3)@[Pd214]4+ in the solution was determined through the comparison of the signal intensity of [2.2]paracyclophane by 1H NMR spectroscopy. Solutions of n-Bu4NX (X = ClO4, BF4, PF6, and OTf) with concentrations varying based on their binding strength (n-Bu4NClO4: 40 mM, n-Bu4NBF4: 200 mM, n-Bu4NPF6: 400 mM, n-Bu4NOTf: 400 mM) were prepared in CD3NO2. Increasing amounts (from 10 to 100 μL) of an anion solution was added and 1H NMR spectra were recorded. The amount of n-Bu4NX added was controlled in order to obtain (NO3)@[Pd214]4+ and the new cage encapsulating a different anion in equilibrium, to enable quantification. The relative binding constants (Kcomp) was defined by Eq. (4).

$$K_{{\mathrm{comp}}} = \frac{{\left[ {\left( {{\mathrm{X}}^-} \right)@{\mathrm{cage}}} \right][{\mathrm{NO}}_3^-]}}{{\left[ {{\mathrm{NO}}_3^-@{\mathrm{cage}}} \right][{\mathrm{X}}^-]}}$$
(4)

As the total concentration of NO3 is four times the initial concentration of the cage before the addition of n-Bu4NX ([(NO3)@cage]0), [NO3] can be expressed by Eq. (5).

$$\left[ {{\mathrm{NO}}_3^-} \right] = 4\times \left[ {\left( {{\mathrm{NO}}_3^-} \right)@{\mathrm{cage}}} \right]_0-\left[ {\left( {{\mathrm{NO}}_3^-} \right)@{\mathrm{cage}}} \right]$$
(5)

[X] in Eq. (4) is expressed by Eq. (6)

$$\left[ {\mathrm{X}}^{-} \right] = \left[ {n {\hbox{-}} {\mathrm{Bu}}_{4}{\mathrm{N}}^{+}} \right] - \left[ {\left( {\mathrm{X}}^{-} \right)@{\mathrm{cage}}} \right]$$
(6)

Thus, using Eqs. (5) and (6), Eq. (4) is expressed by Eq. (7).

$$K_{\mathrm{comp}} = \frac{{\left[ {\left( {{\mathrm{X}}^{-}} \right) @ {\mathrm{cage}}} \right]\cdot (4\left[ {({\mathrm{NO}}_3^{-})@{\mathrm{cage}}} \right]-[\left( {({\mathrm{NO}}_3^{-})} \right)@{\mathrm{cage}}])}}{{\left[ {{\mathrm{NO}}_{3}^{-} @ {\mathrm{cage}}} \right]\cdot (\left[ {n\hbox{-}{\mathrm{Bu}}_{4}{\mathrm{N}}^{+}} \right] - [\left( {{\mathrm{X}}^{-}} \right)@{\mathrm{cage}}])}}$$
(7)

As all the concentrations in Eq. (7) can be determined by 1H NMR spectroscopy, Kcomp were calculated using Eq. (7). The data used for the calculation of the Kcomp values for X = ClO4, BF4, and PF6 are summarized in Supplementary Table 8. The Kcomp values for OTf were not determined as the binding strength difference between this anion and NO3 was too large, and no peaks for (OTf)@[Pd214]4+ were observed even after the addition of 1200 eq. of n-Bu4NOTf to (NO3)@[Pd214]4+.

### Competition experiment for NO3− between the cage and the macrocycle

The (NO3)@[Pd212]4+ macrocycle was prepared from [PdPy*2](OTf)2 and 1 in the presence of n-Bu4NNO3 in CD3NO2 at 298 K. To this solution, a solution of 1 eq. of (OTf)@[Pd214](OTf)3 in CD3NO2 was added. 1H NMR spectra shown in Supplementary Fig. 21 show peaks corresponding to (NO3)@[Pd214]4+ and [Pd212]4+, indicating that the NO3 anion initially bound to the [Pd212]4+ macrocycle was selectively encapsulated by the [Pd214]4+ cage, demonstrating the stronger affinity of NO3 for the cage rather than for the macrocycle.

### X-ray analysis of (NO3−)@[Pd214]4+

The single crystal was immersed in and coated with Paraton N oil (Hampton Research Corp.), and mounted on a MicroMountTM (MiteGen LLC). Diffraction data of the single crystal were collected on a SuperNova single-crystal X-ray diffractometer with an Eos CCD detector (Rigaku Oxford Diffraction) at 294 K under ambient conditions, using Cu Kα (λ = 1.54184 Å) radiation monochromated by multilayer mirror optics. Bragg spots were integrated using the CrysAlisPro program package (Rigaku Oxford Diffraction). An empirical absorption correction based on the multi-scan method using spherical harmonics was implemented in the SCALE3 ABSPACK scaling algorithm. The structure was solved by an intrinsic phasing method on the SHELXT program52 and refined by a full-matrix least-squares minimization on F2 executed by the SHELXL program53, using the Olex2 software package (OlexSys Ltd)54. The structure was determined as an inversion twin having the polar space group of Aea2 by using a twin law matrix: −1 0 0 0 1 0 0 0 −1, where the Flack parameter55 was given as 0.004(3). Thermal displacement parameters where refined anisotropically for all non-hydrogen atoms. All the hydrogen atoms were located at calculated positions and the parameters were refined with a riding model. For counter anions, we could find a disordered NO3 molecule in the [Pd214]4+ cage and two OTf molecules, which are crystallographically equivalent to each other, in the clearance of the crystal. However, another anion could not be satisfactorily solved due to a terrible disorder. The data were corrected for disordered electron density by using the PLATON SQUEEZE method56. The crystal structure is shown in Fig. 5 and Supplementary Figs. 18 and 19. Crystallographic data and structural characteristics of the cage are summarized in Supplementary Tables 18 and 19, respectively. Crystallographic information file for (NO3)@[Pd214]2+ is provided in Supplementary Data 1. The X-ray crystallographic coordinates for structures reported in this Article have been deposited at the Cambridge Crystallographic Data Centre (CCDC), under deposition number CCDC 1935011. These data can be obtained free of charge from the CCDC via http://www.ccdc.cam.ac.uk/data_request/cif.

### Numerical analysis of self-assembly process of the cage

For the numerical analysis of self-assembly process (NASAP), a reaction network for the self-assembly of the [Pd214]4+ cage, which is indicated as (2,4,0), is constructed as follows. Starting from the final cage (2,4,0), the reaction path is traced back to the substrates, that is, [PdPy*4]2+ and 1. In this back propagation process, all the directly accessible molecular species are considered as the intermediates. Note that in this network model the intermediate species consisting of more components than the cage (2,4,0) are excluded. With this procedure taken, it is found that the total of 29 molecular species (including both the reactants and the products themselves) construct a minimal reaction network composed of 68 reactions, each of which contain the forward and backward processes. The resultant minimal reaction network is shown in Supplementary Fig. 22.

Although we call it minimal, this reaction network turns out to be so large that it is impossible to assign rate constants to each reaction and to search for the parameters in such a vast parameter space to fit the experimental results best. Therefore, we divided the whole network into five classes possessing similar characteristics and defined the rate constants in Fig. 7a:

Type I: For the coordination of a free ditopic ligand 1 releasing a leaving ligand Py*: k1 [min–1 M–1] and k–1 [min–1 M–1] for the forward and backward reactions, respectively.

Type II: For the formation of dinuclear species from two mononuclear species, with Pd(II) ions bridged by a ditopic ligand 1: k2 [min–1 M–1] and k–2 [min–1 M–1] for the forward and backward reactions, respectively.

Type III: For the cyclization in dinuclear species: k3 [min–1] and k–3 [min–1 M–1] for the forward and backward reactions, respectively.

Type IV: For the cross-linking in dinuclear species: k4 [min–1] and k–4 [min–1 M–1] for the forward and backward reactions, respectively.

Type V: For the final reaction (cage completion): k5 [min–1] and k–5 [min–1 M–1] for the forward and backward reaction, respectively.

Note here that each rate constant is defined per reaction site, based on the modeling procedure reported previously. So the actual reaction rate for each reaction is estimated as the above constant multiplied by the total number of available combinations. For example, for a ligand reaction between [PdPy*4]2+ and 1 to produce [Pd1Py*3]2+ and Py*, the rate constant is given as k1 times 4 (the number of Pd–Py* bonds in [PdPy*4]2+) times 2 (the number of coordination sites in 1), i.e.,

$$\left[ {{{\mathrm{PdPy}}^*}_{4}} \right]^{2 + } + {\bf{1}} \mathop{\longrightarrow}\limits^{{8k_{1}}}\left[ {\mathrm{Pd}}{\bf{1}}{{\mathrm{Py}}^*}_{3}\right]^{2 + } + {\mathrm{Py}}^*$$
(8)

We adopted this setting to explicitly distinguish the structural difference among the species with the same composition.

In order to numerically track the time evolution of the existence ratios for both reactants and products and the (〈n〉, 〈k〉) values, we have adopted a stochastic approach based on the chemical master equation, the so-called Gillespie algorithm57,58,59,60. In this algorithm, for all the possible N reactions including molecular species Sai, Sbi, Sci, …,

$${\it{S}}_{{\it{ai}}}{\it{ + S}}_{{\it{bi}}}{\it{ + \ldots }} \to {\it{S}}_{{\it{ci}}}{\it{ + \ldots }}\left( {{\it{i = 1, \ldots , N}}} \right)$$
(9)

The total reaction rate Rtot is calculated as

$$R_{{\mathrm{tot}}} = r_1 + r_2 + \ldots + r_N$$
(10)
$$r_i = k_i\left[ {S_{ai}} \right]\left[ {S_{bi}} \right] \ldots .$$

Starting from the initial time t = 0, at each instant t, which one of the reactions to occur is determined with the uniform random number, $$s_1 \in (0,1)$$ as

if $$s_1 \le \frac{{r_1}}{{R_{{\mathrm{tot}}}}}$$, then reaction 1 occurs,

if $$\frac{{r_1}}{{R_{{\mathrm{tot}}}}} < s_1 \le \frac{{r_1 + r_2}}{{R_{{\mathrm{tot}}}}}$$, then reaction 2 occurs,

if $$\frac{{r_1 + \ldots + r_{N - 1}}}{{R_{{\mathrm{tot}}}}} < s_1 \le 1$$, then reaction N occurs.

Another uniform random number $$s_2 \in (0,1)$$ is independently given to fix the incremental dt as

$${\it{dt}} = {\mathrm{ln}}(1/{\it{s}}_2)/R_{{\it{{\mathrm{tot}}}}}$$
(11)

Time is updated as t = t+dt, together with the update of the numbers of corresponding molecular species, i.e., $$S_{ai} \to S_{ai} - 1$$, $$S_{bi} \to S_{bi} - 1$$, $$S_{ci} \to S_{ci} + 1$$, …. The reason why this approach traces the chemical reactions and actually works well is given in the literatures in detail, along with the practical way to implement it57,58,59,60.

With the initial conditions (numbers of species), 〈PdPy*40 = 1,200, 〈10 = 2,400, 〈others〉0 = 0, rate constant search was performed in a ten-dimensional parameter space (k1, k–1, k2, k–2, k3, k–3, k4, k–4, k5, k–5). The Avogadro number and the volume of the simulation box were set to be NA = 6.0 × 1023 and V = 2.0 × 10–18 L, respectively, which gives the same concentrations as the experiments were carried out under. After the rate constant search was finished, refined simulations were performed for some rate parameter sets that give existence ratios and (n, k) plot in good agreement with the experimental counterparts. The adequacy of the fitting to the experimental data was evaluated from the residual sum of squares (RSS) with the average of the experimental data, obtained from three runs. For all the time steps ti at which the experimental data of existence ratios RexS and parameters nex and kex are available, RSS’s are calculated with the numerically obtained values RnuS and so on as (note that S = [PdPy*4]2+, 1, [Pd214]4+, or Py*),

$${\mathrm{RSS}}_1 = \mathop {\sum }\limits_{t_i} \mathop {\sum }\limits_S \left( {R_{{\mathrm{nu}},t_i}^S - R_{{\mathrm{ex}},t_i}^S} \right)^2$$
(12)
$${\mathrm{RSS}}_2 = \mathop {\sum }\limits_{t_i} \left( {n_{{\mathrm{nu}},t_i} - n_{{\mathrm{ex}},t_i}} \right)^2 + \mathop {\sum }\limits_{t_i} \left( {k_{{\mathrm{nu}},t_i} - k_{{\mathrm{ex}},t_i}} \right)^2$$
(13)

A representative result is shown in Fig. 7b–d, in which the initial particles and the volume of the simulation box were set to be a hundred times larger than the rough parameter search, i.e., 〈PdPy*40 = 120,000, 〈10 = 240,000, and V = 2.5 × 10–16 L. A similar behavior was confirmed with several runs for the particle numbers given above.