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Molecular exclusion limits for diffusion across a porous capsid

Abstract

Molecular communication across physical barriers requires pores to connect the environments on either side and discriminate between the diffusants. Here we use porous virus-like particles (VLPs) derived from bacteriophage P22 to investigate the range of molecule sizes able to gain access to its interior. Although there are cryo-EM models of the VLP, they may not accurately depict the parameters of the molecules able to pass across the pores due to the dynamic nature of the P22 particles in the solution. After encapsulating the enzyme AdhD within the P22 VLPs, we use a redox reaction involving PAMAM dendrimer modified NADH/NAD+ to examine the size and charge limitations of molecules entering P22. Utilizing the three different accessible morphologies of the P22 particles, we determine the effective pore sizes of each and demonstrate that negatively charged substrates diffuse across more readily when compared to those that are neutral, despite the negatively charge exterior of the particles.

Introduction

Physical barriers, and the molecular communication across those barriers, are fundamental themes in biology1,2,3. Among other features, biological barriers provide cellular and sub-cellular separation with controlled permeability. The diffusion of molecules across barriers is often selective and is usually made possible via channels or pores that can discriminate based on molecular properties such as size, charge, and chemical makeup4,5,6,7,8. Pores, gates, and channels are often responsible for homeostasis by controlling chemical fluctuations across the barrier. Porous barriers are not exclusive to biological systems and they are used extensively in many materials fields such as catalysis, energy, gas storage, semi-conductors, purification, and separations, among others9,10,11,12. The advantages of porous materials often align with the advantages provided by pores in biological systems.

A eukaryotic cell provides many sophisticated examples of porous barriers in the form of membranes. The cell itself has a defining semi-permeable membrane and internally contains many membrane-bound organelles where diffusion across is controlled via channels. An example is the nucleus, where the nuclear pore complex limits the diffusion of molecules smaller than 9 nm in either direction13. Bacterial cells are also defined by semi-permeable membranes, but sub-cellular bacterial microcompartments, such as the carboxysome, are assembled from proteins and package enzymes that facilitate specialized metabolic reactions. The carboxysome is a proteinaceous compartment where the porous shell acts as a molecular barrier and is thought to discriminate between O2 and CO2 by selectively limiting diffusion of O2 through the pores14,15,16. While channels and pores are key features that serve as communications portals across the barriers17,18,19, the mechanisms of diffusion across biological pores remains a major topic of interest.

Viruses and virus-like particles (VLPs) provide additional examples of porous biological systems. Many viral capsids comprise porous protein shells whose function is to protect their encapsulated genome while allowing the diffusion of water and ions across the pores20. The capsids of VLPs are structurally similar to the viruses from which they are derived but are non-infectious and can be repurposed to sequester a variety of different cargos including proteins, enzymes, small molecules, polymers, and even other protein compartments21,22,23,24,25. For example, VLPs derived from the Salmonella typhimurium virus bacteriophage P22 serve as versatile porous protein cages that can be repurposed to encapsulate a diverse array of cargo molecules. P22 VLPs self-assemble into 56 nm T = 7 icosahedral capsids from 420 units of a coat protein (CP) and ~100–300 copies of a scaffolding protein (SP). The SP templates assembly of P22 and, upon formation of the particles, is encapsulated as a cargo26,27,28,29,30. Genetically fusing other proteins to the SP has allowed for the encapsulation of cargo molecules in high copy-number, with local concentrations mimicking that of the intracellular environment, while maintaining the native morphology of the P22 particles31,32,33,34,35.

P22 VLPs provide a unique system to study the molecular selectivity for diffusion across a porous barrier connecting the interior of a compartment to the bulk environment. These particles offer the advantage of the controlled encapsulation of enzymes and provide access to three different particle morphologies, each with a different pore size. The first of these morphologies is the procapsid (PC). In both the infectious phage and the VLP system, PC is the initial structure formed by SP templated assembly and has ~3 nm pores located at the center of the hexameric CP units26,30. In the infectious phage, a morphological transformation takes places during DNA packaging resulting in an expansion of the PC structure to a more angular particle 62 nm in diameter but with smaller pores36,37. This morphogenesis can also be induced in VLPs with either heat or chemical treatment and the resulting particles are referred to as the expanded morphology (EX)27,38,39. The third morphology, known as the wiffle-ball (WB), is accessed by heating the VLP capsids at 75 °C39. The diameter of WB is the same as EX, but the loss of all 12 of the pentameric units in the icosahedral capsid creates extremely large 10 nm pores. While there are static approximations of these three structures (PC, EX, and WB) and some information available on the modeled dynamics of other capsids in an aqueous environment40,41, the dynamics and effective porosity of P22 are not resolved.

In this study, we investigate the porosity of P22 VLPS and develop a methodology to study diffusion across porous barriers. Since barriers have been known to contribute to the selectivity of enzymes, we also determine how diffusion across the P22 pores contributes to this. We use the activity of an encapsulated enzyme to probe the access of different synthetic substrates across the porous barrier of the P22 VLP. By introducing substrates of different sizes, we can experimentally determine how the size-dependent porosity of the P22 particle limited molecular access to the encapsulated enzymes (Fig. 1). The three different P22 morphologies each exhibit different gating selectivity thresholds due to their different pore sizes. We demonstrate that although the biochemically determined effective pore sizes for P22 particles are comparable to the pore size calculated from the available cryo-EM structures, after the side chain electron density is taken into account the effective pore sizes appear much larger than the P22 model, suggesting that the capsid might indeed be dynamic and undergo a “breathing” mechanism. Furthermore, an examination of the role of substrate charge reveals that negatively charged substrates were capable of traversing across the P22 barrier more readily than the neutral substrates. Although the exterior of the P22 VLPs exhibits a net negative charge, the electrostatic environment immediately surrounding the pores shows that there are both negatively and positively charged regions that can account for differences in charge selectivity. The importance of electrostatic effects on diffusion becomes even more pronounced upon alteration of the P22 charge distribution through the construction of a series of mutants. We see that after removing the positive regions found around the pore interior, the diffusion of substrates across the pore is inhibited. Overall, this study provides a generalizable approach to probe the effective permeability of artificial nano-reactors and contributes to fundamental insights into how compartments regulate substrate diffusion.

Results and discussion

Size discrimination of substrates diffusing across the P22 capsid shell

Probing the porosity of P22 VLPs revealed that there is a discrete pore size that limits chemical communication and discriminates between differently sized molecules traversing across the capsid. This allowed us to calculate the effective pore size for the different morphological forms of P22 VLPs. The estimated pores sizes were in good agreement with the pore sizes calculated from the available cryo-EM reconstruction models but differed once we accounted for the steric contributions from the amino acid side chains, suggesting that the flexible regions of the pores may undergo some size fluctuations.

From structural models, the three morphologies of P22 VLPs all contain distinctly different pore sizes (shown in Fig. 3a–c modeled using UCSF Chimera48). The PC morphology (PDB: 2xyy) has pores that are modeled to be approximately 4.2 × 1.9 nm, positioned at the center of the hexamers (Fig. 3a). Upon transformation to the expanded (EX) form (PDB: 5uu5), the pores become much smaller due to a conformational shift of the CPs (Fig. 3b). Lastly, the WB structure (PDB: 3iyh) has large, approximately 10 nm, pores (Fig. 3c) as well as smaller pores present at the center of hexamers. Although these pore sizes have been modeled in the cryo-EM reconstructions, such structural models are static averages and reflect neither the potential dynamic nature of these particles (and pores) in solution, nor account for contributions from the flexible side chains.

To account for the electron density and steric contributions from the side chains around the pores, we then used HOLE to model the pores for the PC and EX P22 capsids (Fig. 4a, b)49. This method resulted in average pore sizes of 1.9 and 1.3 nm for PC and EX, respectively. HOLE used a Monte Carlo algorithm to simulate the route a sphere would take in order to squeeze through the pore (Fig. 4c, d) and also calculate a theoretical diameter of the pore as a sphere diffused across (Fig. 4e, f). The effective diameter calculated in Fig. 4e for PC ranged from 1.3 to 1.8 nm, while for EX (Fig. 4f) it ranged from 0.15 to 0.6 nm. These simulations all indicate much smaller pores size when the side chains are considered, versus when they are excluded. To confirm this observation was due to the side chains rather than the differences in the software programs, we also modeled the PC pore excluding the side chains (Fig. 4a). Without the side chains, HOLE shows a diameter of 4.4 nm, which is very close to the pore sizes calculated using Chimera. These results further underlined the need to determine the effective pore size a small molecule encounters when diffusing across the P22 particle.

AdhD enzymes within the P22 morphological variants behaved as a catalytically active macromolecular bait. A reaction mixture containing both the modified NADH-Neg and the substrate, acetoin, was used. Thus, NADH molecules able to traverse the capsid shell and access the inside of the VLPs would be oxidized to NAD+-Neg and the reaction would proceed as shown in Fig. 3d. If NADH-Neg was not able to access the interior, there would be no reaction. The oxidation of NADH-Neg was monitored by the decrease in the characteristic absorbance of NADH at 340 nm. To account for any differences in the enzyme reaction with different NADH-Neg generations, the activities of the encapsulated enzymes were compared to that of free enzymes with the same substrates and the kinetic parameters are presented as a ratio of free/encapsulated.

Probing the effects of charge on diffusion of molecules across the P22 VLP

In order to determine the effects of electrostatics on the gating of molecules passing across the VLP barrier, we synthetically neutralized the negative charges on the dendrimers and also varied the ionic strength of the buffer. For this study positively charged dendrimers were not used because at the concentrations of the NADH-Neg conjugates required for these experiments, the particles are known to coalesce into a bulk aggregated material53,54.

Charge neutralization of the NADH-Neg molecules was achieved using the reaction outlined in Fig. 5e, where the carboxylate-terminated dendrimers were activated with EDC and methyl amine was added to form terminal amides forming NADH-Neu. DLS and zeta potential were measured in order to confirm the size-integrity of the molecules and verify that the negative charge had been altered. DLS revealed that size of the dendrimers remained comparable after the reaction, and the dendrimer population remained monodisperse after purification (Supplementary Fig. 19). Measurements of zeta potential showed that the new dendrimers were no longer negatively charged and became slightly positively charged (Supplementary Fig. 19) across all generations, likely due to the protonation of some of the internal tertiary amines55.

Upon modeling the electrostatic potential of P22 particles, we saw that there were positively charged regions around the interior of the P22 pores that could facilitate interactions between NADH-Neg and the pores, even though the net charge of the P22 particle is negative. The results showing the charge density around the pores of the PC P22 can be found in Supplementary Movie 1 and Fig. 6a, c. Although side chain information was uniformly truncated in the PC structure deposited in the PDB, at 3.8 Å resolution there is high confidence in the protein backbone location and rudimentary side chain positioning. Consequently, we used Coot software to manually add probable rotamer conformers in order to generate an electrostatic map of the molecular surface of P22 using PyMOL and APBS. Although the exact rotamer positioning is uncertain, the overall surface charges likely remain valid using this technique. The asymmetric unit with this encoded information is shown in Fig. 6a, c and in Supplementary Movie 1. With this electrostatic information, the charge density around the outer surface of the pore appeared negative, while the charge density on the inside surface of the pore appears positive (blue). The inset in Fig. 6a shows the charge distribution across the pore, along with the electric field lines.

The electric field lines provide the likely path a charged molecule might follow within the electrostatic field, a phenomenon previously observed in ferritin protein cages and acetylcholine esterase58,61,62,63. The insert in Fig. 6a provides a model for the electric field surrounding the pore area, where the very negatively charged exterior transitions into a positively charged interior. This generates a charge gradient between the exterior and interior of the capsid. At the ionic strength conditions used in these reactions, it is likely that the dendrimers can get close to the capsid before experiencing the electric field. Once within the field, we hypothesize that the charged molecules will diffuse away from the point charges on the VLP following the trajectory of the field lines, until encountering the positively charged field on the interior region of the pore, which would effectively pull the molecules into the particle. With an increase in ionic strength, the Debye length decreases, effectively weaking the electrostatic interactions between the pores and the NADH molecules. This was supported by the data that showed the NADH-NegHS and NADH-NeuHS molecules having very similar trends in activity and the same calculated effective pore sizes.

To test the validity of our hypothesis, we made two different mutants of the P22 CP subunits that, when assembled into the P22 VLP capsid, changed the charge distribution on either the particle exterior or interior surfaces. Each new mutant contained two mutations: D143N/D357N and K184Q/R203S. The first set of mutations removed negative charges on the exterior both around the pore and regions further away, while the latter mutation removed two positive charges on the interior of the capsid at the region close to and within the pore. The particles were characterized using the same methods as the particles described in earlier sections of Results and discussion (Methods and Supplementary Figs. 2830). Zeta potential measurements revealed that the D143N/D357N mutant had a less negative zeta potential (−19.8 mV) compared to wt P22 (−23.8 mV), while the K184Q/R203S mutant exhibited a more negative zeta potential (−25.2 mV). The electrostatic surfaces and their respective field lines were also calculated, showing a very clear difference in charge distribution both on the interior and exterior surfaces (Fig. 6d, e). D143N/D357N remained positively charged on the inside; however, the exterior showed a much less negative surface (Fig. 6e) compared to wt P22 (Fig. 6c). The field lines for K184Q/R203S reflected a much more significant change to the landscape where the positively charged patches seen on the inside in wtP22 completely disappeared (Fig. 6d).

Testing the diffusion limits of these mutants revealed that when the charges of the positive inner pore were neutralized, molecules previously capable of diffusing across the pore were no longer able to do so. For this experiment, we chose generation NADH-Neg3.5 and NADH-Neu3.5 because in earlier experiments we saw that the negative substrate could access the interior of the wtP22, while the neutral substrate could not. The kinetic plots and extracted kinetic constants using NADH and each of the mutants can be found in Supplementary Fig. 31, while the data collected using NADH-Neg3.5 and NADH-Neu3.5 are shown in Fig. 7. The NADH-Neg3.5 showed very little activity when used together with the negative mutant (K184Q/R203S), while with the positive mutant (D143N/D367N) showed slightly enhanced activity, albeit with some inhibition compared to wtP22 (Supplementary Fig. 31a and Fig. 7). These data are consistent with our earlier hypothesis in that the positive charges located within the pore facilitate the diffusion of charged molecules across the pore and their removal in the K184Q/R203S mutant inhibited the uptake and oxidation of the NADH-Neg3.5. The activity ratio (kcat for free:kcat encapsulated) for K184Q/R203S with NADH-Neg3.5 is much higher compared to values collected on wtP22 (Fig. 7). In contrast the ratio values for D143N/D357N are slightly lower than for the wtP22 indicating more access of the substrate across the pore in response to the diminished negative charge of the capsid exterior. These charge-based differences disappeared when the ionic strength of the medium was increased, due to charge screening, where both mutants and wtP22 capsids exhibited roughly the same activity ratios (Fig. 7).

In summary, this study allowed us to systematically probe molecular diffusion across the porous barrier. Using P22 VLP capsids as a model system we determined the pore sizes for two different particle morphologies; the EX particles (2.7 nm) and PC (4.4 nm). Another morphology, the WB, allowed molecules up to 7.5 nm to diffuse freely across the capsid boundary and served as a control where the barrier to small molecule diffusion was removed. Although there has been speculation on the dynamic nature of these VLPs and simulations that have predicted that the pores may undergo large fluctuations40,41, the lack of high-resolution structural data had limited our knowledge. Our data showed that the molecular cut-off size was close to the size of the pores calculated using the available cryo-EM model. However, when considering the side chains of the residues lining the pores, the data suggested that the pores of P22 are dynamic where molecules twice the theoretical size were able to diffuse across. The effective pore size is highly dependent on the charge of the diffusing molecule and can be tuned by neutralizing the molecular charge or increasing the ionic strength of the reaction buffer; both of which resulted in a decreased effective pore size. These are interesting findings considering that the exterior of P22 is known to be negative and it was the negatively charged dendrimers that were able to diffuse across more readily. Upon analyzing the electrostatic potential around the pores, we suggest a charge-driven mechanism for the negatively charged molecules to diffuse across, where the charge gradients around the pores guide the molecules diffusing across. Porosity is a crucial structural feature present in many biological compartments and often an important principle when designing new functional materials as a way to segregate components. Here we showed how the pores can drastically affect enzymatic activity, perhaps providing further insight into how their presence contributes to the specificity of catalysts in biological systems and those that are synthetically derived. We have presented a systematic study of the molecular porosity and permeability of a compartment and determined the diffusion limitations across a model capsid boundary in terms of both size and charge.

Methods

Site-directed mutagenesis protocol for D143N/D357N and K184Q/R203S P22 CP

The sense and anti-sense primers (Supplementary Table 1) were designed to incorporate the desired mutations and are displayed in Supplementary Information. The two mutations were inserted into each construct simultaneously by mixing four primers in each reaction setup. The setup for each reaction was as follows: 6.6 μL of PCR grade water, 0.6 μL of each primer (10 μM) to a final concentration of 0.3 μM, 1 μL of 10 ng pRSF-Duet plasmid containing the CP gene, and 10 μL of KOD Hot Start Master Mix (0.04 U/μL). The PCR cycling followed the recommended conditions for the KOD polymerase: (1) polymerase activation (2 min at 95 °C), (2) denaturation (2 min at 95 °C), (3) annealing at the lowest Tm °C for 10 s, (4) extension for 125 s at 70 °C (25 s/kbp); followed by steps 2–4 cycled 40 times. Before transforming into cells, the PCR products were incubated with dNTPs for 4 h at 37 °C.

Size-exclusion chromatography with multi-angle light scattering and refractive index detection

MALS coupled with SEC was used to determine the molecular weight of all the constructs. The samples (25 μL/each) were introduced to the mobile phase of an Agilent 1200 HPLC using an autoinjector and separated over WTC-200S 5 μM, 2000 Å, 7.8 × 300 mm column. A 16-angle MALS detector (HELEOS-Wyatt Technology Corporation) and Optilab rEX refractometer (Wyatt Technology Corporation) then detected the scattering and concentration of the injected samples. Astra 5.3.14 software (Wyatt Technology Corporation) was used for data analysis with the following equation used to compute the molecular weight:

$$M=\frac{R\left(\theta \right)}{\frac{{4\pi }^{2}{n}_{0}^{2}}{{N}_{A}{\lambda }_{0}^{4}}{\left(\frac{{dn}}{{dc}}\right)}^{2}{cP}(\theta )}$$
(1)

Where $$R\left(\theta \right)$$ is the excess Rayleigh ratio from solute, $${n}_{0}$$ is the solvent refractive index, $${N}_{A}$$ is Avogadro’s number, $${\lambda }_{0}$$ is the wavelength of incident light, $$\left(\frac{{dn}}{{dc}}\right)$$ is a material-specific refractive index increment where in the case of proteins the value is 0.1850 mL/g, $$M$$ is the molar mass, $$c$$ is the solute concentration in w/v, and finally $$P(\theta )$$ is the form factor.

Transmission electron microscopy

For obtained microscopy images of the particles, the samples were prepared with a 0.1 mg/mL concentration and 10 μL was applied to formvar coated grids (Electron Microscopy Sciences). The droplet was covered and allowed to incubate at room temperature for 5 min. The droplet was removed using filter paper and grid was washed with 10 μL of distilled water. The grids were then incubated with 10 μL of 1% uranyl acetate for staining, and subsequently removed after a 5 min incubation. JEOL 1010 transmission electron microscope was used for imaging with a 100 kV accelerating voltage.

Agarose gel shift assay

The native gel for the shift assay was prepared by adding 0.4 g of agarose to 50 mL of 40 mM Tris, 5 mM acetate, 1 mM EDTA, pH 8.2 to result in a 0.8 % (w/v) concentration. Bromophenol blue-containing loading buffer was added to all samples prior to loading. The shift assay was run for 2 h at 75 V with the running buffer matching the gel buffer above. The gel was stained by Coomasie blue, rinsed with water, and incubated with destaining solution. UVP MultDoc-IT Digital Imaging System was used to obtain the gel images.

P22 expansion to wiffle-ball morphology

The P22 protein concentration was adjusted to 1 mg/mL using 50 mM sodium phosphate, 100 mM sodium chloride, pH 7.0 buffer, and heated at 75 °C for 20 min. The samples were cooled and usually a precipitate, containing aggregated proteins, formed that was removed using centrifugation. Ultracentrifugation (215,619xg-Sorvall WX Ultra Series) was used to recover the intact WB particles.

Expansion using SDS

The P22 protein concentration was adjusted to 1 mg/mL using 50 mM sodium phosphate, 100 mM sodium chloride, pH 7.0 buffer, and a 7 mM concentration of SDS solution was prepared in the same buffer. The SDS solution was added slowly to the P22-containing solution to result in final concentrations of 0.5 mg/mL of protein and 3.5 mM SDS. Incubation proceeded for 5 min at room temperature and then immediately ultra-centrifuged twice to remove SDS and any broken particles.

To ensure maximum homogeneity of P22 particles, all samples were also purified over a CsCl gradient. The gradient was prepared using 1.2 and 1.4 g/mL CsCl solutions in sodium phosphate buffer and then mixed using Biocomp Gradient Master 108 preset setting for 12–14% (w/w). Protein concentration was adjusted to be between 8 and 10 mg/mL and 750 μL of solution was carefully added to the top of the mixed gradient. The samples were then ultra-centrifuged 287,472 × g for 2 h and resulted in a band midway through the sample tube containing the particles.

Activity assays were carried out on Agilent 8454 Uv-Vis Spectrophotometer at room temperature. The buffer used for kinetics studies was 50 mM sodium phosphate, 100 mM NaCl, pH 7.0, unless otherwise specified. For the AdhD activity assay, P22 PC, EX, and WB, encapsulating SP-AdhD cargo, were adjusted to a concentration of 650 nM of encapsulated SP-AdhD, and the substrate (Acetoin) concentration was varied from 1 to 200 mM. The concentration of NADH variants was 220 μM (ε340 = 6.22 mM−1cm−1). It is likely that this concentration is not accurate due to the NADH modifications; however, since we were mainly focused on comparing free to encapsulated, this was not significant. There was a total of seven different concentrations for each kinetic assay, and each reaction shown in the Supplementary Information documents was repeated in triplicate. To ensure that minimal free enzyme was present in the samples before monitoring kinetics of the encapsulated enzymes, the P22 solution was ultra-centrifuged and purified over a Ni-NTA column to remove any his-tag equipped free AdhD-SP enzymes before each batch of reactions and used immediately. Kinetics assay done at higher ionic strength (HS) was done in pH 7 50 mM sodium phosphate, 400 mM sodium chloride buffer. Everything else was kept constant. The Michaelis–Menten constant (KM) and the turnover rate (kcat) values were extracted using a nonlinear regression Eq. (2).

$$f(C)=\frac{v[C]}{{K}_{\rm{M}}+[C]}$$
(2)

where [C] refers to the substrate concentration, since the concentration of NADH remained constant, this is referring to the concentration of acetoin. The v variable refers to the maximum velocity at a given [C]. The Michaelis–Menten constant, KM, is the [C] at 1/2 vmax.

Aqueous pore model prediction

The model used here was adapted from Renkin et al. and Timney et al.13,50. In these equations, the pores in the particles were assumed to be cylindrical and the dendrimers traversing across were considered as spheres. There were two equations combined that considered two different scenarios. Equation (3) describes a situation in which the dendrimers pass across the VLP pores without interacting with the edges of the pore.

$$A={{A}_{0}\left(1-\frac{r}{{r}_{0}}\right)}^{2}$$
(3)

The effective pore area (A) can be calculated by multiplying the theoretical cross section of the pore (A0) by the dendrimers probability to enter the pore: ($$1-\frac{r}{{r}_{0}}$$)2, where r is the radius of the dendrimer and r0 is the radius of the pore. The second scenario corrects for the friction that may arise between the pore of the particle and the dendrimers:

$$\frac{A}{{A}_{0}}={1-2.104\left(\frac{r}{{r}_{0}}\right)+2.09\left(\frac{r}{{r}_{0}}\right)}^{3}{+.95\left(\frac{r}{{r}_{0}}\right)}^{5}$$
(4)

Multiplying the steric effects of the dendrimers entering the pore and the frictional contribution of the interaction between the VLP pore wall and the dendrimer (4), we can calculate the permeability coefficient (P):

$$P=\left({\left(1-{\frac{r}{{r}_{0}}}\right)}^{2}\right)\left(\left(\right.1-2.104\left({\frac{r}{{r}_{0}}}\right)+2.09{\left({\frac{r}{{r}_{0}}}\right)}^{3}+.95{\left({\frac{r}{{r}_{0}}}\right)}^{5}\right)$$
(5)

The permeability coefficient describes the rate of diffusion (cm/s), while we measured the rate of enzyme turnover as a function of dendrimer size. Because we directly compared free and encapsulated enzyme, the ratio between the two directly correlated with the diffusion across the porous membrane and corrected for effects arising from the dendrimer-NADH conjugates accessing the active site. Rearranging (5), we fit the data corresponding to the kcat ratio of free: encapsulated as a function of dendrimer size using (6).

$$f(r)=\frac{P}{\left({\left(1-\frac{r}{{r}_{0}}\right)}^{2}\right)\left(\left(\right.1-2.104\left(\frac{r}{{r}_{0}}\right)+2.09{\left(\frac{r}{{r}_{0}}\right)}^{3}+.95{\left(\frac{r}{{r}_{0}}\right)}^{5}\right)}$$
(6)

Sigmoidal curve fitting for pore size prediction

The sigmoidal curve is typically used as a dose-response curve to identify cooperativity within a system and predict the likelihood of a binding event. In our system it was used to predict the likelihood of a substrate accessing the encapsulated enzyme. The basal region of the curve showed the expected linear response where increasing substrate size slowly decreases the kcat when measured using the encapsulated enzyme. This was likely a result of friction between the pore wall and the substrate. The asymptotic, or the maximum, region of the curve provided us with the sizes of substrates no longer gaining access to the enzyme. The slope of the fast growth phase, or the Hill Slope, and slow growth phase base allowed us to calculate the inflection point and predict the pore size. We interpret the transition between the slow growth and fast growth phases (inflection point) as a shift from substrate diffusion across the pore being limited by friction to a system limited by sterics when the substrate size approached the pore size. Equation (7) was used to fit the data and calculate the required values,

$$f\left(d\right)={\rm{Base}}+\left(\frac{{\rm{Max}}}{1+{{\rm{e}}}^{\left(\frac{{d}_{{\rm{half}}}-d}{{\rm{Slope}}}\right)}}\right)$$
(7)

where d was the size of the dendrimer, the base was the value at which the curve crosses the y-intercept, max established the maximum value at which the slope approached zero, dhalf is the diameter at the midpoint between the base and max and is used to establish the Hill Slope, labeled as slope in (7). The d at which the base line and slope line intersect is called the inflection point, and also the theoretical pore size.

Electrostatic surface of the pores

To map the electrostatic surface of the PC P22 pores, the coordinates of PDB 2xyy were loaded into COOT where amino acids containing truncated side chains were mutated to the same amino acids with intact side chains. The lowest energy conformer was computed and a modified 2xyy PDB file was established. The structural information from the PDB file was then used to generate a PQR file using APBS online software. This software used the Poisson–Boltzmann calculations to compute an electrostatic map of CP surface. PyMOL was then used to visualize the molecular surface using the PDB generated by COOT and the APBS PQR file. The movie was made using ImageJ software.

The first step of the reaction was the bromination of NAD+ at the C8 position of adenine. NAD+ (5.00 g, 7.55 mmol) was dissolved in pH 4.5, 50 mM sodium acetate buffer (60 mL) at room temperature. Reaction was stirred under argon as bromine (2.0 mL, 39.05 mmol) was added dropwise to solution. Reaction was allowed to proceed overnight. Note: the reaction still worked when completed without argon. We have found that a precipitate formed upon the addition of bromine that contributed to a lower yield of our product. The completion of the reaction was monitored by changes in the UV absorbance where the λmax shifted from 259 to 265 nm (Supplementary Fig. 3) and further verified using H1 NMR through the disappearance of the C8 proton peak (Supplementary Fig. 4)46. For this step it was important to protect NAD from electrophilic attack at the C5 position of the nicotinamide; therefore, the oxidized form (NAD+) was used. The mixture was washed with carbon tetrachloride until the organic layer showed little to no color. The aqueous phase was recovered and dialyzed into water overnight using a 500 MW cut-off cellulose dialysis tubing (Spectrum Labs). The resulting solution was lyophilized for 48 h and product was recovered as a light-yellow powder (3.09 g, 52% yield). 1H-NMR (400 MHz, D2O) $${\rm{\delta }}$$ 9.25 (s, 1H), 9.10 (d, 1H), 8.75(d, 1H), 8.20 (t, 1H), 8.1 (s, 1H), 6.10 (d, 1H), 5.95 (d, 1H), 4.0–4.5 (m, 9H).

At this point in the synthesis NAD+ was reduced to NADH, which is more thermally stable as the C2 and C6 of nicotinamide of NAD+ are especially prone to nucleophilic attack, which was important to avoid in the following step. 8-NAD+-Br (351.3 mg, 0.487 mmol) was dissolved in 18 mL of 1.3% (v/w) sodium bicarbonate buffer using a two-neck RBF. Solution was stirred while bubbling argon gas through solution for 30 min. Using other neck of RBF, sodium dithionite (175.6 mg) was added while continuing to bubble argon. Resulting mixture was stirred under argon gas for 3 h at room temperature, monitored by the appearance of a peak at 340 nm in the UV spectrum (Supplementary Fig. 3) and verified using H1 NMR through the upfield shift of the nicotinamide protons (Supplementary Fig. 5). The remaining sodium dithionite was oxidized by removing the flow of argon and stirring while exposed to air for 1 h. The completion of reaction was monitored by measuring the A265 to A340 ratio and reaction stopped when the ratio approached ~3. The product was recovered by precipitating with 10× cold acetone (stored in −20 °C) and centrifuging at 4500 × g for 5 min. The supernatant was decanted through a filter. Remaining pellet was dissolved in water and lyophilized for 48 h. Product was recovered as a beige solid (430 mg). 1H-NMR (400 MHz, D2O) $${\rm{\delta }}$$ 8.1 (s, 1H), 6.82 (s, 1H), 6.18 (d, 1H), 5.80 (d, 1H), 4.0–5.0(m, 11H), 2.9 (m, 2H).

8-NADH-Br (321 mg, 0.408 mmol) was added to 6 mL DMSO and heated to 60 °C. In a different flask, 2,2′-(ethylenedioxy)bis(ethylamine) = linker (1.285 g, 8.32 mmol) was dissolved in 6 mL of DMSO and also heated to 60 °C. The solution containing 8-NADH-Br was added to solution containing linker. Resulting mixture was stirred under a condenser at 80 °C for 6 h. Reaction was cooled to room temperature, and the acetone precipitation was repeated by adding 120 mL of cold acetone and precipitate was recovered using centrifugation (4500 × g, 5 min). Pellet was dissolved in water and lyophilized to dry. Powder was recovered as a beige solid (318 mg, 60% yield). Yield was calculated based on A340. 1H-NMR (400 MHz, D2O) $${\rm{\delta }}$$ 8.2 (s, 1H), 6.75 (s, 1H), 5.85–6.10 (m, 2H), 5.25 (dd, 1H), 3.8–4.5 (m, 9H), 3.1–3.5 (m, 13H), 2.80 (d, 2H) 1.01 (t, 1H). Calculated HRMS for C27H42O16N9P2 810.2225, observed 810.2223. Calculated HRMS for C27H41O16N9NaP2 832.2044, observed 810.2043. (Supplementary Figs. 6 and 7).

All synthesis for the conjugation proceeded using the same reaction scheme with varying stoichiometric amounts of dendrimer. The theoretical number of NADH molecules per dendrimer was adjusted using the molar ratios displayed in Table 1. The concentrations of NADH were determined using A340. 8-NADH-NH2 amount was adjusted to 50 mg (61.6 µmols) and kept constant for all reactions. As a representative example: Gen. 4.5 dendrimer (202 mg, 7.7 µmols, 650 µL) was dissolved in 3 mL of pH 4.7, 100 mM MES buffer. EDC (0.0145 g, 75 µmols) was added to the reaction and stirred for 1 h. 8-NADH-NH2 was dissolved in the MES buffer (3 mL) and added to solution containing dendrimers. Mixture was stirred overnight and purified using cellulose dialysis tubing. Molecular weight membrane cutoffs varied from 1500 to 14,000 Da depending on the size of dendrimers. Product was lyophilized and stored at 4 °C until use.

All synthesis for the conjugation proceeded using the same reaction scheme with varying stoichiometric amounts of methyl amine and EDC. The molar ratios for EDC and methyl amine were in ten-fold excess of theoretical dendrimer terminal carboxylic acid groups. As a representative example: NADH-Gen 3.5 (3.85 µmols) was dissolved in pH 4.7, 100 mM MES buffer with EDC (0.468 g, 2.464 mmols) and stirred for 1 h at room temperature. Methyl amine (0.166 g, 2.464 mmols) was added to reaction and stirred overnight and purified using the same type of dialysis tubing as was used in the synthesis of NADH-GenXX dendrimers.

Data availability

The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary information files. Data are also available from the corresponding author upon reasonable request.

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Acknowledgements

This work was funded by a grant from the Human Frontier Science Program (HFSP) 4124801. E. S. was partially supported by the Graduate Training Program in Quantitative and Chemical Biology under Award T32 GM109825 and Indiana University. T. D. was supported in part by the National Science Foundation through grant 1720625. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Where indicated, molecular graphics and analyses performed with UCSF Chimera, developed by the Resource for Biocomputing, Visualization, and Informatics at the University of California, San Francisco, with support from NIH P41-GM103311. We thank the IU Electron Microscopy Center, the Nanoscale Characterization Facility, and the Physical Biochemistry Instrumentation Facility for access to their instrumentation.

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E. S. and T. D. developed the methodology for probing the porosity of the virus-like particles. E. S. prepared the protein constructs, synthesized the NADH-dendrimer conjugate, and analyzed and collected all data pertaining to the two. Simulation data using COOT and HOLE were computed and provided by B. L. Theoretical data of the electrostatic surface were collected and analyzed by E. S. T. D. conceived and oversaw the project. E. S. and T. D. wrote the manuscript.

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Correspondence to Trevor Douglas.

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Selivanovitch, E., LaFrance, B. & Douglas, T. Molecular exclusion limits for diffusion across a porous capsid. Nat Commun 12, 2903 (2021). https://doi.org/10.1038/s41467-021-23200-1

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