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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:
[email protected].
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K.; Davis,P. Nat. Photon. 2008, 2, 728–732.
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© XXXX American Chemical Society
DOI: 10.1021/la9000413Langmuir XXXX, XXX(XX), 000–000 A
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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
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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.
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N. R.;Beton, P. H. Nature 2003, 424, 1029.
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S.; Alivisatos,A. P. Adv. Mater. 2007, 19, 4183–4188.
(30) Lewis, P. A.; Smith, R. K.; Kelly, K. F.; Bumm, L. A.;
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K.;Yamashita, I. Langmuir 2007, 23, 1615–1618.
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(34) Yamashita, I. Thin Solid Films 2001, 393, 12–18.
DOI: 10.1021/la9000413 Langmuir XXXX, XXX(XX), 000–000
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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.
DOI: 10.1021/la9000413Langmuir XXXX, XXX(XX), 000–000
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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.
DOI: 10.1021/la9000413 Langmuir XXXX, XXX(XX), 000–000
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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
6
. 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
DOI: 10.1021/la9000413Langmuir XXXX, XXX(XX), 000–000
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