A quantitative model for a nanoscale switch accurately predicts thermal actuation behavior
Kyle Crocker, Joshua Johnson, Wolfgang Pfeifer, Carlos Castro, Ralf Bundschuh
JJournal Name
A quantitative model for a nanoscale switch accuratelypredicts thermal actuation behavior † Kyle Crocker, a Joshua Johnson, bc Wolfgang Pfeifer ad , Carlos Castro d , and RalfBundschuh ∗ abe f g Manipulation of temperature can be used to actuate DNA origami nano-hinges containing goldnanoparticles. We develop a physical model of this system that uses partition function analysis of theinteraction between the nano-hinge and nanoparticle to predict the probability that the nano-hinge isopen at a given temperature. The model agrees well with experimental data and predicts experimentalconditions that allow the actuation temperature of the nano-hinge to be tuned over a range oftemperatures from ◦ C to ◦ C . Additionally, the model reveals surprising physical constraintson the system. This combination of physical insight and predictive potential is likely to informfuture designs that integrate nanoparticles into dynamic DNA origami structures. Furthermore, ourmodeling approach could be expanded to consider the incorporation, stability, and actuation of othertypes of functional elements or actuation mechanisms integrated into nucleic acid devices. In 2006, Paul Rothemund published seminal work on the designof nanostructures out of DNA, developing a technique known asDNA origami . Although early structures were static, expand-ing this technique to produce functional, dynamic structures hasbeen of particular interest, since the use of DNA as a construc-tion material renders the resulting structures naturally well-suitedfor use as machines in biological or synthetic systems. To thisend, significant research has focused on the development of dy-namic nanoscale devices . Indeed, dynamic DNA origami de-vices are being developed for use as drug delivery systems ,as well as molecular biological probes , computing elements,and nanorobots . Interest in these applications has driventhe development of a variety of actuation methods. Actuation ∗ Corresponding author. a Department of Physics, The Ohio State University, Columbus, OH 43210, USA. E-mail:[email protected] b Interdisciplinary Biophysics Graduate Program, The Ohio State University, Columbus,OH 43210, USA. c Department of Chemistry, Imperial College London, Molecular Sciences Research Hub,80 Wood Lane, London W12 0BZ, UK. d Department of Mechanical and Aerospace Engineering, The Ohio State University,Columbus, OH 43210, USA. e Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH43210, USA. f Division of Hematology, Department of Internal Medicine, The Ohio State University,Columbus, OH 43210, USA. g Center for RNA Biology, The Ohio State University, Columbus, OH 43210, USA. † Electronic Supplementary Information (ESI) available. See DOI:00.0000/00000000. can be achieved in a number of ways, such as introduction ofshort oligonucleotides with specifically designed sequences ,or changing environmental factors such as salt conditions ,pH , or temperature .In order to be suitable for use in such applications, however,it is necessary to have precise control over the stimulus response,which remains challenging. To this end, we quantitatively charac-terize the temperature actuation of a DNA origami hinge contain-ing a gold nanoparticle (AuNP), which was previously describedby Johnson et al. . This device consists of two stiff arms con-nected by a flexible vertex, such that the motion around the vertexis restricted to a single angular dimension. A DNA-coated AuNPis attached to the top arm, and complementary DNA strands areaffixed to the bottom arm that anneal to the AuNP to hold thehinge closed. When the temperature is increased, the hybridiza-tion between the AuNP and bottom arm melts to release the hingeinto the open state. The AuNP remains stably attached to the toparm, allowing for repeatable temperature-controlled opening andclosing. This type of system is of interest since AuNP-DNA origamicomposites have many exciting applications, such as in plasmon-ics and nanoelectronics . In particular, the potential for AuNPfacilitated reconfiguration that is both fast and tunable could beimportant in these applications. Another area of interest for thesecomposite devices is that they have the potential to allow pre-cisely controlled local heating and therefore actuation: Althoughthe experiments described here are performed with bulk temper-ature change, the AuNP itself could in principle be locally heatedwith a laser . Thus the study of systems into which such AuNPs Journal Name, [year], [vol.] , a r X i v : . [ q - b i o . B M ] J a n re incorporated is a potentially fruitful area of research.In order to design increasingly complex and useful DNA de-vices, it is necessary to construct predictive models of their func-tion. This has proven to be challenging, however, since environ-mental factors and thermal fluctuations can play an importantrole, often rendering classical solid mechanics approaches com-mon to macroscopic engineering unsuitable . Nevertheless, anumber of computational techniques have shown predictive ef-ficacy. All-atom molecular dynamics (MD) simulations, whichtrack interactions of each atom in a system over time, providedetailed and accurate information about system dynamics .The computational cost is quite high, however, rendering such anapproach practical only for small subsections of DNA devices andshort time scales . In order to study larger systems, one must usea coarse-grained approach. One way to do this is to approximatethe atoms that make up a nucleotide as a single particle and trackthe positions and interactions of many such particles. This ap-proach is taken most notably in the commonly used oxDNA sim-ulation , as well as the recently developed MrDNA model .These approaches significantly extend simulation time-scales, butthey are still typically limited to microsecond or at most millisec-ond timescales, while actuated conformational changes often oc-cur on the second timescale or longer. Another coarse-grainedapproach is to use finite element (FE) modeling to predict DNAstructures. A widely used example of this is CANDO , althoughthis is typically used for shape prediction since it lacks molecu-lar details that govern dynamics. Pan et al. expanded the FEapproach to improve the description of thermal fluctuations inshape, but this still did not extend to large-scale actuated confor-mational changes .Long timescales or large structures may render even suchcoarse-grained approaches computationally unfeasible, particu-larly if one wants to rapidly iterate through many structural vari-ations to guide design. It is therefore desirable to develop evenmore computationally efficient techniques. Here, we focus on theapplication of one such technique, statistical mechanics. Whilesimilar approaches have been used to model DNA strand displace-ment , which is widely used for actuation, application to ac-tuation of devices themselves remain rare despite the increasingneed for computational efficiency to guide design of functionalDNA origami devices. In one of the few examples of applicationto an actual device that we are aware of, Marras et al. use a sta-tistical mechanics approach to model a system in which changesin salt concentration are used to actuate a hinge . Here, we de-velop a thermodynamic model that accurately describes the tem-perature actuation of the nano-hinge device containing an AuNP.To our knowledge, this is the first statistical mechanics model ofa composite DNA origami system, which is a critical step, sincemany applications require the incorporation of NPs or other func-tional elements. Furthermore, we demonstrate that this modelis able to predict actuation temperatures as a function of devicedesign, enabling principled design of devices with desired tran-sition temperatures. Additionally, our model gives surprising in-sights into the system, revealing a limit on the number of simul-taneously bound strands and demonstrating that configurationalentropy and suboptimal energetic states meaningfully impact sys- tem behavior.
In this subsection we describe briefly the experiments by Johnson et al. that provide the basis for our model . The AuNP-hinge sys-tem is shown schematically in Fig. 1(A), with arrows indicatingthat the hinge is opened as temperature is increased and closedwhen temperature is decreased. In Fig. 1(B), averaged data fordifferent overhang strand lengths are shown for hinges with twooverhang strands (left), which we call “bivalent" or three over-hang strands (right), which we call "trivalent". The overhanglengths (6-8 bases) and sequences (all adenine bases) are identi-cal in the bivalent and trivalent cases. The studied DNA origami hinges and AuNP-hinge constructs wereprepared as previously described . Briefly, nM scaffold DNA(p8064) and nM staple strands were pooled in TE-buffer ( mM Tris, mM EDTA, pH . , 5 mM NaCl) supplemented with mM MgCl and subjected to a thermal annealing consisting of min at ◦ C following by hours at ◦ C and cooling to ◦ C.Excess staple strands were removed by centrifugal purification inthe presence of PEG . Conjugation of T ssDNA coated AuNP,prepared as described by Johnson et al. , to the purified DNAhinges was performed by addition of 5-fold excess AuNPs to theresuspended DNA hinges and incubation at ◦ C for 15 minutes.
Thermal profiles of the different constructs were collected ona Cary Eclipse Fluorometer with thermostated multicell cuvetteholder. If not stated otherwise, temperature ramps were setto ◦ C/min and thermal profiles were collected by cycling be-tween the respective minimum and maximum temperatures atleast twice. A reference hinge without AuNP was used to substracttemperature dependent fluorescence effects of the fluorophore.
Negative stain electron microscopy was used to confirm foldingand correct incorporation of AuNPs into the DNA hinges, follow-ing previously described protocols . Purified DNA hinges andAuNP-hinge constructs were adsorbed onto TEM grids (ElectronMicroscopy Sciences, Hatfield, PA), stained using freshly preparedUranyl-formate and imaged on a FEI Tecnai G2 Spirit TEM, oper-ated at kV. Following data collection and the substraction of fluorescencetemperature dependence, the fluorescence is normalized suchthat the average maximum value (corresponding to all openhinges) is equal to one and the average minimum value (corre-sponding to all closed hinges) is equal to zero . Journal Name, [year], [vol.][vol.]
Negative stain electron microscopy was used to confirm foldingand correct incorporation of AuNPs into the DNA hinges, follow-ing previously described protocols . Purified DNA hinges andAuNP-hinge constructs were adsorbed onto TEM grids (ElectronMicroscopy Sciences, Hatfield, PA), stained using freshly preparedUranyl-formate and imaged on a FEI Tecnai G2 Spirit TEM, oper-ated at kV. Following data collection and the substraction of fluorescencetemperature dependence, the fluorescence is normalized suchthat the average maximum value (corresponding to all openhinges) is equal to one and the average minimum value (corre-sponding to all closed hinges) is equal to zero . Journal Name, [year], [vol.][vol.] , ig. 1 Experimental system and temperature actuation data underlyingour model. (A) shows the experimental system: a gold nanoparticle(AuNP) is affixed to the top arm of a DNA origami nano-hinge via long,stable dsDNA strands formed between single stranded DNA (ssDNA)overhangs on the arm and complementary strands of ssDNA coating theAuNP. Shorter overhangs on the bottom arm anneal at low temperaturesand melt at high temperatures. The inset shows a TEM image of a closedhinge with AuNP at room temperature. Scale bar is nm. When thebottom overhang is annealed, the hinge is forced into a closed state wherefluorescence is quenched. The normalized fluorescence in (B) thereforeprovides a measure of the bulk fraction of hinges that are open. The insetsin (B) illustrate the number of overhangs on the bottom arm of the hinge:either two ("bivalent") or three ("trivalent"). The legend indicates thesequence of the bottom arm ssDNA, for instance A corresponds to asequence consisting of 6 adenine bases. All hinge overhangs are madeup of adenine bases, and all AuNP connections are made up of thyminebases. The lengths of bottom arm overhangs vary between 6 and 8 inboth the bivalent and trivalent cases. The normalized data is averaged over all experimental replicatesfor both melting and annealing curves, and this average and thecorresponding standard deviation are shown. Data from slightlydifferent temperatures had to be combined due to fluctuationsduring thermal ramps which were set to collect one data pointevery . ◦ C. Specifically, we use the temperature values of thefirst melting replicate and then identify the closest observed tem-perature values in other replicates to take the average and stan-dard deviation of the corresponding fluorescence values. Thataverage and standard deviation are assigned to the temperaturevalue of the first melting replicate. In order to estimate the errordue to the temperatures not lining up exactly among the repli-cates, we identified the maximum discrepancy in temperaturevalues among replicates where fluorescence values are averaged,and compared the expected change in fluorescence according toour model to the experimental noise. We found that this worstcase estimate of systematic error induced by averaging data fromslightly different temperature values is on the order of the noisein the experimental measurements. Therefore, we concluded thatany interpolation over temperature values from different repli-cates is unnecessary, as it would effectively constitute interpola-tion over noise.
In this section, we describe how the experimental system is mod-eled, both at a conceptual and mathematical level.
To relate the experimental readout to a calculable property of thesystem, we note that the fluorescence of nano-hinges, when nor-malized between and , is equal to the fraction of open nano-hinges. Furthermore, for a system in thermodynamic equilibrium,the fraction of open hinges gives the probability that an individ-ual hinge is open. We therefore assume the system is in thermo-dynamic equilibrium and create a thermodynamic model of anindividual hinge. As discussed in more detail in section 3.1, a thermodynamicmodel requires enumeration of allowed states and correspondingfree energies. The experimentally observable state in this systemis whether the hinge is open or closed, so we consider the mi-crostates of the system that correspond to these macrostates. Wetreat an open hinge as consisting of only a single state, captur-ing the effect of the many physical microstates in the closing freeenergy parameters, the hinge closing enthalpy change ∆ H cl andthe hinge closing entropy change ∆ S cl . For a closed hinge, weenumerate the possible binding states more explicitly as shown inFig. 2. First, any number of the bottom overhangs on the nano-hinge can be involved in base pairing with the ssDNA strands onthe AuNP (Fig 2A). For any given base-pairing interaction, anyconsecutive stretch of adenines on the bottom overhang can bindto any consecutive stretch of equal length of thymines on the ss-DNA strands on the AuNP (Fig 2B). Each set of consecutive basepairs is associated with a base pairing enthalphy ∆ H bp and en-tropy ∆ S bp .We do not consider any base-pairing states that involve bulgesor internal loops, i.e., unpaired bases internal to a base-pairedregion, which have a prohibitively high free energy cost. Vari-ous experimental results indicate that the free energy cost of suchstates is at least on the order of kJ/mol. Tanaka et al. findthat free energies for single A and T bulges at ◦ C are . ± . kJ/mol and . ± . kJ/mol, respectively. Longer bulges are ex-pected to be similarly costly, as reported by Turner and Matthewsin the context of RNA . The cost of interior loops can be approx-imated using mismatch parameters for an ACA/TTT sequence,which Allawi et al. find to be . ± . kJ/mol . Similarly,Peyret et al. find AA and TT mismatches to have energy costs onthe order kJ/mol . At a cost of at least kJ/mol, the Boltz-mann factor corresponding to a 7 base polyA-polyT section of ds-DNA drops from ≈ to below upon the creation of a bulgeor interior loop. In order to compute the probability that the system resides in thestates specified above, it is necessary to determine the free ener-gies associated with each state and to compute a partition func-tion. The free energy of each of these microstates is determinedby the number of paired bases and the energy required to close
Journal Name, [year], [vol.] , ig. 2 Schematic illustration of closed nano-hinge states. The AuNP isindicated by the gold sphere, the bottom hinge overhangs by blue lines,top arm overhangs by red lines, and the AuNP strands by green lines.The gray helices represent the bottom arm of the nano-hinge. (A) givesthe states on the strand binding level, while (B) gives the states on thebase pairing level. In (A) we note that there is one way to bind threeconnections (first panel), three ways to bind two connections (secondpanel), three ways to bind one connection (third panel), and one way tobind zero connections (fourth panel). This is described mathematically inEq. (4). In (B), there are states with no fraying (first panel), states withfraying from the poly-T end (second panel), states with fraying from thepoly-A end (third panel), and states with fraying from both ends (fourthpanel). This is described mathematically in Eq. (5). the hinge. Since the temperature is variable, both the enthalpicand entropic parts of the free energies must be considered. Usingthe open state as a reference free energy G open = , we can write G i ( T ) = ∆ H cl − T ∆ S cl + N i ( ∆ H bp − T ∆ S bp ) . (1)for each closed microstate i , where T is temperature, N i is thenumber of base stacks (since the base pairing energy is associatedwith the energetic favorability of stacking two consecutive basepairs); ∆ H cl and ∆ S cl are the hinge closing enthalpy and entropy,respectively; and ∆ H bp and ∆ S bp are the base pairing enthalpyand entropy, respectively. The partition function for this system isthen Z = + ∑ i exp (cid:2) − G i ( T ) / k B T (cid:3) , (2)where the sum is over all microstates of the system. Since we treatthe hinge as consisting of a single open state with free energy G open ≡ , the probability that a hinge is in an open state is givenby p open = / Z . (3)In order to calculate Z more explicitly, we have to considerthe partition function of the closed states, denoted Z cs . Sinceall closed states are multiplied by a Boltzmann factor corre-sponding to the free energy cost to close the hinge, denoted by ∆ G cl = ∆ H cl − T ∆ S cl , the primary challenge is to account for allpossible base pairing states, as illustrated in Fig. 2. To do this foran arbitrary number of overhangs per hinge with potentially dif-fering numbers of bases per overhang, we first denote overhangsof different lengths (i.e. different numbers of bases) by subscript j , and the number of type j overhangs by N c , j . For instance, if wehave a hinge with three overhangs, two of which with six bases(type 1) and one of which with eight bases (type 2), we wouldhave N c , = and N c , = . We then need to account for all possi-ble choices of actually realized connections n j of type j subject tothe constraint ∑ j n j ≤ N c , max , where N c , max is the maximum num-ber of bound connections per state. The number of possibilities tochoose n j out of N c , j total available connections must also be con-sidered, introducing a binomial coefficient for each j . Combiningthe above yields Z cs = exp (cid:0) − ∆ G cl / k B T (cid:1) × min { N c , max , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × min { N c , max − n , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × ··· (cid:35)(cid:35) (4)where Z S , j is the partition function describing all of the possiblebase pairing interactions for a single connection of type j . Inparticular, Z S , j ≡ N S , j ∑ i = ( N T − i + )( N A , j − i + ) exp (cid:20) − ∆ G term + i ∆ G bp k B T (cid:21) (5)where N S , j is the maximum number of stacks in the type j du-plex, N T is the maximum number of stacks available to thepoly-T strand, and N A , j is the maximum number of stacks avail-able to the poly-A strand of type j . Thus, N S , j = min { N T , N A , j } . ∆ G bp = ∆ H bp − T ∆ S bp is the free energy of a single stack, and ∆ G term = (cid:2) . kJ/mol − T ( . kJ /(mol K) ) (cid:3) is the terminalbase pairing energy . Note that N T − i + is the number ofpositions on the poly-T strand at which i consecutive stacks canbind, and that N A , j − i + is the number of positions on the poly-Astrand at which i consecutive stacks can bind. Thus, their productis the total multiplicity of the state with i bound stacks.Taking everything together, we therefore have Z = + exp (cid:0) − ∆ G cl / k B T (cid:1) × min { N c , max , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × min { N c , max − n , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × ··· (cid:35)(cid:35) (6)with p open = / Z , where for clarity the ∆ G ’s are not written asfunctions of temperature, but they retain the temperature depen-dence indicated in Eq. (1). In order to relate the model to the data, we fit the openingprobability, p open , to the experimental normalized fluorescenceby varying the four energetic parameters within physically real-istic bounds (i.e. ≤ ∆ H cl ≤ ∞ and − ∞ ≤ ∆ S cl , ∆ H bp , ∆ S bp ≤ ).This fit is performed via a non-linear least squares minimizationfor this bounded set of parameters using a Trust Region Reflectivealgorithm , which is implemented using the Python SciPy pack- Journal Name, [year], [vol.][vol.]
Journal Name, [year], [vol.] , ig. 2 Schematic illustration of closed nano-hinge states. The AuNP isindicated by the gold sphere, the bottom hinge overhangs by blue lines,top arm overhangs by red lines, and the AuNP strands by green lines.The gray helices represent the bottom arm of the nano-hinge. (A) givesthe states on the strand binding level, while (B) gives the states on thebase pairing level. In (A) we note that there is one way to bind threeconnections (first panel), three ways to bind two connections (secondpanel), three ways to bind one connection (third panel), and one way tobind zero connections (fourth panel). This is described mathematically inEq. (4). In (B), there are states with no fraying (first panel), states withfraying from the poly-T end (second panel), states with fraying from thepoly-A end (third panel), and states with fraying from both ends (fourthpanel). This is described mathematically in Eq. (5). the hinge. Since the temperature is variable, both the enthalpicand entropic parts of the free energies must be considered. Usingthe open state as a reference free energy G open = , we can write G i ( T ) = ∆ H cl − T ∆ S cl + N i ( ∆ H bp − T ∆ S bp ) . (1)for each closed microstate i , where T is temperature, N i is thenumber of base stacks (since the base pairing energy is associatedwith the energetic favorability of stacking two consecutive basepairs); ∆ H cl and ∆ S cl are the hinge closing enthalpy and entropy,respectively; and ∆ H bp and ∆ S bp are the base pairing enthalpyand entropy, respectively. The partition function for this system isthen Z = + ∑ i exp (cid:2) − G i ( T ) / k B T (cid:3) , (2)where the sum is over all microstates of the system. Since we treatthe hinge as consisting of a single open state with free energy G open ≡ , the probability that a hinge is in an open state is givenby p open = / Z . (3)In order to calculate Z more explicitly, we have to considerthe partition function of the closed states, denoted Z cs . Sinceall closed states are multiplied by a Boltzmann factor corre-sponding to the free energy cost to close the hinge, denoted by ∆ G cl = ∆ H cl − T ∆ S cl , the primary challenge is to account for allpossible base pairing states, as illustrated in Fig. 2. To do this foran arbitrary number of overhangs per hinge with potentially dif-fering numbers of bases per overhang, we first denote overhangsof different lengths (i.e. different numbers of bases) by subscript j , and the number of type j overhangs by N c , j . For instance, if wehave a hinge with three overhangs, two of which with six bases(type 1) and one of which with eight bases (type 2), we wouldhave N c , = and N c , = . We then need to account for all possi-ble choices of actually realized connections n j of type j subject tothe constraint ∑ j n j ≤ N c , max , where N c , max is the maximum num-ber of bound connections per state. The number of possibilities tochoose n j out of N c , j total available connections must also be con-sidered, introducing a binomial coefficient for each j . Combiningthe above yields Z cs = exp (cid:0) − ∆ G cl / k B T (cid:1) × min { N c , max , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × min { N c , max − n , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × ··· (cid:35)(cid:35) (4)where Z S , j is the partition function describing all of the possiblebase pairing interactions for a single connection of type j . Inparticular, Z S , j ≡ N S , j ∑ i = ( N T − i + )( N A , j − i + ) exp (cid:20) − ∆ G term + i ∆ G bp k B T (cid:21) (5)where N S , j is the maximum number of stacks in the type j du-plex, N T is the maximum number of stacks available to thepoly-T strand, and N A , j is the maximum number of stacks avail-able to the poly-A strand of type j . Thus, N S , j = min { N T , N A , j } . ∆ G bp = ∆ H bp − T ∆ S bp is the free energy of a single stack, and ∆ G term = (cid:2) . kJ/mol − T ( . kJ /(mol K) ) (cid:3) is the terminalbase pairing energy . Note that N T − i + is the number ofpositions on the poly-T strand at which i consecutive stacks canbind, and that N A , j − i + is the number of positions on the poly-Astrand at which i consecutive stacks can bind. Thus, their productis the total multiplicity of the state with i bound stacks.Taking everything together, we therefore have Z = + exp (cid:0) − ∆ G cl / k B T (cid:1) × min { N c , max , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × min { N c , max − n , N c , } ∑ n = (cid:34)(cid:18) N c , n (cid:19) Z n S , × ··· (cid:35)(cid:35) (6)with p open = / Z , where for clarity the ∆ G ’s are not written asfunctions of temperature, but they retain the temperature depen-dence indicated in Eq. (1). In order to relate the model to the data, we fit the openingprobability, p open , to the experimental normalized fluorescenceby varying the four energetic parameters within physically real-istic bounds (i.e. ≤ ∆ H cl ≤ ∞ and − ∞ ≤ ∆ S cl , ∆ H bp , ∆ S bp ≤ ).This fit is performed via a non-linear least squares minimizationfor this bounded set of parameters using a Trust Region Reflectivealgorithm , which is implemented using the Python SciPy pack- Journal Name, [year], [vol.][vol.] , ge . Additionally, we performed a discrete optimization overthe maximal number of overhangs N c , max ∈ { , , } available forsimultaneous binding. Python code implementing the model is available at https://github.com/bundschuhlab/PublicationScripts/tree/master/NanoswitchTActuationPrediction . Data isavailable upon request.
In this section, we will first give a short high level overview ofour model and then demonstrate how it agrees with the experi-mental data and expectations based on the literature. Next, weextract mechanistic insights about nano-hinge actuation. Lastly,we demonstrate that the model can be used to guide the designof nano-hinges that can be actuated over a wide range of temper-atures.
We formulate a thermodynamic model for actuation of the hingecontaining an AuNP as shown in Fig. 1(A). Specification of a ther-modynamic model requires enumeration of the allowed states ofthe system, the free energies associated with each state, as wellas a relationship between these states and experimental observ-ables. The macroscopic observable here is the open (fluorescing)or closed (quenched) state of the hinge. We model the system ashaving a single open state, representing many microstates, whichhas some unknown free energy cost to transition into the closedstate. This free energy cost captures the contribution from allopen microstates and consists of an enthalpic component ∆ H cl and an entropic component ∆ S cl . When the hinge is closed, thereare many binding microstates available, but these states are eas-ier to enumerate: when closed, the hinge overhangs are allowedto anneal to the AuNP DNAs, and every combination of consec-utive base pairing stacks is allowed. This is shown schematicallyin Fig. 2. Fig. 2(A) shows the strand-level allowed binding states,and Fig. 2(B) shows the base pair-level allowed binding states.Fraying is allowed from the AuNP strand end, the hinge overhangend, and from both ends. Additionally, the strands are allowedto slide relative to each other without penalty, so that any com-bination of consecutive bases can anneal (all the way up to theAuNP). Since the hinge overhangs are poly-A and the AuNP over-hangs are poly-T, each stack of two consecutive base pairs thatforms is associated with the same base pairing free energy withenthalpic component ∆ H bp and entropic component ∆ S bp . Giventhese definitions of the states and their free energies, the partitionfunction of the system and thus the probability of a hinge to be inthe open state and fluorescing can be calculated as a function oftemperature (see section 2.3.3). We fit the temperature dependent opening probability predictedby the thermodynamic model to the experimental normalized flu-orescence using a non-linear least squares minimization for theenthalpies and entropies ∆ H cl , ∆ S cl , ∆ H bp , and ∆ S bp and discreteoptimization over the maximal number of overhangs N c , max ∈{ , , } available for simultaneous binding (see section 2.4). Totest the predictive power and robustness of the model we firstfit to bivalent (two overhangs on the bottom arm of the hinge)and trivalent (three overhangs on the bottom arm of the hinge)data simultaneously using all five fit parameters. We then fit tobivalent and trivalent data separately varying the four energy pa-rameters but keeping constant the value of N c , max = found to beoptimal in the simultaneous fit.A comparison of these fits to the data is shown in Fig. 3, andthe best fit parameters are summarized in Table 1. In Fig. 3, row(A) shows the simultaneous fit to the bivalent and trivalent data,row (B) shows the fit to the bivalent data and the predictions forthe trivalent system using the bivalent fit parameters, and row(C) shows the fit to the trivalent data and the predictions for thebivalent system using the trivalent fit parameters. Each panelshows the average root mean squared difference (RMS) betweenthe data and model. In all cases, the model agrees well with thedata with a maximum RMS of . for the fit data and of . forthe predicted curves. Fig. 3
Model with N c , max = fit to experimental data. In row (A),the model is fit to bivalent and trivalent data simultaneously. In row(B), the model is fit to the bivalent data (solid lines), and the trivalentmodel with bivalent fit parameters is compared to trivalent data (dashedlines). In row (C), the model is fit to the trivalent data (solid lines), andthe bivalent model with trivalent fit parameters is compared to bivalentdata (dashed lines). Each panel contains the average root mean squareddifference ( RMS ) between the model and the experimental data. Therightmost column shows the ratio between the fit base pairing parametervalues and the expected base pairing parameter values . We want to emphasize that, while the experimental data is
Journal Name, [year], [vol.] , able 1 Best fit parameter values: Model parameters used for curvesshown in Fig. 3 in units of kJ/mol for enthalpies and kJ/mol K forentropies.
Data fit ∆ H cl ∆ S cl ∆ H bp ∆ S bp N c , max Tri. & Bi. ± − . ±− . − . ± . − . ± . ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . separated by valency rather than overhang length, the experi-mental actuation curves for the bivalent and trivalent cases dif-fer (as shown in Fig. S1) and thus provide independent tests forthe model. The fits in Fig. 3(B-C) show that only fitting to ei-ther the bivalent or trivalent data allows prediction of the otherwith nearly the same quality as fitting to both. Thus, the va-lency based difference is accurately captured by the model with-out further adjustment of its parameters. We conclude that thethermodynamic model faithfully describes the entire temperaturedependence of nano-hinge actuation for six different experimen-tal conditions spanning two different valencies and three differentoverhang lengths using five fit parameters. While we treat the enthalpy ∆ H bp and entropy ∆ S bp of the basepairing as fit parameters, these have been independently mea-sured by SantaLucia from melting experiments on short DNAoligomers and have been used for decades to quantitatively de-scribe DNA melting. Therefore, it is illustrative to compare ourbest fit parameters to SantaLucia’s values. The rightmost columnof Fig. 3 shows that the ratio of our best fit parameters to San-taLucia’s values with corrections (details in ESI section 5.1†) forsalt concentration for each of the three fits is close to one. Asa further test of the appropriateness of the observed base pair-ing parameters, we fit the model again while keeping the basepairing enthalpy ∆ H bp and entropy ∆ S bp at SantaLucia’s literaturevalues, corrected for salt conditions (details in ESI sec-tion 5.1†). These fits are shown in Supplementary Fig. S2, andthe fit parameters are given in Table S1. We find an excellentfit if only the base pairing enthalpy is fixed to its literature valueand a reasonable fit if base pairing enthalpy and entropy are bothfixed at their literature values. These fits are especially reasonablewhen considering that the effects of divalent salt remain difficultto quantify and mostly affect the entropy .The good agreement between literature values of the base pair-ing parameters and the best fit parameters of our model is inter-esting, since other studies have found that the presence of a DNAorigami device can have significant impact on base pairing freeenergy . One possible explanation is that in the experimentsunderlying these earlier studies the base pairing occurs in a muchmore geometrically constrained context, which is avoided by thepresence of the AuNP coated with longer DNA strands in the ex-periments that are modeled here. Also, the corrections in thesestudies are directly to the base pairing free energy while weconsider enthalphy and entropy separately to model the entiretemperature dependence. Since the free energy results from a delicate balance between enthalpic and entropic contributions,the free energy may be more sensitive than enthalpy or entropyalone. We conclude that our fit parameters and literature val-ues for the base pairing parameters agree well, providing furtherindependent validation of our model. For our fitting we use the data published by Johnson et al. aver-aged over both experimental replicates and direction of temper-ature change, with the width of the curves corresponding to thestandard deviation over all of these data sets (see section 2.2.2for details). Although we model this system as an equilibriumprocess, it is important to note that there is hysteresis in the ex-perimental data between the annealing, which exhibits a slightlylower transition temperature, and the melting, which exhibitsa slightly higher transition temperature. In order to verify thatthe average of these melting and annealing curves is a good ap-proximation to an equilibrium condition, we perform the ther-mal actuation experiment at two different rates of temperaturechange: ◦ C/min (as previously done by Johnson et al. ) and . ◦ C/min. For these experiments, we replaced the AuNP withdouble-stranded DNA linkers to avoid potential AuNP degrada-tion with extended time at elevated temperatures . These ex-periments reveal that as the rate is decreased, the hysteresis alsodecreases and both the melting and annealing curves approachthe average of the fast rate curves. Additionally, the average of theslow rate curves is similar to the average of the fast rate curves.This data, shown in Fig. S3, illustrates that the averaging is areasonable approximation to the equilibrium conditions.Nevertheless, we also test the model in the two extreme as-sumptions that the true equilibrium is either the melting or an-nealing data. These fits are shown in Fig. S4, with best fit param-eters reported in Table S2. While these fits result in RMS valuesthat are somewhat higher than the fits to averaged data (0.07and 0.08 for bivalent and trivalent, respectively), the values ofthe base pairing parameters still agree well with literature values,particularly when relying on just the annealing. We conclude thatour observations concerning the validity of the thermodynamicmodel are robust to the details of the treatment of the experi-mentally observed hysteresis. The parameter N c , max of our model is the maximal number ofoverhangs that participate in base pairing simultaneously. In thebivalent case, only two overhangs are present, but since in thetrivalent case three overhangs are present, it is somewhat surpris-ing that N c , max = provides the best fit to our model. In orderto evaluate if N c , max = really provides a significantly better fitof the experimental data than the more natural N c , max = , we fitthe model again varying the free energy fit parameters ∆ H cl , ∆ S cl , ∆ H bp , and ∆ S bp but keeping the maximal number N c , max = ofsimultaneous overhangs fixed. These fits are shown in Supple-mentary Fig. S5, with parameter values given in Table S3. The Journal Name, [year], [vol.][vol.]
Data fit ∆ H cl ∆ S cl ∆ H bp ∆ S bp N c , max Tri. & Bi. ± − . ±− . − . ± . − . ± . ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . separated by valency rather than overhang length, the experi-mental actuation curves for the bivalent and trivalent cases dif-fer (as shown in Fig. S1) and thus provide independent tests forthe model. The fits in Fig. 3(B-C) show that only fitting to ei-ther the bivalent or trivalent data allows prediction of the otherwith nearly the same quality as fitting to both. Thus, the va-lency based difference is accurately captured by the model with-out further adjustment of its parameters. We conclude that thethermodynamic model faithfully describes the entire temperaturedependence of nano-hinge actuation for six different experimen-tal conditions spanning two different valencies and three differentoverhang lengths using five fit parameters. While we treat the enthalpy ∆ H bp and entropy ∆ S bp of the basepairing as fit parameters, these have been independently mea-sured by SantaLucia from melting experiments on short DNAoligomers and have been used for decades to quantitatively de-scribe DNA melting. Therefore, it is illustrative to compare ourbest fit parameters to SantaLucia’s values. The rightmost columnof Fig. 3 shows that the ratio of our best fit parameters to San-taLucia’s values with corrections (details in ESI section 5.1†) forsalt concentration for each of the three fits is close to one. Asa further test of the appropriateness of the observed base pair-ing parameters, we fit the model again while keeping the basepairing enthalpy ∆ H bp and entropy ∆ S bp at SantaLucia’s literaturevalues, corrected for salt conditions (details in ESI sec-tion 5.1†). These fits are shown in Supplementary Fig. S2, andthe fit parameters are given in Table S1. We find an excellentfit if only the base pairing enthalpy is fixed to its literature valueand a reasonable fit if base pairing enthalpy and entropy are bothfixed at their literature values. These fits are especially reasonablewhen considering that the effects of divalent salt remain difficultto quantify and mostly affect the entropy .The good agreement between literature values of the base pair-ing parameters and the best fit parameters of our model is inter-esting, since other studies have found that the presence of a DNAorigami device can have significant impact on base pairing freeenergy . One possible explanation is that in the experimentsunderlying these earlier studies the base pairing occurs in a muchmore geometrically constrained context, which is avoided by thepresence of the AuNP coated with longer DNA strands in the ex-periments that are modeled here. Also, the corrections in thesestudies are directly to the base pairing free energy while weconsider enthalphy and entropy separately to model the entiretemperature dependence. Since the free energy results from a delicate balance between enthalpic and entropic contributions,the free energy may be more sensitive than enthalpy or entropyalone. We conclude that our fit parameters and literature val-ues for the base pairing parameters agree well, providing furtherindependent validation of our model. For our fitting we use the data published by Johnson et al. aver-aged over both experimental replicates and direction of temper-ature change, with the width of the curves corresponding to thestandard deviation over all of these data sets (see section 2.2.2for details). Although we model this system as an equilibriumprocess, it is important to note that there is hysteresis in the ex-perimental data between the annealing, which exhibits a slightlylower transition temperature, and the melting, which exhibitsa slightly higher transition temperature. In order to verify thatthe average of these melting and annealing curves is a good ap-proximation to an equilibrium condition, we perform the ther-mal actuation experiment at two different rates of temperaturechange: ◦ C/min (as previously done by Johnson et al. ) and . ◦ C/min. For these experiments, we replaced the AuNP withdouble-stranded DNA linkers to avoid potential AuNP degrada-tion with extended time at elevated temperatures . These ex-periments reveal that as the rate is decreased, the hysteresis alsodecreases and both the melting and annealing curves approachthe average of the fast rate curves. Additionally, the average of theslow rate curves is similar to the average of the fast rate curves.This data, shown in Fig. S3, illustrates that the averaging is areasonable approximation to the equilibrium conditions.Nevertheless, we also test the model in the two extreme as-sumptions that the true equilibrium is either the melting or an-nealing data. These fits are shown in Fig. S4, with best fit param-eters reported in Table S2. While these fits result in RMS valuesthat are somewhat higher than the fits to averaged data (0.07and 0.08 for bivalent and trivalent, respectively), the values ofthe base pairing parameters still agree well with literature values,particularly when relying on just the annealing. We conclude thatour observations concerning the validity of the thermodynamicmodel are robust to the details of the treatment of the experi-mentally observed hysteresis. The parameter N c , max of our model is the maximal number ofoverhangs that participate in base pairing simultaneously. In thebivalent case, only two overhangs are present, but since in thetrivalent case three overhangs are present, it is somewhat surpris-ing that N c , max = provides the best fit to our model. In orderto evaluate if N c , max = really provides a significantly better fitof the experimental data than the more natural N c , max = , we fitthe model again varying the free energy fit parameters ∆ H cl , ∆ S cl , ∆ H bp , and ∆ S bp but keeping the maximal number N c , max = ofsimultaneous overhangs fixed. These fits are shown in Supple-mentary Fig. S5, with parameter values given in Table S3. The Journal Name, [year], [vol.][vol.] , imultaneous fit to all six experimental conditions is of low qual-ity (RMS = . in both the bivalent and trivalent cases). Fittingthe bivalent and trivalent data alone gives good fits with RMSvalues of 0.05 and 0.06, respectively (note that N c , max = is iden-tical to N c , max = for the bivalent case since only two overhangscan bind). However, when only one of these data sets is fit, theprediction of the data set excluded from the fit is completely in-consistent with the experimental results. In addition, the best fitvalues for the base pairing enthalpy and entropy are very incon-sistent with SantaLucia’s literature values except for the case ofthe fit to the bivalent data alone, where N c , max = and N c , max = are equivalent. We therefore conclude that even in the presenceof three overhangs, simultaneous binding must be constrained toat most two of these overhangs.For completeness, we also show the fits for a maximal number N c , max = of simultaneously binding overhangs in SupplementaryFig. S6, with parameter values in Table S4. These fits and/orthe agreement with literature expectation of their resulting basepairing parameter values are poor, so we conclude that more thanone overhang must be involved in simultaneous binding.Although a determination of the mechanism of the constraintof at most two simultaneously bound overhangs is beyond thescope of this study, there are a number of possible explanations.Previously, Shi et al. observed a confinement effect via molecularsimulation that is in general favorable to base pairing . How-ever, Marras et al. observed a significant decrease in base pair-ing favorability due to strand confinement upon comparison of astatistical mechanical model to experiment , and Jonchhe et al. observed a large decrease in stability of duplex DNA when a DNAhairpin is confined to a 15x15 nm nanocage . They attributedthis reduced stability to reduced water activity that results fromthe strong alignment of water molecules with the charged envi-ronment . These studies suggest that localization of the over-hang DNA strands due to hybridization could introduce a confine-ment effect that decreases the favorability of base pairing, whichin turn could set an effective strand limit on binding. In our case,the decrease in base pairing favorability could be driven by phys-ical distortions of the preferred dsDNA geometry in a confinedspace (i.e. the the dsDNA may be forced into unfavorable bendsor twists in order to bind properly). Additionally, there may bean increasingly high energy cost to bring more negative charges(both the DNA bases and AuNP are negatively charged) into asmall space, in particular if the screening is reduced by confine-ment, as suggested by Jonchhe et al. There also may be singlestranded AuNP strands within the space enclosed by the threehinge overhangs, the hinge arm, and the bottom surface of thenanoparticle. If this were the case, the configurational entropyof these strands could decrease sharply upon binding of a thirdstrand, rendering this energetically unfavorable.It is also possible that there are simply only two AuNP DNAstrands available to bind to the overhangs, or that the the numberof available AuNP strands in the sample have some distributionsuch that they on average behave as if two strands are available.Such an effect could originate from a low density of DNA on theAuNP. Finally, the geometry of the connections on top of the hingemay fix the position of the AuNP relative to the bottom hinge arm in such a way that only (effectively) two strands are able to accessthe bottom arm overhangs.
Having a validated thermodynamic model allows elucidation ofthe mechanisms of actuation, including the difference betweenthe bivalent and the trivalent case. Since the maximum numberof simultaneously bound overhangs is N c , max = , it may be sur-prising that there is a difference between the bivalent and triva-lent cases at all, since the lowest free energy state is the samein both cases, corresponding to two overhangs being fully basepaired with complementary strands on the AuNP. Yet, the experi-mental data clearly differs between the bivalent and the trivalentcase, and the model quantitatively captures this difference. Themodel reveals that the difference between the bivalent and thetrivalent case comes solely from configurational entropy: in thebivalent system there are two possible ways one overhang canbind the AuNP and one way two overhangs can bind the AuNP; inthe trivalent system there are three possible ways one overhangcan bind the AuNP and three possible ways two overhangs canbind the AuNP. Thus, there are more microstates in the trivalentsystem than in the bivalent system, even if the energetics of thesestates are identical. In fact, if all three overhangs were able tobind, a much larger change in actuation temperatures would beexpected (Fig. S5). Therefore, we conclude that the additionalconfigurational entropy in the trivalent case causes the shifts ofthe actuation points toward higher temperatures. The thermodynamic model also elucidates the role of sliding andfraying in the binding between hinge overhangs and strands at-tached to the AuNP. As shown in Fig. 2(B) the model takes intoaccount fraying of the base pairing between hinge and AuNP over-hangs at each end as well as arbitrary sliding of the two strandsrelative to each other (since they are homopolymers). Supple-mentary Fig. S7 (parameters in Table S5) shows fits for variantsof the model in which fraying and sliding (A), fraying (B), andsliding (C) are not allowed (see ESI section 5.2†). While the RMSvalues are similar to those in Fig. 3, the agreement with SantaLu-cia’s base pairing parameters is significantly worse for the casesin which fraying is not allowed, suggesting suboptimal annealingstates are important in regulating the thermal actuation.Interestingly, however, the sliding states do not seem to playas important a role. The RMS values are again similar to those inFig. 3, but here the entropic base pairing parameter is closer to ex-pectation while the enthalpic one shows a bigger discrepancy. Weare more confident about the expected value of the enthalpic basepairing parameter, since it should not be impacted by salt ;hence, we think it likely that sliding is having some impact thatthe model is capturing. Since the extent to which sliding influ-ences the actuation is unknown, we conclude that it is prudent toleave sliding states in the model.
Journal Name, [year], [vol.] , .8 Quantitative model allows design of devices with arbi-trary transition temperatures The model shows excellent agreement with the data, and is ableto predict the change in actuation response due to a change inoverhang valency and length. This predictive power may be use-ful to achieve actuation at a desired temperature, avoiding costlyexperimental trial and error. the model predicts that manipula-tion of overhang design parameters can be used to achieve ac-tuation at essentially any desired temperature in the range fromabout ◦ C to ◦ C as shown Fig. 4(A). The predicted actuationtemperatures of each design are included in Table 2.To validate the ability to guide design, we show in Fig. 4(B)that three of these predicted actuation curves agree very well(RMS = . ) with experimental data that were not used in modeldevelopment. These data correspond to AuNP nano-hinges with:two 9-base polyA overhangs ( A , ), previously published by John-son et al. ; two 6-base overhangs and one 8-base overhang( A , , ); and two 8-base overhangs and one 6-base overhang( A , , ). The two latter data sets are original to this work and thustheir raw melting and annealing curves are shown in Fig. S8. Weconclude that the model can be used to design overhang combi-nations with essentially arbitrary actuation temperatures in therange from about ◦ C to ◦ C . Fig. 4 (A) Temperature spread of predicted actuation curves for hy-pothetical nano-hinges with variable overhang length and valency. Acomplete description of the designs sorted by actuation temperature canbe found in Table 2. Solid and grayed lines are the curves fit to data(trivalent and bivalent 6, 7, and 8 base overhangs), while dashed linesindicate predictions. The violet, pink, and maroon dashed lines indicatepredictions validated (in the same colors) in (B). As mentioned above,(B) gives validation of the A , , , A , , , and A , predictions via a com-parison to experimental data (plotted with thin lines in the same colors)which were not used to create or fit the model. We also considered the possibility of extrapolation of the modelto higher and lower temperatures. Johnson et al. , however, didnot observe transitions in the case of a trivalent hinge with 9-baseoverhangs ( A , , ) up to ◦ C . This disagrees with the model Table 2
Actuation temperatures (i.e. the temperatures at which themodel predicts that the hinge is equally likely to be open or closed) ofhinges shown in Fig. 4. Each number corresponds to the length of anoverhang, so designs with two numbers are bivalent hinges and designswith three numbers are trivalent hinges. Actuation temperatures canbe tuned to within ◦ C of any desired temperature between ◦ C and ◦ C Design T act ( ◦ C) Design T act ( ◦ C)6,6 30.4 6,6,8 38.85,6,6 32.2 7,7,7 39.96,7 33.7 8,8 40.66,6,6 34.2 6,8,8 41.95,6,7 35.1 7,8,8 42.87,7 36.4 8,8,8 43.84,5,9 37.2 9,9 43.8 prediction of an equilibrium actuation temperature of . ◦ C for A , , , so it would seem that the model should not be extrapolatedto temperatures above the previously indicated ◦ C .Since the nano-hinges themselves melt around ◦ C , itis not entirely unexpected that the model begins to break down athigher temperatures. As the temperature increases and individualsections of the hinge begin to melt, the hinge may become moreflexible, effectively decreasing ∆ G cl and thus increasing the prob-ability of closed states. Additionally, some local melting of theconnections that hold the nano-hinge overhangs in place couldgrant them more flexibility and remove the N c,max = constraint.If this constraint were removed with all other best fit parame-ters kept the same, the model would predict A , , actuation at . ◦ C , which is consistent with the experimental data.At lower temperatures, Johnson et al. did not observe actua-tion in either a bivalent or trivalent hinge with 5-base overhangs( A , and A , , ) down to ◦ C . This represents a discrepancyof only a few degrees with the model prediction of equilibriumactuation temperatures of . ◦ C and . ◦ C for A , and A , , ,respectively. Since it is unlikely that the nano-hinge changes sig-nificantly at the lower end of the temperature range, this dis-crepancy could be due to slower kinetics at lower temperatures,which may result in hinges never fully closing on experimentaltimescales. The fast, accurate, and predictive thermodynamic DNA origamiactuation model developed in this work offers a viable alterna-tive to computationally costly molecular dynamics modeling inthe design of dynamic DNA origami devices. We have shownthat not only is it useful as a design tool, but it is able to pro-vide mechanistic insight into the actuation process that suggestfuture avenues of experimental research. In particular, determi-nation of the precise physical origin of the N c,max = constraintmay increase understanding of composite DNA nano-structures.For example, experiments that examine the effect of variation ofdensity of the DNA coating the AuNP would be interesting, al-though this is challenging to control precisely in practice. Further-more, the creation of increasingly complex dynamic DNA devicesnecessitates increasingly computationally efficient modelling , ofwhich statistical mechanics is likely to be an important part. Thistype of model is in principle applicable to any device that is actu- Journal Name, [year], [vol.][vol.]
Actuation temperatures (i.e. the temperatures at which themodel predicts that the hinge is equally likely to be open or closed) ofhinges shown in Fig. 4. Each number corresponds to the length of anoverhang, so designs with two numbers are bivalent hinges and designswith three numbers are trivalent hinges. Actuation temperatures canbe tuned to within ◦ C of any desired temperature between ◦ C and ◦ C Design T act ( ◦ C) Design T act ( ◦ C)6,6 30.4 6,6,8 38.85,6,6 32.2 7,7,7 39.96,7 33.7 8,8 40.66,6,6 34.2 6,8,8 41.95,6,7 35.1 7,8,8 42.87,7 36.4 8,8,8 43.84,5,9 37.2 9,9 43.8 prediction of an equilibrium actuation temperature of . ◦ C for A , , , so it would seem that the model should not be extrapolatedto temperatures above the previously indicated ◦ C .Since the nano-hinges themselves melt around ◦ C , itis not entirely unexpected that the model begins to break down athigher temperatures. As the temperature increases and individualsections of the hinge begin to melt, the hinge may become moreflexible, effectively decreasing ∆ G cl and thus increasing the prob-ability of closed states. Additionally, some local melting of theconnections that hold the nano-hinge overhangs in place couldgrant them more flexibility and remove the N c,max = constraint.If this constraint were removed with all other best fit parame-ters kept the same, the model would predict A , , actuation at . ◦ C , which is consistent with the experimental data.At lower temperatures, Johnson et al. did not observe actua-tion in either a bivalent or trivalent hinge with 5-base overhangs( A , and A , , ) down to ◦ C . This represents a discrepancyof only a few degrees with the model prediction of equilibriumactuation temperatures of . ◦ C and . ◦ C for A , and A , , ,respectively. Since it is unlikely that the nano-hinge changes sig-nificantly at the lower end of the temperature range, this dis-crepancy could be due to slower kinetics at lower temperatures,which may result in hinges never fully closing on experimentaltimescales. The fast, accurate, and predictive thermodynamic DNA origamiactuation model developed in this work offers a viable alterna-tive to computationally costly molecular dynamics modeling inthe design of dynamic DNA origami devices. We have shownthat not only is it useful as a design tool, but it is able to pro-vide mechanistic insight into the actuation process that suggestfuture avenues of experimental research. In particular, determi-nation of the precise physical origin of the N c,max = constraintmay increase understanding of composite DNA nano-structures.For example, experiments that examine the effect of variation ofdensity of the DNA coating the AuNP would be interesting, al-though this is challenging to control precisely in practice. Further-more, the creation of increasingly complex dynamic DNA devicesnecessitates increasingly computationally efficient modelling , ofwhich statistical mechanics is likely to be an important part. Thistype of model is in principle applicable to any device that is actu- Journal Name, [year], [vol.][vol.] , ted by melting/annealing of DNA duplexes, and similar methodshave been shown to be applicable to other dynamic structures .Attempts to use statistical mechanics methods to model a widerrange of devices, as well as the incorporation of kinetics usingtransition matrices acting on microstates (such as the ones de-fined here) to capture non-equilibrium effects, are important ar-eas of future research. Conflicts of interest
There are no conflicts to declare.
Acknowledgements
We thank the Winter lab for providing nanoparticles for the ad-ditional experiments and valuable feedback on the work. Thismaterial is based upon work supported by the National ScienceFoundation under Grant No. DMR-1719316 to RB and by the De-partment of Energy under Grant no. DE-SC0017270 to CC.
References
Nature , 2006, , 297–302.2 M. DeLuca, Z. Shi, C. E. Castro and G. Arya,
Nanoscale Horiz. ,2020, , 182–201.3 A. E. Marras, L. Zhou, H.-J. Su and C. E. Castro, PNAS , 2015, , 713–718.4 S. M. Douglas, I. Bachelet and G. M. Church,
Science , 2012, , 831–834.5 H. Ijäs, I. Hakaste, B. Shen, M. A. Kostiainen and V. Linko,
ACS Nano , 2019, , 5959–5967.6 P. Ketterer, E. M. Willner and H. Dietz, Sci. Adv. , 2016, ,e1501209.7 D. Zhao, J. V. Le, M. A. Darcy, K. Crocker, M. G. Poirier, C. Cas-tro and R. Bundschuh, Biophys. J. , 2019, , 2204–2216.8 J. V. Le, Y. Luo, M. A. Darcy, C. R. Lucas, M. F. Goodwin, M. G.Poirier and C. E. Castro,
ACS Nano , 2016, , 7073–7084.9 T. Gerling, K. F. Wagenbauer, A. M. Neuner and H. Dietz, Sci-ence , 2015, , 1446–1452.10 E. Kopperger, J. List, S. Madhira, F. Rothfischer, D. C. Lamband F. C. Simmel,
Science , 2018, , 296–301.11 S. Nummelin, B. Shen, P. Piskunen, Q. Liu, M. A. Kostiainenand V. Linko,
ACS Synth. Biol. , 2020, , 1923–1940.12 F. C. Simmel, B. Yurke and H. R. Singh, Chem. Rev. , 2019, , 6326–6369.13 D. Y. Zhang and G. Seelig,
Nat. Chem. , 2011, , 103–113.14 B. Yurke, A. J. Turberfield, A. P. Mills, F. C. Simmel and J. L.Neumann, Nature , 2000, , 605–608.15 A. E. Marras, Z. Shi, M. G. Lindell III, R. A. Patton, C.-M.Huang, L. Zhou, H.-J. Su, G. Arya and C. E. Castro,
ACS Nano ,2018, , 9484–9494.16 C. Mao, W. Sun, Z. Shen and N. C. Seeman, Nature , 1999, , 144–146.17 J. M. Majikes, L. C. Ferraz and T. H. LaBean,
BioconjugateChem. , 2017, , 1821–1825.18 D. Liu, A. Bruckbauer, C. Abell, S. Balasubramanian, D.-J.Kang, D. Klenerman and D. Zhou, J. Am. Chem. Soc. , 2006, , 2067–2071. 19 T. Li and M. Famulok,
J. Am. Chem. Soc. , 2013, , 1593–1599.20 S. Modi, M. Swetha, D. Goswami, G. D. Gupta, S. Mayor andY. Krishnan,
Nat. Nanotechnol. , 2009, , 325–330.21 P. M. Arnott and S. Howorka, ACS Nano , 2019, , 3334–3340.22 V. A. Turek, R. Chikkaraddy, S. Cormier, B. Stockham, T. Ding,U. F. Keyser and J. J. Baumberg, Adv. Funct. Mater. , 2018, ,1706410.23 J. A. Johnson, A. Dehankar, J. O. Winter and C. E. Castro, Nano Lett. , 2019, , 8469–8475.24 N. Liu and T. Liedl, Chem. Rev. , 2018, , 3032–3053.25 T. Bayrak, N. S. Jagtap and A. Erbe,
Int. J. Mol. Sci. , 2018, ,3019.26 A. O. Govorov and H. H. Richardson, Nano today , 2007, ,30–38.27 C.-Y. Li, E. A. Hemmig, J. Kong, J. Yoo, S. Hernández-Ainsa,U. F. Keyser and A. Aksimentiev, ACS Nano , 2015, , 1420–1433.28 N. Wu, D. M. Czajkowsky, J. Zhang, J. Qu, M. Ye, D. Zeng,X. Zhou, J. Hu, Z. Shao, B. Li et al. , J. Am. Chem. Soc. , 2013, , 12172–12175.29 J. Yoo and A. Aksimentiev,
PNAS , 2013, , 20099–20104.30 T. E. Ouldridge, A. A. Louis and J. P. Doye,
J. Chem. Phys. ,2011, , 02B627.31 P. Šulc, F. Romano, T. E. Ouldridge, L. Rovigatti, J. P. Doyeand A. A. Louis,
J. Chem. Phys. , 2012, , 135101.32 B. E. Snodin, F. Randisi, M. Mosayebi, P. Šulc, J. S. Schreck,F. Romano, T. E. Ouldridge, R. Tsukanov, E. Nir, A. A. Louis et al. , J. Chem. Phys. , 2015, , 06B613_1.33 R. Sharma, J. S. Schreck, F. Romano, A. A. Louis and J. P.Doye,
ACS Nano , 2017, , 12426–12435.34 C. Maffeo and A. Aksimentiev, Nucleic Acids Res. , 2020, ,5135–5146.35 D.-N. Kim, F. Kilchherr, H. Dietz and M. Bathe, Nucleic AcidsRes. , 2012, , 2862–2868.36 K. Pan, W. P. Bricker, S. Ratanalert and M. Bathe, Nucleic AcidsRes. , 2017, , 6284–6298.37 N. Srinivas, T. E. Ouldridge, P. Šulc, J. M. Schaeffer, B. Yurke,A. A. Louis, J. P. Doye and E. Winfree, Nucleic Acids Res. , 2013, , 10641–10658.38 P. Irmisch, T. E. Ouldridge and R. Seidel, J. Am. Chem. Soc. ,2020, , 11451–11463.39 E. Stahl, T. G. Martin, F. Praetorius and H. Dietz,
Angew.Chem. , 2014, , 12949–12954.40 F. Tanaka, A. Kameda, M. Yamamoto and A. Ohuchi,
Biochem-istry , 2004, , 7143–7150.41 D. H. Turner and D. H. Mathews, Nucleic Acids Res. , 2010, ,D280–D282.42 H. T. Allawi and J. SantaLucia Jr, Nucleic Acids Res. , 1998, ,2694–2701.43 N. Peyret, P. A. Seneviratne, H. T. Allawi and J. SantaLucia, Biochemistry , 1999, , 3468–3477. Journal Name, [year], [vol.] , PNAS , 1998, , 1460–1465.45 M. A. Branch, T. F. Coleman and Y. Li, SIAM Journal on Scien-tific Computing , 1999, , 1–23.46 P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber-land, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson,W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wil-son, K. Jarrod Millman, N. Mayorov, A. R. J. Nelson, E. Jones,R. Kern, E. Larson, C. Carey, ˙I. Polat, Y. Feng, E. W. Moore,J. Vand erPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Hen-riksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H.Ribeiro, F. Pedregosa, P. van Mulbregt and SciPy 1.0 Contrib-utors, Nat. Methods , 2020, , 261–272.47 R. Owczarzy, B. G. Moreira, Y. You, M. A. Behlke and J. A.Walder, Biochemistry , 2008, , 5336–5353.48 M. T. Record and T. M. Lohman, Biopolymers: Original Re-search on Biomolecules , 1978, , 159–166. 49 D. Erie, N. Sinha, W. Olson, R. Jones and K. Breslauer, Bio-chemistry , 1987, , 7150–7159.50 Z. Shi and G. Arya, Nucleic Acids Res. , 2020, , 548–560.51 F. Li, H. Zhang, B. Dever, X.-F. Li and X. C. Le, BioconjugateChem. , 2013, , 1790–1797.52 S. Jonchhe, S. Pandey, D. Karna, P. Pokhrel, Y. Cui, S. Mishra,H. Sugiyama, M. Endo and H. Mao, J. Am. Chem. Soc. , 2020, , 10042–10049.53 S. Jonchhe, S. Pandey, T. Emura, K. Hidaka, M. A. Hossain,P. Shrestha, H. Sugiyama, M. Endo and H. Mao,
PNAS , 2018, , 9539–9544.54 X. Wei, J. Nangreave, S. Jiang, H. Yan and Y. Liu,
J. Am. Chem.Soc. , 2013, , 6165–6176.55 C. E. Castro, F. Kilchherr, D.-N. Kim, E. L. Shiao, T. Wauer,P. Wortmann, M. Bathe and H. Dietz,
Nat. Methods , 2011, ,221.
10 | 1–14
Journal Name, [year], [vol.][vol.]
Journal Name, [year], [vol.][vol.] , Supplementary material
We calculate the salt corrected base pairing parameter values us-ing the melting temperature correction T m ( Mg + ) = T m ( + )+ a + b ln [ Mg + ]+ f GC ( c + d ln [ Mg + ])+ ( N bp − ) (cid:26) e + f ln [ Mg + ] + g ( ln [ Mg + ]) (cid:27) (7)given by Owczarzy et al. , where a , b , c , d , e , f , and g are exper-imentally determined constants. We use the magnesium salt cor-rection since our and Johnson et al. ’s experiments are performedin the presence of . mM free magnesium. Since we care aboutthe change in base pairing energy for a generic internal base, weignore edge effects by taking the limit as N bp approaches infinity.Furthermore, our sequences consist solely of A’s and T’s, so thefraction of G and C bases f GC = . Thus the melting temperaturein our case is given by T m ( Mg + ) = T m ( + ) + (cid:0) . − . [ . ] (cid:1) × − (8)where we have substituted the appropriate constants for a and b .Since the melting temperature is the temperature at which thebase pairing energy change is 0, we can write the melting temper-ature in terms of the base pairing entropy and enthalpy changeas T m = ∆ H bp / ∆ S bp . (9)Assuming that salt concentration only impacts base pairing en-tropy , the corrected base pairing entropy change ∆ S bp , c isgiven by ∆ S bp , c = ∆ H bp T m ( Mg + ) (10)with T m ( Mg + ) from Eq. (8). In order to disentangle the effects of fraying and sliding of theDNA duplexes between the overhangs and the DNA strands onthe AuNP, we use alternative models that exclude each of themseparately or both of them. Fraying and sliding enter the origi-nal model via the multiplicities in Eq. (5). We thus here providealternative versions of Eq. (5) for the three different cases thatexclude fraying and/or sliding.
In order to exclude fraying and sliding, Eq. (5) becomes Z S , j = exp (cid:20) − ∆ G term + N S , j ∆ G s k B T (cid:21) , (11)since there is now only a single state allowed with free energy ∆ G term + N S , j ∆ G s . In order to exclude only fraying, Eq. (5) becomes Z S , j = ( | N T − N A , j | + ) exp (cid:20) − ∆ G term + N A , j ∆ G s k B T (cid:21) , (12)where the prefactor is simply the number of positions available tocompletely paired strands. In order to exclude only sliding, Eq. (5) becomes Z S , j = N S , j ∑ i = ( N S , j − i + ) exp (cid:20) − ∆ G term + i ∆ G s k B T (cid:21) , (13)where the sum is over the number of possible base paired stacks,and the prefactor gives the multiplicity of each of these stacks. Table S1
Fixed base pairing parameters in units of kJ/mol for enthalpiesand kJ/mol K for entropies. Values without errors are fixed.
Data fit ∆ H cl ∆ S cl ∆ H bp ∆ S bp N c , max Enthalpy fixed ± . − . ± . − . − . ± . ± . − . ± . − . − . Table S2
Melt and anneal only parameters in units of kJ/mol for en-thalpies and kJ/mol K for entropies.
Data fit ∆ H cl ∆ S cl ∆ H bp ∆ S bp N c , max Melt ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . Table S3 N c , max = fit parameters in units of kJ/mol for enthalpies andkJ/mol K for entropies. Data fit ∆ H cl ∆ S cl ∆ H bp ∆ S bp N c , max Tri. & Bi. ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . Table S4 N c , max = fit parameters in units of kJ/mol for enthalpies andkJ/mol K for entropies. Data fit ∆ H cl ∆ S cl ∆ H bp ∆ S bp N c , max Tri. & Bi. ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . Table S5
No fraying and no sliding fit parameters in units of kJ/mol forenthalpies and kJ/mol K for entropies.
Data fit ∆ H cl ∆ S cl ∆ H bp ∆ S bp N c , max No slide or fray ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . ± − . ± . − . ± . − . ± . Journal Name, [year], [vol.] , .4 Supplemental figures Fig. S1
Effect of overhang strand valency on actuation curves. The ex-perimental data (thin lines) and model fits (thick lines) are identical to theones shown in Fig. 3(A) but rather than separating the data by valency,each panel shows the bivalent (orange) and the trivalent (turquoise) casefor (A) 6 base polyA overhangs, (B) 7 base polyA overhangs, and (C) 8base polyA overhangs.
Fig. S2
Model with N c , max = fit to experimental datasets. In (A),base pairing enthalpy ∆ H bp is fixed at the SantaLucia expected value,and in (B) both the base pairing enthalpy and the salt corrected basepairing entropy ∆ S bp are fixed. In both cases, the model is fit to bivalentand trivalent data simultaneously. Each panel contains the average rootmean squared difference ( RMS ) between the model and the average ofthe experimental data. The rightmost column shows the ratio betweenthe fit base pairing parameter values and the salt corrected expectedbase pairing parameter values given by Eq. 10. Since the outlined barsare fixed to the expected values, they have an actual to expected valueratio of 1.
Fig. S3
Comparison of melting and annealing curves for linker DNA(with no AuNP) with varied rates. These experiments are similar tothose with DNA nano-hinge containing AuNP, except that the duplexesfrom the top arm are extended and the AuNP removed. These duplexeshave polyT single-stranded portions on the ends that mimic the singlestranded polyTs that normally coat the AuNP. In particular, fabricationincluded hinges with 10 fold excess of linkers ( nM) using the samefolding protocol as for all other hinges. The left panel shows melting andannealing curves for two linkers annealing to 8 base overhangs, while theright panel shows melting and annealing curves for three linkers annealingto 7 base overhangs. For the red curves the temperature is changed at arate of ◦ C / min , while for the green curves the temperature is changedat a rate of . ◦ C / min .
12 | 1–14
Journal Name, [year], [vol.][vol.]
Journal Name, [year], [vol.][vol.] , ig. S4 Model with N c , max = fit to experimental datasets containingonly melting (A) and only annealing (B) data. In both cases, the modelis fit to bivalent and trivalent data simultaneously. Each panel containsthe root mean squared difference ( RMS ) between the model and theaverage of the experimental data. The rightmost column shows the ratiobetween the fit base pairing parameter values and the expected basepairing parameter values . Fig. S5
Model with N c , max = fit to experimental data. In row (A),the model is fit to bivalent and trivalent data simultaneously. In row(B), the model is fit to the bivalent data (solid lines), and the trivalentmodel with bivalent fit parameters is compared to trivalent data (dashedlines). In row (C), the model is fit to the trivalent data (solid lines), andthe bivalent model with trivalent fit parameters is compared to bivalentdata (dashed lines). Each panel contains the average root mean squareddifference ( RMS ) between the model and the average of the experimentaldata. The rightmost column shows the ratio between the fit base pairingparameter values and the expected base pairing parameter values . Fig. S6
Model with N c , max = fit to experimental data. In row (A),the model is fit to bivalent and trivalent data simultaneously. In row(B), the model is fit to the bivalent data (solid lines), and the trivalentmodel with bivalent fit parameters is compared to trivalent data (dashedlines). In row (C), the model is fit to the trivalent data (solid lines), andthe bivalent model with trivalent fit parameters is compared to bivalentdata (dashed lines). Each panel contains the average root mean squareddifference ( RMS ) between the model and the average of the experimentaldata. The rightmost column shows the ratio between the fit base pairingparameter values and the expected base pairing parameter values . Fig. S7
Model with N c , max = with exclusion of fraying and/or sliding fitto experimental bivalent and trivalent data simultaneously. In row (A),both fraying and sliding are disallowed. In row (B), sliding is allowed,but fraying is disallowed. In row (C), fraying is allowed, but sliding isdisallowed. Each panel contains the average root mean squared difference( RMS ) between the model and the average of the experimental data. Therightmost column shows the ratio between the fit base pairing parametervalues and the expected base pairing parameter values . Journal Name, [year], [vol.] , ig. S8 Comparison of melting and annealing curves for mixed hinges.The left panel shows melting (solid) and annealing (dotted) curves fortwo experimental replicates of a nano-hinge with two 6-base overhangsand one 8-base overhang, and the right panel shows melting (solid) andannealing (dotted) curves for one experimental replicate of a nano-hingewith one 6-base overhang and two 8-base overhangs.
14 | 1–14
Journal Name, [year], [vol.][vol.]