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Letter

Random Number Generation by aTwo-Dimensional Crystal of Protein Molecules

Yasuhiro Ikezoe, Song-Ju Kim, Ichiro Yamashita, and Masahiko HaraLangmuir, Article ASAP • Publication Date (Web): 10 March 2009

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Random Number Generation by a Two-Dimensional Crystalof Protein Molecules

Yasuhiro Ikezoe,*,† Song-Ju Kim,‡ Ichiro Yamashita,§, ) and Masahiko Hara†,‡,^

†Flucto-Order Functions Asian Collaboration Team, RIKEN Advanced Science Institute, 2-1 Hirosawa,Wako, Saitama 351-0198, Japan, ‡Flucto-Order Functions Asian Collaboration Team, RIKEN AdvancedScience Institute, Fusion Technology Center, Hanyang University, 17 Haengdang-dong, Seongdong-gu,

Seoul 133-791, Korea, §Advanced Technology Research Laboratories, Panasonic Corporation, 3-4Hikaridai,Seika-cho, Soraku-gun, Kyoto 619-0237, Japan, )Graduate School of Materials Science,

Nara Institute of Science and Technology, 8916-5 Takayama, Ikoma, Nara 630-0192, Japan and^Department of Electronic Chemistry, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku,

Yokohama 226-8502, Japan

Received January 6, 2009. Revised Manuscript Received February 26, 2009

Wediscuss 2D and binary self-assemblies of proteinmolecules using apo-ferritin and holo-ferritin, which haveidentical outer-shell structures but different inner structures. The assemblies do not show any phase separationbut form 2D monomolecular-layer crystals. Statistical analyses showed a random molecular distribution in thecrystal where the molar ratio was conserved as it was in the solution. This molecular pattern is readily prepared,but it is neither reproducible nor predictable and hence can be used as a nanometer-scale cryptographic device oran identification tag.

Introduction

Random number sequences play key roles in our daily livesin a highly complicated telecommunication environment andare an important element in computer simulations in thenatural sciences, mathematics, economics, and so on. Toensure security in the information technology or the validityof simulation results, it is important to use an unpredictableand aperiodic random number sequence. Many kinds ofmethods have been known to obtain such sequences. Someare based on physical phenomena such as radiation,1 lasers,2,3

hydrodynamics,4 and quantum effects.5 Others are based onmathematical algorithms such as the linear congruentialalgorithm,6 theBlum-Blum-Shubalgorithmbased onprimefactorization,7 the Mersenne twister algorithm based on themaximumlength sequence,8 the cellular automaton,9,10 and soon. Although the methods based on physical phenomenashould be random in principle, unfortunately the observationprocesses or instruments adopted often accompany nonran-dom factors. However, the methods based on algorithmsimplemented on digital computers produce intrinsically de-terministic and periodic sequences. Therefore, these randomnumber sequences are called pseudorandom numbers.

Here we report a new random number generator based onthe self-assembly of molecules. The molecule used here isferritin, a protein molecule that has a quasi-spherical hollowshell structure (12nmouter diameter and7nm inner diameter)composed of 24 subunits (∼450 kDa) and is capable of storingboth iron hydroxide in vivo and a variety of inorganiccompounds, such as magnetic materials11-13 or semiconduc-tors,14,15 in vitro. This molecule is also expected to contributeto diverse applications such as electronic devices,16,17 cata-lysts,18-20 and diagnostics.13 Ferritin is roughly classified intotwo species: apo-ferritin (AF), without any core materials inthe molecular cage, and holo-ferritin (HF), which stores aninorganic nanoparticle inside. Except for the inner difference,these molecules have exactly identical chemical structures.Now we consider the difference in the intermolecular interac-tion between these molecules. The presence of the nanoparti-cle in HF would induce the conformation change and electricfield modification within the protein shell. However, such adiscrepancy between HF and AF should be relaxed along theoutward direction in the thick protein shell. Therefore, it

*Towhom correspondence should be addressed. Phone:+81-48-462-4421. Fax: +81-48-462-4695. E-mail: y.ikezoe@riken.jp.

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© XXXX American Chemical Society

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would mostly disappear at the molecular surface. The inter-molecular interaction is governed mainly by the outermostsurface of the molecules, and thus there would be a smalldiscrepancy in the interaction between HF and AF. In thisletter,we show some2Dcrystals of ferritinmolecules obtainedfrom solution including both HF and AF molecules in acertain ratio. Our experiments revealed no phase separations.However, we obtained 2D monomolecular-layer crystalswhere the molar ratio was conserved as it was in the solution.In addition, statistical analyses found the microscopic mole-cular distribution to be random. Especially for crystals ob-tained from solution including equimolar AF and HF, weobtained a uniform random number sequence by indexing 0and 1 for AF and HF, respectively. We know of manyexcellent reports on the binary self-assemblies of nanometer-scale or micrometer-scale particles in 3D or 2D in which theordered structures, superlattices, or phase separations havebeen well discussed.21-30 However, we first discuss the ran-domness in the ordered structure in the binary assembly ofnanoparticles, and we show a fundamental Monte Carlosimulation utilizing the random molecular distribution.

Experimental Section

In our experiments, we used a genetically synthesized ferritincalled N1-LF.31 The HF molecule with an indium oxide nano-particle was obtained from the AF molecules using a methodpreviously reported (Supporting Information S1).32 N1-LF has amore hydrophobic surface than the native ferritin, which facil-itates crystallization as previously reported.33 The crystallizationprocedure is summarized as follows. A 300 μL droplet of anaqueous solution ofN1-LF (2 μg/mL) is formed on a hydrophilicSiO2 surface surrounded by a hydrophobic surface and then isslowly dried. The adsorbed molecules on the water surface areconcentrated in the vicinity of the perimeter of the droplet byoutward convective flow during the drying process, and even-tually the 2Dcrystal grows at thewater surface perpendicularly tothe contact line. After the water is entirely evaporated, the 2Dcrystal is left on the SiO2 surface (Supporting Information S2).The random molecular mixture in the 2D crystal cannot berealized unless the molecular distribution in the solution or atthe water surface is also random. Therefore, for AF and HFmolecules to be mixed sufficiently evenly on the microscopic

scale, the solution was stirred with a magnetic stirrer for morethan 30 min. Under some unfavorable conditions for crystal-lization such as the fast evaporationofwater, only a disorderedor3D aggregation of molecules was observed, in which we can nolonger discuss themolecular distribution.However, whenever thecrystal was formed, we obtained the random molecular distribu-tion shown in the following section. The crystal obtained wasrobust, and its structure has not been broken for a few weeks aslong as it was kept in a dry environment. However, when thecrystal was exposed to a humid atmosphere, dew drops readilydestroyed the crystal because it was fixed by physical adsorptiononto the Si substrate, not by covalent bonds. It has been knownthat some thermal processes, for example, heating the 2D crystalto remove the protein shell,34 are effective at fixing inorganicnanoparticles directly onto a Si substrate. The details of thecrystallization conditions are described in our previous paper.33

Figure 1 shows a scanning electron microscope (SEM) image ofthe 2D molecular crystal obtained from the solution, includingequimolar HF and AFmolecules. The scale bar is 50 nm, and thewhite spots are indium oxide nanoparticles in the HFs; one whitespot corresponds tooneHFmolecule. The fastFourier transform(FFT) analysis of this image, as shown in the lower left, reveals a6-fold-symmetrical pattern with a lattice constant of approxi-mately 12nm that is equal to themolecular size. ThisFFTpatternis similar to that of the hexagonally close-packed (HCP) mole-cular crystal obtained from the pureHF solution,33 andhence it isthought that the HF molecules in Figure 1 are also on the latticepoints of the HCP molecular crystal. Interestingly, the FFTpattern showed no other patterns except for the above 6-foldone, which indicates that the crystal has no superlattice struc-tures. However, many dark lattice points were simultaneouslyfound in this image. We can surmise that these dark points areoccupied by the AF molecules because the AF molecules areinvisible in the SEM observation as a result of the absence ofinorganic elements. To verify this hypothesis, we investigatedother crystals obtained from the solution including HF and AFmolecules in amolar ratio ofN/1.Wecall the crystal simply theN/1 crystal hereafter.When theN/1 crystal was fabricated, the totalconcentration of ferritin molecules was adjusted to 2 μg/mL.

Results and Discussions

Figure 2a-d shows a series of SEM images of the N/1crystal. The images are the results under the conditions wherethe value ofN is 1, 2, 3, and 4, respectively. The yellow spots inthe right-hand image of each Figure correspond to the darklattice points of the left-hand image. These yellow spots andthe HFmolecules cover the whole area without any defects inthe hexagonally symmetric manner. Besides, the number ofdark spots obviously decreases as the value ofN increases.Wealso investigated the ratio of the number of HF molecules towhole molecules using much larger single-crystal domainscomposed ofmore than 1000molecules (Supporting Informa-tion, S3). As a result, the number ratios of HF in the N/1crystal with N = 1, 2, 3, and 4 were 0.498, 0.642, 0.739, and0.818, respectively. These results are nearly equal to the molarratio in the solution, N/(N + 1). Here, we should emphasizethat the dark spots are not vacancies. If there is a vacancy,then it looks very bright because of the emission of thesecondary electrons from the bare silicon surface (SupportingInformation, S4). Fromall of these observations andanalyses,we can reasonably conclude that the crystal was composed ofHF and AF molecules.

The absence of phase separation implies that the intermo-lecular interaction is identical between molecules. Under sucha condition, the probability distribution of the number of HF

(21) Bartlett, P.; Ottewill, R. H.; Pusey, P. N.Phys. Rev. Lett. 1992, 68, 3801.(22) Kiely, C. J.; Fink, J.; Brust, M.; Bethell, D.; Schiffrin, D. J. Nature

1998, 396, 444.(23) Leunissen, M. E.; Christova, C. G.; Hynninen, A.-P.; Royall, C. P.;

Campbell, A. I.; Imhof, A.; Dijkstra, M.; van Roij, R.; van Blaaderen, A.Nature 2005, 437, 235.

(24) Shevchenko, E. V.; Talapin, D. V.; Kotov, N. A.; O’Brien, S.;Murray, C. B. Nature 2006, 439, 55.

(25) Cheon, J.; Park, J. I.; Choi, J. S.; Jun, Y. W.; Kim, S.; Kim, M. G.;Kim, Y. M.; Kim, Y. J. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 3023–3027.

(26) Zheng, J.; Constantinou, P. E.; Micheel, C.; Alivisatos, A. P.; Kiehl,R. A.; Seeman, N. C. Nano Lett. 2006, 6, 1502–1504.

(27) Shevchenko, E. V.; Ringler, M.; Schwemer, A.; Talapin, D. V.; Klar,T. A.; Rogach, A. L.; Feldmann, J.; Alivisatos, A. P. J. Am. Chem. Soc. 2008,130, 3274–3275.

(28) Theobald, J. A.; Oxtoby, N. S.; Phillips, M. A.; Champness, N. R.;Beton, P. H. Nature 2003, 424, 1029.

(29) Shevchenko, E. V.; Kortright, J.; Talapin, D. V.; Aloni, S.; Alivisatos,A. P. Adv. Mater. 2007, 19, 4183–4188.

(30) Lewis, P. A.; Smith, R. K.; Kelly, K. F.; Bumm, L. A.; Reed, S. M.;Clegg, R. S.;Gunderson, J.D.;Hutchison, J. E.;Weiss, P. S. J. Phys. Chem. B2001, 105, 10630–10636.

(31) Sano, K.; Ajima, K.; Iwahori, K.; Yudasaka, M.; Iijima, S.; Yama-shita, I.; Shiba, K. Small 2005, 1, 826–832.

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(33) Ikezoe, Y.; Kumashiro, Y.; Tamada, K.; Matsui, T.; Yamashita, I.;Shiba, K.; Hara, M. Langmuir 2008, 24, 12836–12841. (34) Yamashita, I. Thin Solid Films 2001, 393, 12–18.

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molecules surrounding one HF molecule should obey thebinomial distribution as described below.

PN, k ¼ C6kRkNð1-RNÞ6-k ð1Þ

Here,PN, k,Ck6, andRN are the probability that the number of

HFs surrounding one HF is k in theN/1 crystal, the binomialcoefficient expressed as 6!/(k!(6- k)!), and the number ratio ofthe HF molecules to the whole molecules in the N/1 crystal,respectively. Figure 3a-d shows a probability distributionserieswith respect to the number ofHFs surrounding oneHF.The blue diamonds and red circles show calculated data fromeq 1 and experimental data, respectively. In all cases, theexperimental data are in good agreement with the calculated

data. These results serve as evidence that the intermolecularinteractions are identical, which means that the moleculardistribution is truly random and has the potential to generatea random number sequence.

Figure 1. SEM image of a binarymolecular crystal obtained from an equimolar HF andAF solution. The lower -left inset is an FFT image, andthe scale bar is 50 nm. Thewhite spots are indiumoxide nanoparticles in theHFmolecules. The dark area is occupied by theAFmolecules that areinvisible in the SEM observation.

Figure 2. Series of SEM images of theN/1 crystals. Yellow spots inthe right-hand images correspond to the dark lattice points of the left-hand image.

Figure 3. Probability distribution of the number of HF mole-cules surrounding one HF in each crystal. (a-d) Calculated data(blue diamonds) and experimental data (red circles) for the 1/1, 2/1,3/1, and 4/1 crystals, respectively.

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Figure 4a shows the procedure to obtain a bit sequencefrom a lozenge-shaped single-crystal domain in 1/1 crystal.At first, we indexed 0 and 1 for AF and HF molecules,respectively, as shown in the left-hand image. The 2D crystalused in the following analyses is amuch larger crystal domain,consisting of 2500 (=50� 50)molecules. Thematrix of 0 and1 on the right-hand side explains the molecular distributionand allows us to generate a 1D bit sequence consisting of 2500zeros or ones. Considering the geometrical symmetry ofthe crystal, three sequences;Seq1, Seq2, and Seq3;wereobtained along the crystal axis, and the randomness of thesequences was independently investigated. We executed fourfundamental and popular statistical tests: the monobit test,the poker test, the runs test, and the long-run test (SupportingInformation, S5),which are based on documents published bythe National Institute of Standards and Technology (NIST),Federal Information Processing Standards (FIPS) 140-2,35,36

and NIST Special Publication 800-22.37,38 Figure 4b shows

one of the results of the statistical test;the poker test, inwhich the frequency distributionof the 16possible 4-bit values(from 0 to 15) is shown. The calculated data were obtainedunder the condition that 4-bit values appear equally. Inaddition to the results of Seq1-Seq3, another result with4-bit values is simultaneously shown. These values wereobtained from the molecular matrix with 4 (= 2 � 2)molecules shown in the dotted-line square in Figure 4a. Thisis a unique point in our system; the crystal has two dimen-sions, and the random number can also be generated in a 2Dmanner. These experimental data show small, unbiaseddeviations from the calculated data. Table 1 shows all of theresults of the statistical tests. The requirement for each testwas recalculated to adjust to our 2500-bit sequence. Fromthese results, it was found that all of the sequences;Seq1,Seq2, Seq3, and the (2 � 2) matrix-based sequence;pass therequirements and can be concluded to be random. Strictlyspeaking, no bit sequences that pass the statistical tests canbe concluded to be random. We should conclude that non-randomness was not detected for any of the bit sequen-ces. However, in this letter, we call the sequence simply arandomnumber sequence in order to avoid the above complexexpression.

Let us now look at the term “random” from the point ofview of materials science. For example, a phrase such as“random mixture of molecules” is often found on a varietyof occasions;39,40 however, whether it is truly randomhasbeendiscussed very little. Considering the different interactionsamongdifferent species, a truly randommixture is impossible.A binary or multicomponent mixture of molecules, atoms, ornanoparticles basically shows phase separation or an alter-native ordered structure. However, we first and quantitativelydiscussed the random molecular distribution in the crystal onthe molecular scale with the aid of statistical tests. In thepresent case, randomness was realized because the differencesin intermolecular interactions between the different kinds ofmolecules were removed by the unique molecular structure: athick protein shell that conceals the inner difference. Further-more, another important and notable point here is that themolecular assembly retains the HCP crystal structure regard-less of the molar ratio. This result also explains that theAF and HF molecules are indistinguishable from each otherby their outer surface characteristics.

Finally, we show that the obtained random number can beused effectively in computer simulations. We chose a simpleand fundamental simulation: a problem regarding squaring acircle. The expected convergent value here is π (SupportingInformation, S6). We prepared four 16-bit random numbersequences from the above 1D 0 and 1 sequences and a 2Dmatrix. The average and standard deviations of the resultsof 100 experiments are shown in Table 2, where all of theaverage values obtained are within a margin of 1% errorcompared to the true π value. The probability distribution inthe simulation should be described simply by a binomialdistribution with the probabilities p and q (= 1- p). Hence,the expected value, E, and the standard deviation, σ, derivedfrom N points are p and (pq/N)1/2, respectively. Under thecondition that p andN are equal toπ/4 and 2500, respectively,the value of σ/E is approximately 0.01 (= 1%). Therefore,

Figure 4. Random number generation from the 2D crystal ob-tained form a 1/1 solution. (a) A small part of the 2D crystal withindices of 0 and 1 that correspond to the AF and HF molecules,respectively. The right-handmatrix shows the molecular distributionof 0 and 1. The bit sequence consisting of 1’s and 0’s was generatedalong each crystal axis. The (2 � 2) matrices were also used toinvestigate the 4-bit value distribution. (b) Frequency distribution ofthe 4-bit values.

(35) http://csrc.nist.gov/publications/fips/fips140-2/fips1402.pdf .Although the four statistical tests used here have been omitted from thedocument, they are still used frequently to estimate randomness of a bitsequence.

(36) Kim, S. J.; Umeno, K.; Hasegawa, A. ISM Rep. Res. Educ. 2003, 17,326–327.

(37) http://csrc.nist.gov/groups/ST/toolkit/rng/documents/SP800-22rev1.pdf.

(38) Kim, S. J.; Umeno, K.; Hasegawa, A. Tech. Rep. IEICE 2003, 103,21–27.

(39) Krisovitch, S. M.; Regen, S. L. J. Am. Chem. Soc. 1992, 114,9828–9835.

(40) Takami, T.; Delamarche, E.; Michel, B.; Gerber, C.; Wolf, H.;Ringsdorf, H. Langmuir 1995, 11, 3876–3881.

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each experimental error is in good agreement with the calcu-lated one. From this analysis, we can conclude that thesimulation was successfully executed and, as a result, thesimulated π value reasonably converged to the true π value.These results indicate that the random number sequenceobtained from the binary protein crystal has no unfavorablebias and is capable of practical use.

Conclusions

We have successfully fabricated 2D monomolecular-layercrystals composed of binary kinds of proteinmolecules;apo-ferritin (AF) and holo-ferritin (HF);and demonstrated ananometer-scale random number generator (RNG) originat-ing from the randommolecular distribution in the 2D binarycrystal.RNGsbasedonphysical phenomenaormathematicalalgorithms are well known. However, our study is the firstdemonstration of RNGs based on chemical characteristics.

We not only investigated the randomness of the moleculardistribution from some statistical tests but also showed thevalidity of the use of our RNG for aMonte Carlo simulation.We have so far succeeded in fabricating a single crystal with asize of approximately 100 μm2, incorporating one millionmolecules. Now let us consider that such a large crystalconsists of equimolar AF and HF molecules. The number ofmolecular distribution patterns in the crystal becomes astro-nomically large because it reaches the 210

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. Although thebinary 2D crystal can be readily prepared, it is neitherreproducible nor predictable. Therefore, this molecular pat-tern can be regarded as a kind of microscopic fingerprint andthus would be used for a nanometer-scale cryptographicdevice or an identification tag.

Acknowledgment. We thank K. Tamada, T. Matsui, andN. Matsukawa for discussions and comments regarding thiswork. This study was supported in part by the LeadingProject of the Ministry of Education, Culture, Sports,Science, and Technology, Japan.

Supporting Information Available: Description of ferritin,2D crystals, statistical tests, and a Monte Carlo simulation.This material is available free of charge via the Internet athttp://pubs.acs.org.

Table 1. Statistical Analyses of the Randomness of the Molecular Distribution

monobit test poker test runs test long-run test

requirement of X 1185 < X < 1315 4.60 < X < 32.80 1186 < X < 1314 X < 17

seq1 1245 13.64 1251 16seq2 1245 18.81 1245 12seq3 1245 27.67 1245 162 � 2 matrix 1245 15.44 N. A. N. A.

Table 2. Results of Monte Carlo Simulations

obtained π value average standard deviation

seq1 3.137 0.022seq2 3.151 0.018seq3 3.120 0.0194 � 4 matrix 3.150 0.019

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