Designing Membrane-Reshaping Nanostructures Through Artificial

bioRxiv preprint doi: https://doi.org/10.1101/2020.02.27.968149; this version posted February 28, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Designing membrane-reshaping nanostructures through artiﬁcial evolution

Joel C. Forster1,2, Johannes Krausser1,2, Manish R. Vuyyuru1, Buzz Baum1,2, and Andela¯ Šaric´ 1,2

1Institute for the Physics of Living Systems, University College London, Gower Street, London, WC1E 6BT, United Kingdom 2MRC Laboratory for Molecular Cell Biology and Department of Cell and Developmental Biology, University College London, Gower Street, London, WC1E 6BT, United Kingdom

In this paper we combine the rules of natural evolution with brane. In this way we determine the optimal ligand pattern molecular dynamics simulations to design a nanostructure with for this task. We show that this approach identified non-trivial a desired function. We apply this scheme to the case of a ligand- nanoparticle designs, whose kinetic and thermodynamics fea- covered nanoparticle and evolve ligand patterns that promote tures we have analysed to determine how they drive efficient efficient cell uptake. Surprisingly, we find that in the regime of particle internalisation. low ligand number the fittest structures are characterised by ligands arranged into long one-dimensional chains that pattern the surface of the particle. We show that these chains of ligands pro- vide particles with high rotational freedom and they lower the Simulation Model. The nanoparticle is modelled as a rigid free energy barrier for membrane crossing. This demonstrates body, made up of a central particle that carries ligands on its the efficacy of artificial evolution to identify non-intuitive design surface to facilitate binding to a fluid membrane, as shown in rules and reveals a new principle of design that can be used to Fig. 1 (b). To curve the membrane and form a vesicle (1, 2) inform artificial nanoparticle construction and the search for a minimal interaction strength between the ligands and the inhibitors of viral entry. membrane is required. Importantly, for a given interaction strength between ligands and the membrane, the efficiency Molecular simulations | Artificial intelligence | Biophysics | Nanotechnology of membrane crossing depends on the arrangement of lig- Correspondence: [email protected] ands on the nanoparticle. For instance, in the limiting case The mechanistic rules that govern the function of the where all the ligands are clustered at one pole, the nanopar- nanoscale machinery of life remain poorly understood. Our ticle binds strongly to the membrane, but cannot be wrapped intuition for how nanostructures in nature operate is guided by the membrane and internalised. by experimental observation interpreted through mechanistic models. Such a strategy rests on the researchers’ ability to To account for different ligand arrangements the nanoparticle navigate the large phase space of possible model ingredients was fully covered by 72 ligand sites, while N of those are ac- and parameters to capture the key physics behind complex tive sites, which are able to bind to the membrane beads via biological phenomena. a generic Lennard-Jones potential of a depth , see Fig. 1 (b) Here, we set out to take a reverse approach: instead of deduc- and Methods. The membrane was modelled using a single ing the design rules of nanostructures by observing nature, particle-thick model (3), which reproduces the correct me- we specify the function that the nanostructure should perform chanical properties of biological membranes and is capable and use a computer model to evolve its design. To this effect of fusion and fission. Simulations were run using Langevin we couple the principles of biological evolution to molecu- dynamics within the LAMMPS molecular dynamics pack- lar dynamics (MD) simulations. The computational protocol age (4). For more details see Methods. starts with a random population of nanostructure designs and measures how well the individual structures perform in MD simulations, as shown in the schematic in Fig. 1 (a) . We Evolutionary Algorithm. To explore efficient nanoparticle then use an evolutionary algorithm (GA) to mutate and breed designs, we kept the total number of active ligands and their the fittest members of the population and repeat the process binding strength to the membrane constant throughout evolu- to evolve better designs. The MD-GA procedure is iterated tion. The ligand design was turned into a 1D single bit arrays, until the nanostructures generated by rounds of mutation and as illustrated in Fig. 1 (b), which enabled the following ge- selection can perform the task efficiently. By observing the netic algorithm operations: tournament selection, two point structures as their performance evolves, we can identify the crossover, and shuffling mutation. To avoid premature fixa- key features needed for a desired function. This reverse ap- tion, the population was split into independent "demes" and proach can be used to identify engineering principles that we permitted to trade individuals every generation (5, 6). would not have a priori postulated based purely on intuition. As a measurement of the "fitness" of the membrane-crossing Thus it represents an orthogonal way of exploring the physi- nanoparticle we used the degree of wrapping by the mem- cal mechanisms of nanostructure design. brane. Budding is typically achieved after a particle binds, As a proof of principle, we apply the framework to study become wrapped by membrane, and detaches from the the ability of a ligand-covered nanoparticle to cross a mem- mother membrane. The fitness function used to analyse the

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sshown as , Evolution bioRxiv preprint doi: https://doi.org/10.1101/2020.02.27.968149; this version posted February 28, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

(a) (b)

Fig. 2. Performance of the evolved nanoparticles. (a) The mean population fitness < F >, normalised by the mean fitness of a random population of particles with the same and N, < Fr >. These curves measure how much the particle is wrapped by the membrane and how quickly it crosses it (Eq. (1)) , shown here for three examples of the active ligand number N and ligand-membrane interaction strength . The fitness value increases as the evolution proceeds for all populations. (b) The fraction of the entire population that successfully crossed the membrane at various values of the active of ligands N and ligand-membrane binding energy . In the region around N ≈ 200kT (marked by black boxes) the population is made up of a mixed collection of budding and non-budding particles; here the nanoparticle design plays a crucial role. rangement for efficient particle budding. common nodes and indicates the spatial connectivity of ligands across the whole nanoparticle; in general, the lower the Structural Characterisation. Several representative exam- number of subgraphs the higher the connectivity for a fixed ples of fit and unfit design are given in Fig. 3 (a). The dif- number of vertices. The population averages of these prop- ference between good and poorly performing particles is vis- erties were taken across constant values of N to produce the ible by eye, with the fit particles showing linearly conencted mean density D and subgraph number S of designs at con- ligands, while poorely performing particles have ligands dis- stant N. The same can be calculated for random particle de- connected from each other or well clustered. signs at constant N (which have not undergone evolution), To characterise nanoparticle designs in an unbiased way, we giving the mean density D and mean subgraph number S . needed a quantitative measure to asses their structure. To this r r end, we translated the ligand positions on the surface of the particle into nodes of a network. The weights of the edges Identifying successful designs. The average normalised between the nodes are set to the inverse of the great arc dis- graph density D/Dr and subgraph number S/Sr for both tance between the ligands, as represented in Fig. 3 (b). This the successful and unsuccessful population are shown in method of graph construction maps the surface of the par- Fig.3 (c) and (d), where Dr and Sr are the correspond- ticles in a rotationally invariant way. Since the closer lig- ing quantities for a uniformly random graph. The insets in ands are to one another, the greater the weight of the edge Fig.3 (c) and (d) illustrate a representative particles from connecting them, the network can be thought of as encoding each respective population. the likelihood of cooperative binding between nodes. Once The main structural difference between the successful and networks were constructed, they were pruned by removing non-successful designs are best visible for low and interme- edges which were above a threshold of 3.3/σ0, σ0 being the diate ligand numbers, in our case below N = 35, where the MD unit length, which effectively retains only nearest and number of possible variations in design is the largest. As next-to-nearest (second order) neighbours. shown in Fig. 3 (c), successful designs tend to have lower We found that two topological graph properties act as good graph densities, meaning that the ligands have a lower aver- structural descriptors that distinguish between successful and age number of neighbours at given number of active ligands unsuccessful designs: the graph density D and the number N. In contrast, Fig. 3 (d) clearly indicates that such parti- of disconnected subgraphs S. The graph density of particle cles also have lower average subgraph numbers. Therefore 2|Ei| hzii i di is defined as d = = , where |Ei| and |Vi|(|Vi|−1) |Vi|−1 ligands in successful particles tend to be connected with one |Vi| are the number of the particle network’s edges and ver- another by only few edges, covering a large angular range tices, respectively, and hzii is the average degree of all nodes across the nanoparticle surface. Taken together, this indicates in the network. The graph density represents the ratio of the that successful designs are characterised by patterns in which number of edges to the number of possible edges in an equiv- each ligand is connected to all other ligands via chains, like alent complete graph, and can be understood as a measure of those visible in Fig.3 (a) and in the inset in Fig.3 (c). Con- how clustered and connected the points in the network are: versely, isolated "patches" of ligands perform very poorly a low network density corresponds to one with large ligand (Fig.3 (a)). Interestingly, designs that have ligands uniformly spacing, while a high density represents greater ligand clus- distributed across the nanoparticle also show poor perfor- tering. The number of disconnected subgraphs in a particle mance, with budding times on average 30% longer than that i, si, is number of separate regions of a graph that share no of the evolved succesful designs.

Forster et al. | Designing membrane-reshaping nanostructures through artiﬁcial evolution bioRχiv | 3 bioRxiv preprint doi: https://doi.org/10.1101/2020.02.27.968149; this version posted February 28, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

(a) (b) Structure Characterisation

A A

Good B Designs δAB B

1 Poor w= AB

Designs δ (c) (d)

Fig. 3. Structural analysis of the evolved particles. (a) Examples of evolved nanoparticle designs that exhibit good and poor performances. (b) The patterns of ligands on the particle are converted into a network, where the network edge weights are the reciprocal of the great arc distance between two ligands δ. This is repeated for all pairs of ligands with a distance δ < 3.3σ. Such a network is here visualised by a Fruchterman-Reingold projection(7). (c) The average normalised graph density for the budding and non-budding particles across the whole population. The insets show sample particles in each regime. (d) The average normalised subgraph size of budding and non-budding populations. The inset networks are two samples of budding and non-budding particles from the N = 23, = 10.0 kT dataset, each with 3 subgraphs; the differences in size distribution and the average connectivity of the subgraphs is visible to the eye.

Explaining successful designs. In searching for a physi- a particle buds, as opposed to for its full run length trun, the cal explanation as to why the long chains of ligands exhibit difference between the budding and non-budding population superior membrane-crossing properties we noticed that un- remains. successful designs, in which the ligands are clustered into This difference in the rotational mobility of budding and non- disconnected patches, end up deforming the membrane but budding particles implies that the purpose of the chains of never becoming fully wrapped by it. ligands is to minimise the free energy barrier required for the To better quantify this behaviour we measured the average ro- particle wrapping. Fig. 4 (b) shows the free energy profile 2 tational mean squared displacement ∆θ2 = h[θ (τ) − θ (0)] i for budding for an example of a fit and unfit design, which as particles meet the membrane, bind to it, and deform it carry the same total adhesion energy, computed using um- across the population of budding and non-budding particles. brella sampling with the degree of membrane wrapping as θ (τ) is the angle at time τ between a predefined nanoparti- the reaction coordinate. Interestingly, the free energy mini- cle axis and a predefined vector in the simulation box. The mum for the unfit particle is positioned at ∼ 75% membrane time evolution of the rotational displacement is presented in wrapping and is separated from the full wrapping by a large Fig.4 (a). While at the beginning of the simulation the ro- barrier. The free energy profile for the fit particle on the other tational displacement does not differ much between budding hand reaches the global free energy minimum at full wrap- and non-budding particles, a significant discrepancy appears ping, without encountering any significant energy barriers en as the simulation progresses. Prior to budding, successful route. As this makes clear, evolution selects for designs that particles exhibit much larger rotational freedom compared to exhibit the lowest free energy barrier for membrane wrap- the non-budding particles. This rotational freedom enables ping. particles to explore the transitional states needed to be able to wrap themselves in the membrane. Indeed, the frequency Discussion and conclusions. By combining molecular dy- of budding events shown in Fig. 4 (a) illustrates the relative namics simulations with genetic algorithms, we have been increase in the rotational freedom of budding particles with able to evolve nanostructures optimised for a specific func- respect to non budding particles correlates and how this cor- tion, selected for the ability to bud across the cell membrane. relates with the proportion of the population that have budded Even in this seemingly simple case, this approach revealed at various times. As a control, we checked that if the rota- novel design rules that would not have been easy to guess, tional displacement is measured only until the point at which even for a well trained modeller.

4 | bioRχiv Forster et al. | Designing membrane-reshaping nanostructures through artificial evolution (a) Forster spirit in closest The realisation. im- experimental crucial of an be in will portance that effects mini- kinetic energy the free neglecting the mum, identifying on structure. based crystal was approach colloidal This target a on for algorithm sequences colloids evolutionary reverse-design DNA-grafted to an calculations combined energy (18) free with nanoscale, the al. At et for locomotion. algo- Srinivasan optimise desired evolutionary a a to of environment used At design engine (17) organisim physical function. a ap- specific al within combined et a rithm a Kriegman for such structure scale, used larger a studies evolve parame- few to interaction Only proach of 16). optimisation (15, ters or min- energy (12–14) of purpose imisation the for mainly within simulations strategies computer evolutionary incorporated studies Previous (9–11). (8). viruses arrangements particles some chain linear the on display envelop proteins that membrane-binding lines Analogously, long into pro- arranged low-density membrane-binding tein instance, carry to For explain appear possibly particles nature. lipoprotein as in well found as patterns delivery, artificial ligands of cell design that the for aid believe can nanoparticles factor, therefore here key We identified rules the design is the thermodynamics. crossing its membrane than of ap- rather practical rate any the for (N, Indeed, plication, explored energy. adhesion the purely total all same based since the is kinetics, structures the the on of performance superior The number limited the cycles. to evolutionary due of a be have instead patches may occasional this lig- such role, act- that of functional possible patches chains is small it ligand with While together ands. these found reliable. of often more were wrapping think Chains membrane can making zippers, One like ing gaps. leav- particle rarely few the chains cover ing effectively the which where but branch, ligands, or char- of cross are chains designs en- long Such free by low crossing. acterised a membrane have for barrier that ergy designs nanoparticle the scheme MD-GA identified the numbers, ligand low and from from intermediate particles At particle sample one for on wrapping membrane ligand a one of for function wrapping. trajectory a full rotational as the at energy sample minimum at Free energy minimum a (b) free energy shows events. free inset budding the The the reaches of ( populations. particle count populations averaged the non-budding ensemble shows and the panel budding top for the the mean while the subpopulation, of respective error each standard the represent bars designs. successful Explaining 4. Fig. (which wasnotcertifiedbypeerreview)istheauthor/funder,whohasgrantedbioRxivalicensetodisplaypreprintinperpetuity.Itmade bioRxiv preprint Rotational MSD tal. et einn ebaersaignnsrcue hog rica evolution artificial through nanostructures membrane-reshaping Designing | doi: https://doi.org/10.1101/2020.02.27.968149 a h oainlma qaedslcmn ftebdigadnnbdigpplto,sonfor shown population, non-budding and budding the of displacement square mean rotational The (a) 11kT = ≈ available undera 75% , N ebaewapn hti eaae rmfl rpigb ag are,wietebdigpril ece the reaches particle budding the while barrier, large a by wrapping full from separated is that wrapping membrane 22 = ein carry designs ) nbt ae) h nesso ahpril tterrpeettv ebaewapns h non-budding the wrappings: membrane representative their at particle each show insets The cases). both in

CC-BY-NC-ND 4.0Internationallicense 35% ; this versionpostedFebruary28,2020. (b) neatosbtenteatv atcelgnsadmem- and ligands particle active the between The interactions (4). LAMMPS package dynamics molecular source open Model. Simulation Methods EP- by funded for partially Hub (EP/P020194/1). is SRC Modelling Society which Molecular Royal resources, and the computational Materials AS), UK and the (JK NEPA (AS), grant ERC and the (BB MRC AS), (JCF), EPSRC from support acknowledge We and biological of systems. context engineered other the in and and both lattices, nanostructures structures, iden- filaments, higher-order range in protein a help as such of great nanomachines, principles of design be the can tifying here of developed advantage that approach envision main We the the here. is presented approach pos- way, computational of unbiased the phase-space an large in Indeed, a designs, explore sible consider. efficiently not to did ability studies the previous superior, that novel, rules identified designs design have and possible designs, of unique space possible phase small stud- a (∼ nanoparticle the only of the explored each distribution( of 23), ies ligand importance and nanopar- (20–22) the of shape subject identified the on uptake work ticle of body previous the While the nanostruc- operate. in functional naturally promise where tures this fractions, higher volume that far low show of a regime we pose Here might approach scheme. combined combined the ap- the of limits simula- plication somewhat computer which expensive, where relatively are fractions, tions packing on focused high nanoscale at the at systems studies MD-GA previous the All mechanical desired a response. of packing granular particle produces a that (19), evolve shape to al. simulations et dynamics strat- Miskin molecular evolution with by egy adaptation study matrix the covariance is combined here who presented one the to .Hr eefcieyepoe eso huad of thousands of tens explored effectively we Here 10). iuain eecridotuigthe using out carried were Simulations . The copyrightholderforthispreprint 11 = kT , N 22 = v|5 | bioRχiv h error The . bioRxiv preprint doi: https://doi.org/10.1101/2020.02.27.968149; this version posted February 28, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. brane beads were modelled using a cut-and-shifted Lennard- Parameters for the evolutionary algorithm were chosen to Jones potential, which takes the following form for the inter- discourage premature fixation (the entire population becom- particle distances below the cut-off rc = 1.8σ0 (σ0 being the ing identical). The probability of crossover Pcross was set to MD unit of lengths) and is zero otherwise: 0.5, probability of mutation Pmut was set to 0.2 the independent probability of a genome bit being selected for swapping P was set to 0.15. Any gaps left after crossover and mutind σ 12 σ 6 mutation are filled with clones of selected individuals. A tab- VLJ(r) = 4 − + 0.114. (2) r r ulated form of these parameters is included in Table S1.

Here is the depth of the potential, representing the effec- Analysis. For the structural analysis of the ligand arrange- tive ligand-membrane interaction strength, and σ = σ0 is the ments, graph edge weights were encoded using the reciprocal contact distance between a ligand and a membrane bead. of the great arc distance between each pair of ligands on the The discretised positions of the ligands on the particle surface surface. Weight cutoff for graph pruning was chosen to in- are set by an approximate sphere packing solution(24). The clude (on average) first and second order nearest neighbours. membrane was modelled using the single particle thick model Rotational mean square displacement (Rotational MSD) was developed by Yuan et al. (3) which is capable of fission and calculated by taking the original position of an arbitrary lig- fusion events, with membrane parameters chosen to encode and at timestep 0 and tracking the position of that same ligand for a flat membrane with a bending rigidity of 15kT . Follow- over the course of the simulation run. ing the notation from the original paper (3), we chose the pa- bead−bead Free energy calculation To obtain the free energy profile rameters bead−bead = 4.34kBT,ξ = 4,µ = 3,r = cut as a function of the degree of nanoparticle wrapping um- 1.12σ, where σ = σ0 is also the membrane bead diameter. Simulations were run using a Langevin thermostat (with tem- brella simulations were carried out using the colvars module perature set to 1.0 and damping parameter set to 1.0) within of LAMMPS. The degree of wrapping γwrap is defined through the co- the npH ensemble for trun = 25000 time steps, each of a size ordination number between the central carrier particle and 0.01τ0, τ0 being the MD unit of time. The membrane con- the 2900 membrane particles. Using the reaction coor- tained 2900 particles with initial in plane dimensions 50σ0 by dinate γwrap the simulation trajectories are biased by the 50σ0, and a 200σ0 high box. The membrane centre of mass kγ 2 was tethered to the centre of the box by a spring. Particle harmonic potential Vbias(γwrap,kγ) = 2 γwrap − γwrap,0 , coverage and budding was detected using a spatial clustering where γwrap,0 denotes the position of the umbrella window algorithm in LAMMPS, with the time point at which a signif- and kγ the force constant of the biasing potential. icantly sized (> 30) group of membrane particles break away During umbrella simulations 60 consecutive biasing win- from the membrane bulk (along with the nanoparticle) being dows are set by choosing γwrap,0 in the interval [50,...450] recorded as the time of budding. with kγ = 0.5kT . In each window the centre-of-mass position of the nanoparticle is sampled for 5000 time steps at a Particle Evolution. The genetic algorithm was built in- frequency 0.01 and each full simulation is repeated 200 times house using the DEAP python library(25) and used as a wrap- to obtain statistics. The resulting histograms for the individ- per for LAMMPS simulations. The fitness function used in ual windows are combined subsequently using the weighted the genetic algorithm was chosen to drive greater particle histogram analysis method. coverage and drive faster budding (Eq. (1)). The rewards were set to reflect this, with the reward for budding Rb = 100, for coverage Rc = 1, and for budding time Rt = 10. Unique Acknowledgements particle designs were tested 4 times under different initial ro- We acknowledge support from EPSRC (JCF), MRC (BB and tational positions. AS), the ERC grant NEPA (JK and AS), the Royal Society Each ,N pair had its own evolutionary algorithm instance (AS), the UK Materials and Molecular Modelling Hub for which was split into 4 subpopulations (demes) each with 20 computational resources, which is partially funded by EP- individuals which ran independently in parallel during each SRC (EP/P020194/1). generation until all individuals within them were assigned fitness scores and crossover mutation and cloning had occurred. At this point a single individual was chosen at random to mi- Bibliography grate to another deme and replace the lowest fitness parti- 1. Huajian Gao, Wendong Shi, and Lambert B Freund. Mechanics of receptor-mediated en- cle in that population (with the probability of being selected docytosis. page 6. 2. Tine Curk, Peter Wirnsberger, Jure Dobnikar, Daan Frenkel, and Andela¯ Šaric.´ Controlling determined by its fitness normalised by the sum of all fit- cargo trafficking in multicomponent membranes. Nano Letters, 18(9):5350–5356, 2018. doi: nesses in the deme) and the next generation proceeded in the 10.1021/acs.nanolett.8b00786. PMID: 29667410. 3. Hongyan Yuan, Changjin Huang, Ju Li, George Lykotrafitis, and Sulin Zhang. One-particle- same fashion. Selection tournaments were performed on 3 thick, solvent-free, coarse-grained model for biological and biomimetic fluid membranes. randomly selected particles, with the top particle in the tour- Phys. Rev. E, 82:011905, Jul 2010. doi: 10.1103/PhysRevE.82.011905. 4. Steve Plimpton, Aidan Thompson, Stan Moore, and Axel Kohlmeyer. Lammps documenta- nament being allowed to survive and tournaments repeated tion, May 2017. until the population has been exhausted. Each instance of the 5. Michael Affenzeller, Stephan Winkler, Stefan Wagner, and Andreas Beham. Genetic Algo- rithms and Genetic Programming: Modern Concepts and Practical Applications. Chapman algorithm was run for a total of 35 generations. & Hall/CRC, 1st edition, 2009. ISBN 1584886293, 9781584886297.

6 | bioRχiv Forster et al. | Designing membrane-reshaping nanostructures through artiﬁcial evolution bioRxiv preprint doi: https://doi.org/10.1101/2020.02.27.968149; this version posted February 28, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

6. Gerhard Goos, Juris Hartmanis, Jan van Leeuwen, David Hutchison, Takeo Kanade, Josef Kittler, Jon M Kleinberg, Friedemann Mattern, John C Mitchell, Moni Naor, and et al. Lecture notes in computer science. page 960. 7. Thomas M. J. Fruchterman and Edward M. Reingold. Graph drawing by force-directed placement. Software: Practice and Experience, 21(11):1129–1164, Nov 1991. ISSN 00380644, 1097024X. doi: 10.1002/spe.4380211102. 8. Gang Ren, Gabby Rudenko, Steven J Ludtke, Johann Deisenhofer, Wah Chiu, and Henry J Pownall. Model of human low-density lipoprotein and bound receptor based on cryoem. Proceedings of the National Academy of Sciences, 107(3):1059–1064, 2010. 9. Xiangxi Wang, Jingshan Ren, Qiang Gao, Zhongyu Hu, Yao Sun, Xuemei Li, David J Row- lands, Weidong Yin, Junzhi Wang, David I Stuart, et al. Hepatitis a virus and the origins of picornaviruses. Nature, 517(7532):85, 2015. 10. Jodi A Hadden, Juan R Perilla, Christopher John Schlicksup, Balasubramanian Venkatakr- ishnan, Adam Zlotnick, and Klaus Schulten. All-atom molecular dynamics of the hbv capsid reveals insights into biological function and cryo-em resolution limits. Elife, 7:e32478, 2018. 11. Bärbel Kaufmann, Alan A Simpson, and Michael G Rossmann. The structure of human parvovirus b19. Proceedings of the National Academy of Sciences, 101(32):11628–11633, 2004. 12. Laura Filion and Marjolein Dijkstra. Prediction of binary hard-sphere crystal structures. Physical Review E, 79(4):046714, 2009. 13. Gernot J Pauschenwein and Gerhard Kahl. Clusters, columns, and lamellae—minimum energy configurations in core softened potentials. Soft Matter, 4(7):1396–1399, 2008. 14. Jure Dobnikar, J Fornleitner, and G Kahl. Ground states of model core-softened colloids. Journal of Physics: Condensed Matter, 20(49):494220, 2008. 15. Kathrin Muller, Natan Osterman, Dusan Babic, Christos N Likos, Jure Dobnikar, and Arash Nikoubashman. Pattern formation and coarse-graining in two-dimensional colloids driven by multiaxial magnetic fields. Langmuir, 30(18):5088–5096, 2014. 16. Jian Qin, Gurdaman S Khaira, Yongrui Su, Grant P Garner, Marc Miskin, Heinrich M Jaeger, and Juan J de Pablo. Evolutionary pattern design for copolymer directed self-assembly. Soft Matter, 9(48):11467–11472, 2013. 17. Sam Kriegman, Douglas Blackiston, Michael Levin, and Josh Bongard. A scalable pipeline for designing reconfigurable organisms. Proceedings of the National Academy of Sciences, 2020. 18. Babji Srinivasan, Thi Vo, Yugang Zhang, Oleg Gang, Sanat Kumar, and Venkat Venkata- subramanian. Designing dna-grafted particles that self-assemble into desired crystalline structures using the genetic algorithm. Proceedings of the National Academy of Sciences, 110(46):18431–18435, 2013. 19. Marc Z Miskin and Heinrich M Jaeger. Adapting granular materials through artificial evolution. Nature materials, 12(4):326, 2013. 20. Robert Vácha, Francisco J. Martinez-Veracoechea, and Daan Frenkel. Receptor-mediated endocytosis of nanoparticles of various shapes. Nano Letters, 11(12):5391–5395, Dec 2011. ISSN 1530-6984, 1530-6992. doi: 10.1021/nl2030213. 21. Kai Xiong, Jiayin Zhao, Daowen Yang, Qingwen Cheng, Jiuling Wang, and Hongbing Ji. Co- operative wrapping of nanoparticles of various sizes and shapes by lipid membranes. Soft Matter, 13(26):4644–4652, 2017. ISSN 1744-683X, 1744-6848. doi: 10.1039/C7SM00345E. 22. Drew R. Elias, Andrei Poloukhtine, Vladimir Popik, and Andrew Tsourkas. Effect of lig- and density, receptor density, and nanoparticle size on cell targeting. Nanomedicine: Nanotechnology, Biology and Medicine, 9(2):194–201, Feb 2013. ISSN 15499634. doi: 10.1016/j.nano.2012.05.015. 23. Veronika Schubertová, Francisco J. Martinez-Veracoechea, and Robert Vácha. Influence of ligand distribution on uptake efficiency. Soft Matter, 11(14):2726–2730, 2015. ISSN 1744-683X, 1744-6848. doi: 10.1039/C4SM02815E. 24. N. Sloane. Tables of sphere packings and spherical codes. IEEE Transactions on Informa- tion Theory, 27(3):327–338, May 1981. ISSN 0018-9448. doi: 10.1109/TIT.1981.1056351. 25. Félix-Antoine Fortin, François-Michel De Rainville, Marc-André Gardner, Marc Parizeau, and Christian Gagné. DEAP: Evolutionary algorithms made easy. Journal of Machine Learning Research, 13:2171–2175, jul 2012.

Forster et al. | Designing membrane-reshaping nanostructures through artiﬁcial evolution bioRχiv | 7 bioRxiv preprint doi: https://doi.org/10.1101/2020.02.27.968149; this version posted February 28, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Supplementary Note 1: Genetic algorithm

start

i=0

Generate Initial Populations

Transform population to Set new population Population Generation

Particle Model protein classes equal to offspring

Transform proteins to Mutate offspring LAMMPS input scripts (Pmut)

True False Run LAMMPS n == nmax

Clone offspring

Coarse GrainSimulation Read LAMMPS xyz Mate offspring output data individuals (Pcross)

Measure membrane Tournament selection deformation, determine of ﬁttest individuals, ﬁtness copy to offspring

i++

i == G False

True False Moran Migration

Output ﬁnal data i%M == 0 Migration True

stop

Fig. 5. A program ﬂowchart of the key steps of the genetic algorithm implemented in the study. Here i represents the current generation number and is iterated every time the loop is run. G is the number of generations that the GA is allowed to run for, in this study G = 35. M is the frequency of migration, in this study M = 1, so a migration took place every generation as described in the text. n is the population size and nmax is the target population size, in this case nmax = 20 for each deme. Pcross and Pmut are the probabilities of crossover and mutation occuring, as described in the main text. Here an A == B represents a test for equivalence between A and B, and "A%B" is the modulus operator which ﬁnds the whole number remainder of A divided by B.

8 | bioRχiv Forster et al. | Designing membrane-reshaping nanostructures through artiﬁcial evolution bioRxiv preprint doi: https://doi.org/10.1101/2020.02.27.968149; this version posted February 28, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Parameter Value Rb 100 Rt 10 Rc 1 Pcross 0.5 Pmut 0.2 Pmutind 0.15 G 35 ndemes 4 nmax 20 M 1 ntourn 3 trun 25000 R 4

Table 1. Parameters of the evolutionary algorithm and the simulations run as part of it. Rb, Rt and Rc are the rewards for budding, speed of entry and coverage respectively. These are used in the calculation of ﬁtness seen in equation 1 in the main text. Pcross and Pmut are the independent probabilities that an individual will be selected for crossover or mutation. If an individual is selected for crossover a different individual is selected at random with uniform probability from the rest of the deme population. Pmutind is the independent probability that each bit will be chosen to be swapped with another in the genome of an individual chosen for mutation, for each bit chosen a new position is chosen at random from the rest of the bits in the genome with uniform probability. Ngen is the number of generations, ndemes is the number of separated subpopulations, with nmax being the number of individuals in each deme. The total population of the entire system is ndemes × nmax, for a total of 80 in this case. M is the number of individuals chosen for migration at the end of each generation. ntour is the number of individuals selected for each round of tournament selection. trun is the total runtime of each simulation, and R is the number of randomly oriented repeat. simulations run for each new individual.

Forster et al. | Designing membrane-reshaping nanostructures through artiﬁcial evolution bioRχiv | 9