Computing Nonequilibrium Conformational Dynamics of Structured Nucleic Acid Assemblies

Transcription

1 Supportng Informaton for Computng Nonequlbrum Conformatonal Dynamcs of Structured Nuclec Acd Assembles Reza Sharf Sedeh,, Keyao Pan,, Matthew Ralph Adendorff, Oskar Hallatschek, Klaus-Jürgen Bathe,*, and Mark Bathe*, Department of Mechancal Engneerng and Department of Bologcal Engneerng, Massachusetts Insttute of Technology, Cambrdge, Massachusetts 02139, Unted States Department of Physcs, Unversty of Calforna, Berkeley, Berkeley, Calforna 94720, Unted States These authors contrbuted equally to ths work. Correspondng Authors *E-mal: *E-mal: 1

2 Supportng Text Basepar frayng n the MD smulaton The all-atom MD smulaton allows frayng of double-stranded DNA duplex ends. For a gven bp n a MD frame, ts coarse-graned poston and orentaton are mssng n the frame f t frays. In the entre MD trajectory, the maxmum percentage of frames n whch the j th bp frays s 66.8%, j = 1, 2,, 204. If 1, 2, or 3 bps are removed at each end of the four duplexes, such maxmum percentage decreases to 22.3%, 0.4%, and 0.4%, respectvely. Therefore two bps are removed at each end of the four duplexes, resultng n 188 bps modeled by 196 FE nodes n the downstream analyss of the MD trajectory. Comparson of analytcal and numercal Brownan and vacuum egensolutons of free and cantlevered dsdna Here we compare the analytcal soluton for the relaxaton tmes and BM shapes and the natural frequences and vacuum NM shapes of two-dmensonal 10 nm length free and cantlevered cylnders wth radus 1 nm wth our proposed model wthout hydrodynamc nteractons. We also examne the mpact of hydrodynamc nteractons on the dynamcs of the free and cantlevered cylnders. The 10 nm length cylnders are model representatons of DNA helces wth 29 basepars, whch are modeled wth 28 two-node beam elements. The dynamcal behavor of the cylnders s analyzed n the 2D plane comprsed of the major axs and one of ts two equvalent perpendcular axes. Beads of 1 nm radus are located at the nodal postons to model dampng effects of solvent. The stffness and mass matrces, K and M, are obtaned from the beam model and the frcton matrx, Z, s calculated usng the bead model. For the 2D cylnders, only the drag coeffcents and the components of the mass matrx correspondng to the perpendcular axs are consdered to be non-zero. Although hydrodynamc nteractons between beads are always explctly ncluded n our computatonal model, as explaned n the man text (see Eq. 1), to analyze the mpact of hydrodynamc nteractons on BM shapes and relaxaton tmes we compute results for both cases: a cylnder wth hydrodynamc nteractons and a cylnder wthout hydrodynamc nteractons. The analytcal soluton for the relaxaton tmes, τ, and BM shapes, w, of a free cylnder wth Young's modulus E, mass densty ρ, bendng moment of nerta I, length L, cross-sectonal area A, and perpendcular drag force per unt length γ, s, 1 τ γ 4 EI L p w s L = sn p s L + snh p s L sn p + snh p s + cos p L + cosh p s L cos p cosh p (S1) (S2) where p , p , p , and p π( + 1/2) for > 3. Here, s s the contour length of the cylnder. The analytcal soluton for the natural frequences, f, and vacuum NM shapes, v, of the free cylnder s, 2 2

3 f 1 2 2π EI ρa p L (S3) v s L = sn p s L + snh p s L + sn p snh p cosh p cos p cos p s L + cosh p s L (S4) whle the analytcal soluton for the relaxaton tmes and BM shapes of a cantlevered cylnder s, 3 τ γ 4 EI L q (S5) w s L = cosh(q ) cos(q ) sn(q ) + snh(q ) sn q s L snh q s L + cos q s L cosh q s L (S6) where q , q , q , and q π( 1/2) for > 3. Also the analytcal soluton for the natural frequences, f, and vacuum NM shapes, v, of the cantlevered cylnder s as follows, 2 f 1 2 2π EI ρa q L (S7) v s L = sn q s L snh q s L sn q + snh q cos q + cosh q cos q s L cosh q s L. (S8) In the analytcal soluton, γ s, 4 γ = 4πμ ln(l/d) + 2 ln(2) 1/2 (S9) where d s the dameter of the cylnder and μ s the vscosty of the solvent. To obtan the computatonal modelng results wthout hydrodynamc nteractons, we obtan the perpendcular drag on a cylnder usng the proposed bead model and then dvde the obtaned drag force by the length of the cylnder to calculate the perpendcular drag coeffcent. The frst 10 non-rgd-body BM shapes and ther correspondng relaxaton tmes of the free and cantlevered cylnders (DNA helces) obtaned from the analytcal soluton are smlar to those of the computatonal modelng results n whch hydrodynamc nteractons are absent (Fgure S17). 3

4 Fgure S1. Shapes of the BMs of the DNA tweezer. (a) Three orthogonal vews of the atomc model of the DNA tweezer n the mechancal ground state. (b f) Three orthogonal vews of the atomc models of the frst fve non-rgd-body BMs. 4

5 Fgure S2. Shapes of the NMs of the DNA tweezer. (a) Three orthogonal vews of the atomc model of the DNA tweezer n the mechancal ground state. (b f) Three orthogonal vews of the atomc models of the frst fve non-rgd-body NMs. 5

6 Fgure S3. Shapes of the MD EMs of the DNA tweezer. (a) Three orthogonal vews of the atomc model of the DNA tweezer n the mechancal ground state. (b f) Three orthogonal vews of the atomc models of the frst fve MD EMs. Due to bp frayng, two bps at each duplex end are not rendered n all the panels. 6

7 Fgure S4. Comparson between NMs and BMs of the DNA tweezer. (a) Smlarty between ndvdual modes. (b) Symmetrc mean overlap for the frst 20 non-rgd-body NMs and BMs. The smlarty and symmetrc mean overlap are computed as performed prevously. 5 7

8 Fgure S5. Comparson of MD EMs of the DNA tweezer to the NMs and BMs. (a) Smlarty between ndvdual modes. (b) Symmetrc mean overlap for the frst 20 MD EMs and the frst 20 non-rgd-body NMs and BMs. The smlarty and symmetrc mean overlap are computed as performed prevously. 5 8

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10 Fgure S6. Determnaton of the relaxaton tmes of the BMs of the DNA tweezer. The relaxaton tmes are obtaned by fttng exponental functons (red) to the ACFs (blue) of the BD trajectores projected onto the frst 20 non-rgd-body BMs (upper panels). Also shown are the resduals (lower panels). The shaded area represents the 95% confdence nterval. The ttle of each panel provdes the mean and 95% confdence nterval of the relaxaton tme. 10

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12 Fgure S7. Determnaton of the relaxaton tmes of the NMs of the DNA tweezer. The relaxaton tmes are obtaned by fttng exponental functons to the ACFs (blue) of the BD trajectores projected onto the frst 20 non-rgd-body vacuum NMs (upper panels). Also shown are the resduals (lower panels). The shaded area represents the 95% confdence nterval. The ttle of each panel provdes the mean and 95% confdence nterval of the relaxaton tme. 12

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14 Fgure S8. Determnaton of the relaxaton tmes of the MD EMs of the DNA tweezer. The relaxaton tmes are obtaned by fttng exponental functons (red) to the ACFs (blue) of the BD trajectores projected onto the frst 20 MD EMs (upper panels). Also shown are the resduals (lower panels). The shaded area represents the 95% confdence nterval. The ttle of each panel provdes the mean and 95% confdence nterval of the relaxaton tme. 14

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16 Fgure S9. Comparson between the BD and MD smulatons of the DNA tweezer. The relaxaton tmes are obtaned by fttng exponental functons (red) to the ACFs (blue) of the MD trajectores projected onto the frst 20 BMs (upper panels). Also shown are the resduals (lower panels). The ttle of each panel provdes the value of the relaxaton tme. 16

17 Fgure S10. Comparson of the relaxaton tmes of the BMs of the tweezer to those of the MD EMs. Relaxaton tmes of the frst 20 non-rgd-body BMs as the egenvalues (black), the same relaxaton tmes solved by calculatng ACFs of a BD trajectory projected onto the frst 20 nonrgd-body BMs (red), (a) the frst 20 MD EMs, and (b) the frst 20 BD EMs (blue). The shaded area represents the 95% confdence nterval. 17

18 Fgure S11. Relaxaton tmes of the angles n junctons J1 J6 (a f) calculated from BD and MD trajectores. Numercal ACFs are calculated from the smulated BD trajectores (BD), n whch the shaded area represents the 95% confdence nterval, and from the MD trajectory (MD). Double exponental models are ft to the ACFs calculated from the BD trajectores and the MD trajectory, respectvely (BD Ft and MD Ft). Note that the ft model of the MD ACF n juncton J1 contans a zero term. 18

19 Fgure S12. Shapes of the BMs of the nne-layer rng orgam. (a) Three orthogonal vews of the atomc model of the nne-layer rng orgam n the mechancal ground state. (b f) Three orthogonal vews of the atomc models of the frst fve non-rgd-body BMs. 19

20 Fgure S13. Shapes of the NMs of the nne-layer rng orgam. (a) Three orthogonal vews of the atomc model of the nne-layer rng orgam n the mechancal ground state. (b f) Three orthogonal vews of the atomc models of the frst fve non-rgd-body NMs. 20

21 Fgure S14. Comparson between NMs and BMs of the nne-layer rng orgam. (a) Smlarty between ndvdual modes. (b) Symmetrc mean overlap for the frst 20 non-rgd-body NMs and BMs. The smlarty and symmetrc mean overlap are computed as performed prevously. 5 21

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23 Fgure S15. Determnaton of the relaxaton tmes of the BMs of the nne-layer rng orgam. The relaxaton tmes are obtaned by fttng exponental functons (red) to the ACFs (blue) of the BD trajectores projected onto the frst 20 non-rgd-body BMs (upper panels). Also shown are the resduals (lower panels). The shaded area represents the 95% confdence nterval. The ttle of each panel provdes the mean and 95% confdence nterval of the relaxaton tme. 23

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25 Fgure S16. Determnaton of the relaxaton tmes of the NMs of the nne-layer rng orgam. The relaxaton tmes are obtaned by fttng exponental functons to the ACFs (blue) of the BD trajectores projected onto the frst 20 non-rgd-body vacuum NMs (upper panels). Also shown are the resduals (lower panels). The shaded area represents the 95% confdence nterval. The ttle of each panel provdes the mean and 95% confdence nterval of the relaxaton tme. 25

26 Fgure S17. Brownan and vacuum dynamcs of 10 nm length cylnders restrcted to planar moton at 20 C. The analytcal soluton for the frst 10 non-rgd-body BMs, ther correspondng relaxaton tmes, and also the frst 10 non-rgd-body vacuum NMs of (a) free and (b) cantlevered cylnders are compared wth computatonal modelng results obtaned usng the BMs computed usng the proposed procedure wth and wthout hydrodynamc nteractons. Mode overlap matrces between the frst 10 BMs and vacuum NMs obtaned analytcally and/or 26

27 computatonally are plotted (lower left, lower rght, and upper rght fgures respectvely n (a) and (b)). Table S1. Nucleotde sequences of the nne sngle-stranded DNAs, denoted by A H, of the DNA tweezer. 6, 7 Strand ID A (43 nts) B (59 nts) C (59 nts) D (43 nts) E (22 nts) F (54 nts) G (84 nts) H (22 nts) I (22 nts) Nucleotde sequence from the 5 -end to the 3 -end CGACCGAGCGTGAATTAGTGATCCGGAACTCGCGCAATGA ACC TCAGCTGGCCTATCTAAGACTGAACTCGCACCGCCGGCAT AAGCTATGCGCTCTGCCGC TTAGGAGATGGCACGTTAATGAATAGTCTCCACTTGCATC CGAGATCCGAACTGCTGCC CGAGAGAAGGCTTGCCAGGTTACGTTCGTACATCGTCTGA GTT GGCAGCAGTTCAGGCCAGCTGA GGTTCATTGCGGAGTTCAGTCTTAGATGGATCTCGGATGC AAGGCCTTCTCTCG GGTGCCGAGTTCCGGATCACTAATTCCATAGCTTATGCCG GCACTATTCATTAACGTGTGTACGAACGTAACCTGGCAAT GGAG GCGGCAGAGCGACGCTCGGTCG AACTCAGACGACCATCTCCTAA References 1. Howard, J., Mechancs of Motor Protens and the Cytoskeleton. Snauer Assocates, Inc., Sunderland, Massachusetts, Young, D.; Felgar, R. P., Jr., Tables of Characterstc Functons Representng Normal Modes of Vbraton of a Beam. The Unversty of Texas Publcaton, Austn, Texas, 1949; Vol Janson, M. E.; Dogterom, M. Bophys. J. 2004, 87, Wggns, C. H.; Rvelne, D.; Ott, A.; Goldsten, R. E. Bophys. J. 1998, 74, Bathe, M. Protens 2008, 70, Zhou, C.; Yang, Z.; Lu, D. J. Am. Chem. Soc. 2012, 134, Lu, M.; Fu, J.; Hejesen, C.; Yang, Y.; Woodbury, N. W.; Gothelf, K.; Lu, Y.; Yan, H. Nat. Commun. 2013, 4,