Why molecules move along a temperature gradient
`
`Stefan Duhr and Dieter Braun*
`
`Chair for Appli~d Physic;, Ludwig MaJ!imilians Universitat, Amalienstr<me 5•1, 80799 Mtmich, Germany
`
`Edited by Leo P. Kadanoff, University of Chkago, Chicago, IL, and approved Octob<:r 12, 2006 {rec.eived for review M;~y 26, 2006)
`
`Molecules drift along temperature gradients, an effect called
`thermophoresi$, the Soret effect, or thermodiffuslon.ln liquids, it~
`theoretical foundation is the $Ubject of a long-standing debate. By
`using an all-optkai microfiuidic fluore$cem:e method, we present
`experimental results for DNA and polystyrene beads over a large
`range of partide s!a:e~. salt concentrations, and temperatllri1!5. The
`data support a unifying theory based on solvation entropy. Stated
`in !;imple terms, the Soret meffldrmt i!i given by the negative
`solvation entropy, dh1ided by kT, Thil th!lory predicts the thermod(cid:173)
`iffusion of polystyrenil beads and DNA without any free parame·
`tiH$. We as;ume a local thermodynamic equilibrium of the •oivent
`molecules around the molecule. This as!lumptlon is fulfilled for
`moderate temperature gradients below a fluctuation criterion. For
`both DNA and polystyrene beads, thermophoretk motion d-sange$
`sign at lower temperatures. This thermophilidty toward lower
`temperatures is attributed to an il"lcrea$lng positive sntropy of
`hydration, whereas the generally dominating thermophobkity is
`explained by the negative entropy of ionk !ihielding. The under·
`standing of thermodiffusion set!i the ~tage for detailed probing of
`solvation pmpertie!i of colloids and biomolewles. For exam pi!<!, we
`!iUCCI!5sfu!ly determine the etrectiva charga of DNA and bead> over
`a size range that Is not acces5ibie with electrophoresis.
`
`DNA ! flworescer.ce I micrctluidic ! Soret I th~rmodiffusion
`
`T hermodiffusion has been known for a long time (1), but its
`
`theoretical explanation for mole.cules in liquids is still under
`debate. The search for a theoretical understanding is motivated by
`the fact that thermodiffusion in water might lead to powerful
`all-optical screening methods for biomo!ecules and colloids.
`Equally well, thermodiffu~ion handles and moves molecules all(cid:173)
`optically and therefore can complement well established methods:
`for example, electrophoresis or optical tweezers. For the latter,
`forces of optical tweezers scale with particle volume and limit this
`method to particlr..s of only >500 nm. Electrophoresis does not
`suffer from force limitations but is difficult to miniaturize because
`of electrochemical reactions at the electrodes,
`On the other hand. thermodiffusion allows the microscale
`manipulation of smali particles and molecules. For e.xamplc,
`1,000-bp DNA can be p~>HcrMd ;srbitrmf!y in bulk water (Fig, 1}.
`The temperature pattC'ru "DNA{ hea.tcd by:4 K, •va$ \>i:dtten into
`a water 'mm w[th an infntn::d lm>i?.t <ll~Mll.hlg micnmetip!~. Til~'
`concentration of 1,000-bp DNA was imaged by using a fluores(cid:173)
`cent DNA tag. In an overaH cooled chamber at 3°C, DNA
`accumulates toward the heated letters "DNA" (negative Soret
`effect), whereas at room temperature DNA is thermophobic
`(positive Soret effect) as seen by !he dark letters.
`In the past, the apparent complexity of therrnodiffus1on pre(cid:173)
`vented a full theoretical description. As seen for DNA in Fig. 1,
`molecules characteristically deplete from regions with an increased
`temperature, but they can also show the inverted effect and
`accumulate (2, 3). Moreover, the size scaling of thermodiffusion
`recorded bv thermal f1e.ld flow fractionation showed fractional
`power laws ~ith a variety of exponents that are hard to interpret { 4,
`5). The latter effect might be resolved by revealing nonlinear
`ibermovhoretic drift for the strong thermal gradients used in
`thermai field flow fractionation (our unpublished observations).
`A variety of methods were used to measure thermodiffusion,
`mostly in the nonaqueous regime, ranging from beam deflection
`
`- -20"C
`
`iOOpm
`
`fig. 1. Tnermodiffu>ion m~nipulata; the DNA concentration by small temp~r­
`atured!fferenceswrthinthe bulk$Oil,ticm.A thir.waterfilm i> heat~d by 2 K ;;long
`the letters "DNA" with an infrared la>er. F<:.>r a moled chamber at .3'C, tluo;,,.
`cently tagged DNA accumulntes at the w~•m letters. However, at room temper·
`ature, DNA moves Into thr.; cold, shewing reduced fluorescence. The ch~mber is
`60 I'-m thin, containing SO nM DNA i11 1 mM Tris buffer. ~ve•y 50th bi!S2 pair i>
`labeied with TOTO· 1 {for det~ils, see >upporling ir.fcrmation).
`
`(2, 3, 6), holographic scattering (7-9), electrical heating (10), to
`thermal lensing (11 ). Recently we have developed a fluorescence
`microl'luidic imaging technique (12, 13) that aHows tbe mea(cid:173)
`surement of thermodiffusion over a wide molecule size range
`without artifacts induced by thermal convection. Highly diluted
`suspensions can be measured; therefore, par!ide-particle inter(cid:173)
`actions do not have an influence. We only apply moderate
`temperature gradients. In !.he following study, we used this
`method to confirm a straightforward theoretical explanation of
`thermodiffusion.
`
`Theoretical Approach
`For diluted c.oncentrations, it is generally assumed (14) that the
`thermodiffusive drift velocity v depends linearly on the tempe.r·
`ature gradient VT with a proportionality constant which equals
`the thermodiffusion coefficient D-r: v = -DTVT. In steady state,
`thermodiffusion is balanced by ordinary diffusion, Constant
`diffusion and thermodiffusion coefficients both lead to an
`exponential depletion law (15) c(cu "' exp[-··(DT/D)(T - ?'o)J,
`with the concentration c dependmg on the temperature dtffer·
`ence T - To only. The concentration c is normalized by the
`boundary condition of the concentration co with temperature To,
`The Soret coefficient is defined as ratio ST = D·r/D, which
`determines the magnitude of thermodiffusion in the steady state.
`Although !he above exponential distribution could motivate an
`approach based on Boltzmann equilibrium statistics, it is com(cid:173)
`monly argued that thermodiffusion without exception is a local
`nonequilibrium effect that re.quires fluid dynamics, force fields,
`or particle-solvent potentials (16-20). However, in a previou>
`paper (15), we demonstrated that for moderate temperature
`
`Author contribution!: D. B. d~•igned r~seor<h; S.D. •od D.B. perform<!d rese•rrh; S.D. ond
`DJ3. anajy-ted data; ~nd 5-D. and D.B. wrot1=1 th~ pi!pt~:r.
`Th~ author~ di:!d:!l!re no umiHct of lntere:it.
`'To whom mrre5poc.de~« should!>~ ad~ressed. E-maH: diater.br•un~lmu.rla.
`C 2GDG by lhe NatfonaJ Acedemy of Sdene~~ of th~ USA
`
`w\o.<w.pna~.org/cgl/doi/1 0.1 073/pna;,06038731 OJ
`
`Page 1700
`
`TEVA EXHIBIT 1007
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`a
`
`Fig. 2. Sa it dependence. io) Thermodiffu;ion in water Is domlnat~a by kmk
`>hieldlng (Left) and water hydration (Right}. {i;) Soret weffici<;r;t Sr versus
`De bye ;.,,.,gth for carboxyl-'11odih~d poiy>tymne be~ds ot diaml.!\<:>r 1, 1, 0.5.
`and 0.21-'m. Une~r plot (Udt) ar;d logarithmic plot (,qigl>t). The Smet coeffl·
`dents are de;cribed by Eq. :<with an effective surface d1Mge of ,-•ff = 4,50(1
`ef1,m' ~nowr; from ele,·uophor<e~h. The intercept .Sr(~n~ = 0) ;, fitted with a
`hydration <entmpy p<>r part!de ;uoia;:~ of Sh;o = -1,400 Ji(moH<·;.<m<).
`
`ptay a role and we can introduce an effective surface charge
`density <Torr= Q.rr/A per molecule arell A. From the tempe.r(cid:173)
`ature detiva!ion according to Eq. 1. the ionic contribution to
`the Sor et coefficient is s¥""'") = (A {:ltr~rrA::m)/( 4esok T2
`). A
`similar relation was derived for charged micelles recently (22),
`although without considering the temperature dependence of
`lhe dielectric coeffic:e.nt e. Next, ;he. contribution to the Soret
`coefficient from the hydration entropy of water can be directly
`inferred from the particle-area-specific hydration entropy
`Soyrl = Soya/A, namely s~~yd) = -A,~~yd(T)ikT. Finally, lhe
`contribution from the Brownian motion is derived as ST = 1/T
`by inserting the kinetic energy of the particle G "' kT into Eq.
`1. However, this contribution is very small (ST = 0.0034/K) and
`can be neglected for the molecules under consideration. The
`wmribution~ from ionic shielding and hydration entropy add
`up to
`
`[2J
`
`{3fT~rr
`\
`-,l'oyd + -4
`-T--: X Am;l,
`seo
`/
`The Soret coefficient Sr scales linearly with particle surface A
`and Debye length Arm. We tested Eq. 2 by meusuring ST versus
`salt concentration, temperature, and molecul;; size. In all cases,
`thennodiffm;ion is quantitatively predicted without any fre-e
`parameters. We used fluorescence single-particle tracking to
`follow carboxyl-modified polystyrene beads (catalog no, F-8888,
`Molecular Probes, Eugen<J, OR) wittl diameters of 1.1 and 0.5 at
`25 ;;M dialyzed into 0.5 mM Tri~+J Cl at pH 7.6. Thermodiffusion
`of particle~; ::s0.2 ~-tm is measured by the fluorescenc.e decrease
`that reflects the bulk depletion of the particles (12). The
`chamber thickness of 20 !Lm d;;mped the thermal convection to
`negligible speed!i (15). The experimental design also excludes
`thermal lensing and optical trapping (15). Debye lengths Am:
`were titrated with KCI (see the supporting information, which is
`pubiished on the PNAS web site)o
`
`A ('
`Sr = -k--
`-;
`,
`
`1-
`
`Salt Di!pllndsnte. Fig. 2b show5 the Soret coefficients of polysty·
`r<Jne beads with different !lizes versus ,\Dfl· The So ret coefficients
`
`gradients the thermal nuctuations of :he part.lde are the ba5is for
`a loco:) equ11ibritsm. This al!ows the de:;criptlon of the thermod(cid:173)
`iffusive steadv state by a succession of local Boltzmann laws,
`yielding cleo "= exp[ -·(G(:r) - G(T0))ikT], with G being the
`Gibbs-free enthalpy of the single particle-solvent system. Such
`an approach is valid only if the aemperature gradient VT is below
`a threshold 'YT < (aSr)-l, which is given by the particle
`fluctuations ·,vith the hydrodynamic radius a and Soret coeffi(cid:173)
`cient ST, 1l5 shown recently (1.'5). In the pre:;ent study, temper(cid:173)
`ature gradients below this limit were used so that thermodiffu(cid:173)
`sion is measured at local thermodynamlc equllibrium conditions.
`Local thermodynamic equilihrium allows the derivation of a
`thermodynamic foundation of the Soret codficient. The local
`Boltzmarm distribution relates small concentmtion changes &
`with smBil Gibbs-free energy differences: &/c ""' -oG/kT. We
`equate ihiR relation with a locally linearized thermodiffusion
`steady state given by &/c "' -SroT and thus find the Soret
`coefficient by the temperature derivative of G:
`
`[1]
`
`Whereas the above relation is stJfficient for the following
`derivation, it can be generalized by locally applying the thermo(cid:173)
`dynamic rehiion dG = -SdT + v'dp + p.dN. For singi<J particles
`at a constant pressun:. we find that the Soret coefficient equals
`the negative entropy of the partide-solvmlt system S 1:ccording
`to Sr = -SikT, Thi~ relation is not surprising given that the
`entropy is by ddinition related with the temperature derivative
`of the free enthalpy.
`The above general energe1ic treatment is inh<Jrent in previ(cid:173)
`ously described approaches baoed on local equilibrium (14, 21,
`22), including the successful interpretatlon of thermoelectric
`voltages of diluted electrolytes (23, 24), which are described by
`energies of !nmsfer. Re{:ently, the nonequilJbrium approach by
`Ruckenstein (25) was applied to colloids (26) with the charac·
`teristic length I as:;igned to the De bye length ADH· If in~tead one
`would assign the charaet<Jri;;tic length according to l = 2ai3 with
`the particle radi!JS a, the Rucke.nstein approach would actually
`confirm the above local equilibrium relation (1) for the Sorel
`coefficient Measurements em SDS micelle~ (26) appeared to
`confirm this nonequilibrium approach, but for the chosen par·
`tides the competing parameter choices l ='J.a./3 and C = Arm
`yielded comp::~rabfe value~. Thus, the experiments could not
`distinguish between the competing theories.
`We will use the above local equilibrium relation;; to derive
`the Sore: coefficient for particles larger th'm the Debye length
`in aqueous solutions and put the results to rigorous experi·
`mental tests, Two contributions dominate the panicle entropy
`Sin water (Fig. 74): the entropy ofionic ~bidding (Fig. 2a Lejt)
`and the temperature-sensitive entropy of water hydration {Fig,
`2a Right). The contribution from the entropy of ionic shielding
`is calculated with l he tempenltur<J derivative of the Gibbs" free
`enthalpy (26, 27) Giooio "' Q~rrArm/[2A eeo] with the effective
`charge Qerr and particle surface A. Alternatively, this enthalpy
`can be interpreted as an electrical field energy G;.,,i<• "'
`Q~u![2C] in the ionic !lhielding capacitor C. We neglect the
`particle-particle interactions because the fluorescence ap(cid:173)
`proach allows the measurement of highly diluted systems. To
`obtain the Soret coefficient, temperature derivative~ consider
`the Debye length Arm(!} = v'i(T)s-;;k;i?(ze2cs) and the
`dielectric cor:stant t:(T), Bo:h remperalure derivatives give rise
`to a factor f3 = 1 -
`('T/e)ae/BT, The effective. charge Oorr is
`largely temperature-insensitive, which was confirmed by eke(cid:173)
`trophores.is independently (28). Such a dependence would be
`unexpect<Jd because the strongly adsorbed ions dominate the
`value of the effective charge. Experimentally, we deal with
`colloids exhibiting flat surfiices, i.e., the particle radius is
`larger than Atm- In this case, charge renormalization does not
`
`Duhr and Braun
`
`Page 1701
`
`TEVA EXHIBIT 1007
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`0
`
`10
`6
`Debye Le•1gth ;~0 , [nm]
`
`fig, 3. Temperature d<Jpend;,rKe. !a) Th<> t<>r>•perah:ra dependencl.' is don•·
`!n,;tr;d by the linear {hange In tM hydmtion entropy Sevd· It shifts the ~alt·
`dependentthermodiffu;ion Sr(AoH) to lowe,value>. The partlde si>'e i:; L 1 !!m.
`(b) Th~ Smet w.zffident $, inm.>asl.'> linr;arlywith th<>temperature ;;s expect<Jd
`for a hydratkm entrop~; 5nf0(Y). i; depend; or: the molewl;; >p>;cie;, not it; ;lze.
`a:; se€!r; from the rescaled 5cret coefficients for DNA with different length>.
`
`scale linearly with a small intercept at ADH = 0 and confirm the
`ADwdependence of Eq. 2. For smaller·diameter beads, the Soret
`coefficients scale with the particle surface area A (Fig. 2), as
`expected from Eq. 2. To check whether Eq. 2 also quantitatively
`explains the measured Soret coefiklcnts, we inferred the effec~
`tive charge of the beads by electroptlmesi:! (see supporting
`materials). By using 40-nm beads with identical carbOJ..')'l surface
`modifications at ADH = 9.6 nm, we nuorescently observed
`free-flow electrophoresis and l!orrected for electroosmosis, find·
`ing an effective ;;urface. charge density of o-0rr = 4,500 ± 2,000
`e/p..rn2• Thi.:; va!ue is virtually independent from the used salt
`concentra1kms (28). With rhis inferred effective charge, Eq, 2 fits
`the Soret coefficient for various bead sizes and salt concentra(cid:173)
`tions weil (Fig. 2b, solid lines).
`The intercept ST(ADH = 0), where ionic contributions are zero,
`also scales with particle surface and is described by a hydration
`entropy per particle surface of Shyd = -1,400 J/{mol·K·~<-m2). The
`value matches the literature values for similar surfaces reason(cid:173)
`ably well (29-31). For example, dansyl-alanine, a molecule wlth
`surface groups comparable with polystyrene beads, was mea(cid:173)
`sured to have a hydration entropy (29) of -0.13 J/(moi·K) at a
`comparable temper<lture. Linear scaling with its surface area by
`using a raditw of a "" 2 nm results in a value of Snyd = -2,500
`J/(mol·K·p..m2), in qualitative. agreement with our result The
`hydration entropy is a highly informative molecule parameter
`that is notoriously difficult to measure, yielding an interesting
`application for thermodiffusion.
`
`'hlmperaturl! Oep~ndencl!. Hydration entmpie:; Snyd in water are
`known w increa~e linearly witb decreasing temperatures (29-
`31). Because the slope of the ionic contribution of Srvenus A.m1
`ls with high-precision temperature insensitive for water [f3(T)I
`(eT2) s const), only the intercept is expected to decrease as the
`overall temverature of the chamber is reduced, Thi~ is indeed the
`case, as §een from the temperature dependence of beads with
`diameters of L1 ,um (T = 6-'29"C) (Fig. 3a). We infer from the
`intercept S·r(ADfl "' 0) that the hydration entropy changes sign at
`=20°C. As seen for DNA in Fig. 1, hydration entropy can
`dominate the.rmodiffusion at low tempera:ures and move mol(cid:173)
`ecules toward the heat (O.r < 0).
`The properties of hydration entropy lead to a linear increase
`of S.r over temperatures at a fixed aalt concentration as measured
`for 1.1-f<.m beads and DNA (Fig. 3b). We normalized Sr by
`dividing by .h(30°C) to compensate for molecule ~urface area.
`The slope;; of ST over temperature differ between beads and
`DNA, Howeve.r the slope. does not differ be!ween DNA of
`different size (50 bp versus 10,000 bp). Based on Eq. 2. this is to
`
`be expected because !he temperature dependen~"' of the hydra(cid:173)
`tion entropy depends only on the type of surface of the molecule,
`not its size. We measured the diffu;;ion coefficients of the DNA
`species at the respective tempilr<:ture independently. Witbin
`experimental error, changes in the diffusion codficient D match
`with the change of the ·.vater viscosity without the need to assume
`conformationul changes of DNA over the temperat!lre range.
`Please note that the change of the sign of the DNA Soret
`coefficient is situated near the point of maximal water density
`only by chance. There, the two entropic contributions balance,
`For polystyrene beads at Arm ~ 2 nm for example, the sign
`change is observed at 15°C (Fig. 3a ). An increased Soret
`coefficient over tempemture was reported for aqueous solutions
`before (3), however with a distinct nonlinearity that we attribute
`to remnant particle-particle interactions.
`
`Siie Oeptnds;nc!! of the !lead$, The Soret coefficient was measured
`for carboxyl-modified polystyrene beads in diameters ranging
`from 20 nm to 2 /Lm, Bead:; witb di<lme!ers of 0.2, 0.1, O,G4, and
`0.02 iJ.rH were diluted to concentrations of 10 pM, 15 pM, 250
`pM, and 2 nM, and their bulk fluores(:ence was imaged over time
`to derive D7 and D (12, 15) from the depletion and sub:;equent
`back-diffusion. Larger beads with diameters of 1.9, 1.1, and 0.5
`~·m were diluted w concer!!rlltions of 3.3 aM, 25 aM, and 02 pM
`and measured with single-particle tracking. The solmiom were
`buffered in 1 mM Tris (pH 7.6) with Aml = 9,6 nm. ln all cases,
`interaction§ between particles can be excluded. Care wa:s taken
`to keep the temperature gradient in the local equiHbrium regime.
`We find that the Soret coefficient §Cales with particle surface
`over four order~ of magnitude {Fig. ·1a). Tbe data are described
`well with Eq. 2 with an effective surface charge density of rrorr =
`4,500 e!~<-m' and neglected hydration entropy contribution. The
`5-fold too-low prediction for the smallest particle {20 nm in
`diameter) can be explained by charge renormalization because
`its radius is smaller than itDB·
`The diffusion coefficient D for spheres is given by the Einstein
`relation and scales inversely with mdiu:; D a< 1/a. In~erting Eq. 2
`into ST "" Dy/D, the thermodiffusion coefficient DT is expected
`to scale with particle radius a, This is experimentally confirmed
`over two orders of magnitude (Fig. 4b ). These findings contmdict
`several theoreitcal studies claiming that DT should he indepen(cid:173)
`dent of particle size (16-20, 26), based on ambiguous experi(cid:173)
`mental resul!s from thermal field flow frac:icmation (4) thal
`were probably biased by nonlinear thermodiffusion in large
`thermal gradients (15).
`
`Si~i! Oeps;m:lsm::t of ONA. Whereas polystyrene beads share a very
`narrow size distribution as a common feature with DNA mole(cid:173)
`cules, bends are a much tess complicated model system. Beads
`are rigid spheres that interact with the solvent only at its surface.
`ln addition, the charges reside on the surface, where the
`screening takes place. Thus, the finding that thermodiffusion of
`flexible and homogeneously charged DNA is described equally
`well with Eq. 2 is not readi!y expected and quite interesting (Fig,
`4 c and d).
`We measured DNA with sizes of 50-48,502 bp in 1 mM Tds
`buffer {ii.DH = 9.6 nm) Ht !ow molecule (:oncer:trations be1ween
`1 j.tM (50 bp) and 1 nM (48,502 bp). Only every 50th base pair
`was stained with the TOT0-1 fluorescent dve. The diffusion
`coefficient was measured by back-diffusion after the laser was
`turned off and depends on the length L of the DNA in a
`nontrivilll way. The data are well fitted with a hydrodynamic
`radius scaling a :x L 0·7·5• This scaling represents an effective
`average over two DNA length regimes. For DNA molecules
`longer than =1,000 bp, a scaling of 0.6 is found (32), whereas
`shorter DNA scales with an exponent of =1 (see the supporting
`information),
`We can tfescribe the measured Soret coefficient over three
`
`Duhr am:l Bfilur.
`
`Page 1702
`
`TEVA EXHIBIT 1007
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`::r
`
`"! :ll
`
`71
`.---/ I
`
`.. ~~~~'>••~-:;H'~{.~'~r
`c~
`0.1
`1
`oo~
`r:iam~!ac lurr.]
`
`w'
`1c'
`10'
`DNA Lenglo [bpJ
`
`Si?.e droprondenc<'. (a) for poly>tyrene bead,, the Soret wefflciant
`fig. 4.
`"'al~s witr, the particle !iUt'face ave ;four ord;:r; ot magnitude. Mea;uri!ments
`are described by Eq. 1 with an effective surface charg;; demit;.· of .,..rr = 4,500
`ei;m/ {2) and negligible hydration ;;ntropy. Th~ deviation forth<: bei!d wi•h
`3 dianwter of 20 r.m car; be omd<mtoodfrom ~n increa;ed effective charge due
`to the onset of chargE normali<:~tior. for a :;;,<.01~· (b) Acmrdir>giy, t!w ther(cid:173)
`modiffu;ion r.cefflclent Dr scale> linearly with bo:ad diarnet'<'r. (c) The Scret
`((:!!ffider.t (Jf DNA >cal<os ar.wrding to Sr "''/i.., with the iength L of the DNA
`based on EQ. 2 with <on e!tecrive charge per base pa>ir oHU2 e. (d) Thermod·
`iffu>icn coefficient Dr deuea>e~ over DNA length wilh 0,·" L -<'·'", <:dlJ><OrJ by
`the $t~aling of diffusion rceffident D •> L -v>.
`
`order;; of magnitude of DNA lengths with Eq. 2 if we assume that
`effective charge of the DNA is shielded at the surface of a sphere
`with 1he hydrodynamic radius a. Because of the low salt con(cid:173)
`centration (Amr = 9,6 nm), such globular shielding is reasonable.
`Not only is the experimentally observed scaling of tbe Sore1
`coefficient with the squa~e root of itg length correcdy predicted
`based on Eq. 2 (Sr :X o:rda2 o: U!V 5 IX L05), but the Soret
`coefficient also is fully described in a quantitative manner (Fig.
`4c, solid line), with an effective charge of 0.1:2 e per base,
`matching we!( with li!erature values (33) ranging from 0.05 e/bp
`to 0.3 etllp.
`As shown in Pig. 1kl, the thermodiffusion coefficient for DNA
`drops with DNA length ncconling to DT = DS;: c: Q~rda 3 oc
`L2ff-2·25 oc L -0.25. Thu~, shorter DNA a:::tuaHy drifts faster in a
`temperature gradient than longer DNA. It is important to point
`out that this finding is in no way contradictory to experimental
`findings of a constant DT over polymer length in nonaqueous
`settings (8), According to Eq, 1, the thermodyno1mic relevant
`parameter i:; !be Soret coefficient, which is determined by the
`solvation energetics, The argument (J 9) that polymers have to
`decouple imo monomers to show a constant DT merely becomes
`the special case where the solvation energetics determine both
`STand D with equal but inverted size scaling. In accordance with
`our local energetic equilibrium argument, Sr and not DT dom(cid:173)
`inates thermodiffusion also for nonaqueous polymers near a
`glass transition (8). Here, Sr is constant, where.as D-r and D scale
`according to an increased friction. However, for a system of
`DNA in solution, for which long-ranging shielding couples lhe
`monomer>, a constant DT over polymer length :::annN be as ..
`sumed a priori (Fig. 4d).
`
`~Effective C!mrge, The effective charge Q0 ;r is a highly relevant
`parameter for colloid ~cien:::e, biology, and biotechnology, So far
`it only could be inferred from electrophoresis, restricted to
`
`0. i
`O.QOj O.o1
`fla!ticla Suri<~ce !i->m2J
`
`10
`
`EH~'tive d1arge from thermod;Hmion. Effett!ve rharge i; interred
`Ftg. !>,
`from thermodiffusion u;ing Eq. 3. Poly;;tyre11e beads (:;tQ .. ;;t,OOO nm) (a) and
`[l~U\ (S.0-50.000 bp) (b) W<.'re mea>ur~d ever a l;~rge ~i&e r<mge, whkh i:;
`impn~>ible wi·th elertrophore:;i,, A> e~fJf!(.ted, the effectl11e r.h<orge nf the
`bead~ s{aie> with partids >Urface <ond lineariy with the length of DI\JA.
`
`part1clcs smaller than the Debye !engt!J (a :s 3..\nH) (34),
`Unfortunately, many colloids are outside this regime, As shm•m
`before, a similar size restriction does not hold for thermodli'fu"
`skm. In many case5, the hydration entropy Snyd comribmes <15%
`(Fig. 2) and can be neglected at moderate salt levels. Thus, we
`c;m invert Eq, 2 to obtain the el'feclive charge Q"rr for spherical
`molecules from
`
`[3]
`
`The effective charge derived from thermodiffusion measure"
`ments of polystyrene beads and DNA is plotted in Fig. 5 over
`;everal orders of magnitude in size, The effective :::hBrge of beads
`~:::ale;; linearly with particle ~urfacc, with a slope confirming the
`effective surface charge density of if err= 4,500 el~&m 2, which was
`inferred from electrophoresis only for small particles, Average
`deviations from linear scaling are <8o/! .. (Fig. 5a). The effective
`charge inferred from thermodiffusion mea:mrements of DNA
`using Eq. 3 scales linearly with DNA length with an effective
`charge of 0.12 e/bp. The length scaling is eonfirmed over four
`orders of ma.gnitude wiih an average error of 12% (Fig . .'ib).
`Thus, thermodiffusion can be used to infer the effective charge
`with low error~ for a wide range of particle sizes" This is even
`more interesting for biomolecule characteri:mtion because mea"
`surements of thermodiffusion can be performed all-optically in
`picoliter volumes,
`
`Com:lsa!>ion
`We describe. thermodiffusion, the molecule drift along t;;mper(cid:173)
`ature gradients, in liquids ·,vith a general, microscopic lheory.
`Applied to aqueous solutions, this theory predicts thermodiffu·
`:;ion of DNA and poly:;tyrene beads with an average accuracy of
`20%. We experimenta!ly validate major parameter dependencies
`of the theory: linearity against screening length f..m1 and mole"
`cule hydrodynamic area A, quadratic dependence on effective
`charge, and linear1ty against temperature, Measurements of
`thermodiffusion can be miniaturized to the micrometer :;:::ale
`with the all-optical fluorescence lechnique ;md permit micro·
`acopic temperature diffe.rences to manipulate molecules based
`on their surface properties (Fig. 1 ), The theoretical description
`allows the extraction of §olvation entropy and the effective
`charge of molecules and particles over a wide :>ize range,
`
`Mcsteriai!i and Method!>
`!nfrars:d Temps:ratur!! Col'ltm!. The tempeBture gradients used to
`induce thermodiffusive motions were created by aqueous ab(cid:173)
`sorption of an infrared laser (Furukaw<: Electric, Tokyo, Japan)
`at a wavelength of 1,480 nm and 25 mW of power. Water strongly
`
`Page 1703
`
`TEVA EXHIBIT 1007
`TEVA PHARMACEUTICALS USA, INC. V. MONOSOL RX, LLC
`
`

`
`ubsorbs at this wavelength with an attenuation length of K "' 320
`p.m. The laser beam was moderately focused with a lem of 8-mm
`focal {li:;!ance. '(vpk:aHy, lh~ Hmlp(mtl<H\~ i.n rhtl S<ibtion 'IV!~~;
`raised by 1-2 Kin the bcum cenrerwllh a l/~~~di;l.l11»t<~rot25 t<.m,
`measured with th,~ ttmp~mlur{:·d(:p~~niknl f!m.'>!t~:>.CelJcc s.i~ri~d
`of the dye 2' ,7' obJS(Carl:m:X)'t::lh)'l}o5(fl)·~~"rbo~yfh#JI"(:S(.~CitJ (12).
`Thin chamber heights of l0-20 t<-m and moderate focusing
`removed posRible artifacts from optical trapping, thermal lens·
`ing, and thermal convection (12-). For tempera!ure-dependent
`measurements, both the objective and the microfluidic chip were
`tempered with a thermal bath. Imaging was provided from an
`A:;;ioTech Vario fluorescence microscope (Zeiss, Oberkochen,
`Germany), illuminated with a high·power light-emitting diode
`(Luxeon, Calgary, Canada), and recorded with the CCD camera
`SensiCam QE (PCO, Kelheim, Germany).
`
`Mu~ew~e~. Highly monodisperse and protein-free DNA of 50,
`100, 1,000, 4,000, 10,(}00, and 48,502 bp (Fast Ruler fragments
`and >.-DNA; Fermentas, St. Leon-Rot, Germany) were diluted
`to 50 i;.M base pair concentration, i.e., the molecule concentra(cid:173)
`Hon was between 1 '";Jvi (50 bp) and l nM ( 48,502 bp ). DNA was
`fluore~cently labeled by the intercalating TOT0-1 fluore~cent
`dye (Mole.cu!ur Probes) with a low dye/base pair ra;io of 1:50.
`Carboxyl-modified poiysty rene beads with diameters of 2, 1, 0.5,
`0,2, OJ, 0.04, and 0.02 p.m (cat,;!og nos. F-8888, F-8823, F-8827,
`F-8888, F-8795, F-8823, and F-8827; Molecuiar Probes) were
`dialyzed (Elutll Tube mini; Ferment;>!i) in dlstHled water and
`diluted in 1 mM Tris (pH 7.6) to concentrations between 3.3 aM
`(2- ILm) and 2 nM (0.02 J.l.m).
`
`Co!lcrc>r~tr<~thm !m<~gl!ig Ovr<>r Thni!, Ei1her the method of concen(cid:173)
`tration imaging (12) or single-particle tracking was used tD
`
`measure thermod!ffu~ion at low concentralions, narne,Jy <0.03
`g!Hter for DNA and 10-5 g_niter for beads. At higher concen(cid:173)
`trations, we found profound changes of thermodiffusion coef(cid:173)
`ficients, DNA and polystyrene beads of <0.5 !L!ll in diameter
`were imuged over time (12} by brlght·field fluorescence with a
`X40 oil-immersion objective. Concentrations inferred after cor(cid:173)
`recting for bleaching, inhomogeneous mumination, and temper(cid:173)
`ature-dependent fluore:;cence (12) were fitted with a finite
`element theory, The model captures all de!ail$ of both ther(cid:173)
`modiffusive depletion and back-diffusion to measure D,. and D
`independently (aee 5upporting informa;t!on). Measurement>
`were performed in mkrof!uidic ch!ps 10 p.m in height with
`polydimethylsiloxane on both. sides.
`
`Slngi&-!'>;srtkle Tracking. Polystyrene particles of >0.5 p,m in
`diameter were measured by single-particle tracking due to the
`slow equilibration time and risk that steady-state depletion is
`disturbed by thermal convection. The thermodiffm;ive drif! was
`imaged with a x32 air objective at 4 H:r. at an initial stage of
`depletion in a 20-~m-thick chamber, Averaging over the z
`position of the particles removed effects from thermal convec(cid:173)
`tion. The drift veiocity versus temperature gradient of 400 tracks
`were linearly fitted by v = -D,.VT to infer DT. The diffusion
`coefficients D of the partldea were evaluated based on their
`squared displacement, matching wirhin 10% the. Einstein
`relatiollship.
`
`We thlmk Klam Stierstadt, Jan Dhorn, and WenJ<:r Kohl<~r for digcus(cid:173)
`sions and Julia Morfill and Ve,ronil:a Egger for comments on the
`manuscript. Our Emmy-Noether Group is funded by the Deutschr.
`Forsdmng:;gemeinschaft and hosted by Hermann Gaub.
`
`L L~dwig C (1856} Si!mngber Bayer Alwd W1Ss !!1en Math-l'ialwwiss K/20:539.
`2, Giglk) M, Vendramil!i A (1977) Ploys Rev Lei/ 38:26-30.
`l Jacopi11i S, l'i&ua R (20lB} Europhys Leu 63:247--253.
`• :, Jean SJ, Sd;impf ME, Nyborg A (1997) AMI Chem 69:3442--3450,
`5. Shimdu PM, Lw G, Giddings JC (1995) Anal Chern 67:27(!5-27!3.
`6. Pi:u~a R, Guarino A (2002) l'hys RE'' Lei/ 88:208302,
`7. de Gan~ BJ, Kita R, Muller B, Wieg<md S (2003) J CMm Ploys ll8:8073--SO&l.
`B. Rauch J, Kilhl~r W (20(!2) Phys Re:• Lea 86:185901.
`9. \Vif.gand S1 Kt.'lhh:~r Vl (2002) in Th.~rmal Nunequilibriu.;!'s Phenarm.mtJ ln Fluid.
`Mixlures (Springer, Il~rlin), p 189.
`10. Putnam SA, C:ilim DG {2005) Langmuir 21:5317-5323.
`11. Ru"cor.i R, !saL, Piaw! R (1004) J Opl Soc: Am B 21:605--616,
`12. Dul:r S, AJduini S, B.-sun D (20{)4) Ew P!:ys J E 15:2.77--286.
`l3 Braun D, Lit,.:hab~; A {2002) Phy.r Rtl' Leu 89:188103.
`l4, de Groat SR, Ma<ur J' {l %9) Nonequiiibrium Thermodynamics (North-
`Hol!and, Amstm:lam),
`lS. Duhr S, Bra"n D {2006} Phys Re" Le!i 96:1()3301.
`l6. Emery AH, Jr, Driclamer HG (1955) J Chem l'hys 23:2252--2257.
`i7. H?.m J (l96G) J Appi Phy.r 31:1853-1858.
`18. Mon.m:>Y KI {1999) J E<p Th•or Phys 88;944-946,
`
`19. Schimpf ME, ikmemw SN {2.000) J l'hys Chern 8 104;9935-9942.
`2(). Vail A, Krdrhrlv A, Enge W, Kramer L, Kl\hlel W (2005) Phys fley L<ll
`92:214501 .
`2l. Dhom JKO (201Yl) J CMm l'llys l2():l632-:l64:l.
`2·2. F~ycll~ S, Bkk,~l T, l