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Supplementary Online Material For: Molecular Fluctuations As a Ruler of Force-Induced Protein Conformations

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Supplementary Online Material For: Molecular Fluctuations As a Ruler of Force-Induced Protein Conformations Supplementary Online Material for: Molecular fluctuations as a ruler of force-induced protein conformations Andrew Stannard1*, Marc Mora1*, Amy E.M. Beedle1*, Marta Castro-López1, Stephanie Board1, Sergi Garcia-Manyes1,2¶ 1Department of Physics, Randall Centre for Cell and Molecular Biophysics and London Centre for Nanotechnology, King’s College London, Strand, WC2R 2LS London, United Kingdom. 2The Francis Crick Institute, 1 Midland Road, London NW1 1AT, London, UK. *These authors contributed equally to the work ¶Corresponding author: [email protected] 1 Calculation of the stiffness of the magnetic bead potential Calculating the stiffness of the magnetic bead potential can be achieved with knowledge of the specifications of the magnetic beads and magnets used. The magnetic force on a paramagnetic bead is = ( ) where is the magnetic moment of the bead ∇and is ∙ the magnetic flux density of the magnetic field; the spatial gradient of gives the stiffness of the magnetic bead potential. The magnetic moment of a paramagnetic bead will align to the field inducing the moment, with its magnitude depending on the field strength, i.e. = ( ) . It has been shown1,2 that the flux-density-dependence of the magnetic moment of an M-280 Dynabead (as used in our experiments) can be approximated by the Langevin function � ( ) = [coth( ) ] where = 1.66 × 10 Am is the saturation ⁄ magnetic0 − 0⁄ moment and = 15.5 mT a characteristic flux density−13, see2 Fig. S1A. At low flux densities, , these beads are 0 superparamagnetic ; their magnetisation saturates at high flux densities, . 0 ≪ Our magnetic tweezers set-up uses neodymium-based cylindrical magnets0 ≫ (D33-N52, K&J Magnetics), specified by radius = 2.38 mm, length = 4.76 mm, and surface flux density S = 662 mT; the latter corresponds to a magnetisation of = 1.18 × 10 Am-1 = 6 2 S( + ) . 6 of these magnets are used in total: 3 magnets are aligned magnetically- � parallel2 in face2 1-to-⁄ 2 face contact (to effectively make one magnet of length = 14.3 mm); two of these 0 stack s are aligned⁄ magnetically� -antiparallel in side-to-side contact. In cylindrical coordinates, the azimuthally-symmetric magnetic scalar potential outside a cylindrical magnetic is ( + ) ( , ) = 4 2 ( 2 cos + + ( + ) ) � � � 2 2 2 3⁄ 2 with = 0 corresponding to one0 0 of the0 magnetic − faces. From the scalar potential, the magnetic flux density can be found by = where = 1.26 × 10 H/m is the permeability − of 0free∇ space. Fig. S1B shows the calculation of the magnetic flux density in−6 the plane that transects both magnet stacks. 0 Fig. S1C showing the corresponding paramagnetic force on an M-280 Dynabead in the vicinity of the magnet stacks. Measurements are performed on magnetic beads located on the axis defined by the side-to-side/stack-to-stack contact line, with positioning achieved through lateral movement of the bead-containing sample. The magnet-bead separation, , the perpendicular distance between the bead and the magnetic faces, is controlled by vertical positioning of the magnets. Fig. S1D shows the paramagnetic force as a function of magnet-bead separation along the central axis. The stiffness of the potential experienced by the paramagnetic bead is simply given by the gradient of this force, shown in Fig. S1E over two decades of force, the typical force range used in magnetic tweezers experiments. Also shown is an estimation of the stiffness of a typical protein construct tethered to a bead in our experiments. The force-dependent stiffness of a freely-jointed chain of contour length is 2 ( ) = ( )[1 coth ( ) ( ) ] 2 2 −1 where = 4.04 pN nm is thermal⁄ energy − at room temperature⁄ − , =⁄ 1.1 nm is the Kuhn length we use for protein constructs, and = 100 nm is the order of magnitude of contour length of constructs used throughout here. Over this force range the stiffness of the protein far exceeds that of the magnetic bead potential (by 5 or more orders of magnitude), and thus the ‘trap’ stiffness experienced by magnetic beads in our magnetic tweezers experiments is completely dominated by the protein stiffness, and thus our measurements of ‘magnetic trap’ stiffness are indeed measurements of protein stiffness. By comparison, in the force range of interest here, protein stiffness is at least an order of magnitude less than a typical cantilever using for SMFS AFM (10 pN/nm), where the stiffness of the probe-protein coupled system is dominated by the probe. Fig. S1E also illustrates the insignificant change in paramagnetic force when a protein (that a bead is tethered to) folds/unfolds to lengthen/shorten the magnet-bead separation. For example, at a typical force of ~10 pN, the stiffness of the magnetic bead potential is ~10-5 pN/nm. At this force, the change in average bead position due to a protein L folding/unfolding event is ~10 nm, meaning ~10-4 pN change in force, i.e. only one part in ten thousand, thus insignificant. Figure S1. Single molecule magnetic tweezers experiments allow measurement of the probe- independent stiffness of a protein under force. (A) Estimation of the magnetic moment of an M-280 Dynabead as a function of magnetic flux density. (B) Magnetic flux density in the vicinity of the cylindrical magnets used in our experimental set-up. (C) Combining (A) and (B) gives the position- dependent force on a magnetic bead. (D) The vertical force along the axis defined by the side-to-side contact between cylindrical magnet stacks. (E) Comparing the force-dependent stiffnesses due to the magnetic potential to an approximation to a typical protein construct clearly show the former to be insignificant, and thus the stiffness of the ‘magnetic trap’ is in fact the protein stiffness. Also shown is the stiffness of a typical cantilever used in SMFS AFM experiments, for comparison. 3 Compliance measurement in an ideal chain under a constant force constraint The equipartition theorem applies to harmonic oscillators, such as classical Hookean springs of linear elasticity. In thermal equilibrium, the average potential energy of a spring of constant stiffness is = 2 = 2 2 where = is the mean squared〈〉 displacement⁄ ⁄ from the average equilibrium extension and 2 is 2the thermal2 energy. Using thermodynamic beta = ( ) for convenience, the above equation 〈 〉rearranges− 〈〉 to a simple expression for compliance, = −1, of a classical spring in relation〈〉 to fluctuations in its extension −1 = 2 It is well known that (bio)polymers, and the ideal chain models that can be used to describe them, exhibit nonlinear elasticity – a force-dependent compliance. As such, it is not immediately obvious that the simple relation above still holds for ideal chains. An ideal chain is one in which there are no energetic interactions within the chain itself. As such, for an ideal chain under a constant force constraint, the only energy associated with any given microstate (configuration) is the potential energy under the constraining force, , where is the extension (end-to-end distance) of microstate . In the isothermal-isoforce ensemble, the probability of microstate is − = −1 where the partition function is the sum of Boltzmann factors over all microstates = from which the force-dependent average extension � of an ideal chain can be found as = = −1 Since the average extension of an〈 ideal〉 � chain is a nonlinear � function of force, its compliance (change in average extension per change in unit force) will be force-dependent, defined as = −1 2 −2 This expression can be simplified ≡ 〈〉 using �expressions −for the mean extension � (above), the mean squared extension = = 2 2 −1 2 and the first-order partial differential〈 〉 of� the partition function� with respect to force = = to give � 〈〉 = = 2 2 2 thus, recovering the same compliance expression〈 〉 − as 〈 given〉 by the equipartition theorem applied to a spring of linear elasticity. This expression will hold for any ideal chain model (e.g. freely-jointed, freely-rotating, etc) where (nonlinear) elasticity is purely entropic in origin. 4 Compliance calculation from the freely-jointed chain (FJC) model of polymer elasticity The freely-jointed chain (FJC) model is a facile, yet powerful, way to describe the mechanical behaviour of polymers, including proteins, and is the simplest ideal chain model. An FJC consists of rigid segments of fixed length, – the Kuhn length – connected via free joints (no steric hinderance/restrictions in either radial or azimuthal angles between adjacent segments). Each of these Kuhn segments represents a section of polymer that can be approximated to be behaving as an independent, discrete unit. The contour length (maximum end-to-end separation) of an FJC is simply given by = , and it is a well-known result that the partition function of an FJC under the constraint of a constant tensile force is = [4 ( ) sinh( )] −1 Above we showed in the previous section that = for any ideal chain, thus = ( ) = 〈〉 (ln ) −1 −1 and for the FJC model the force-dependent〈〉 average extension is = {ln[4 ( ) sinh( )]} = [coth( ) ( ) ] −1 −1 −1 where the function〈〉 in squared brackets can be recognised as the Langevin− function of the dimensionless force . Since , an expression for the force-dependent compliance of an FJC is then ≡ 〈〉 = [1 coth ( ) ( ) ] 2 −2 It is relatively trivial, but time-consuming, to− show that this− same result can be obtained by using = = after explicitly finding = ( ) and = from the FJC partition2 function.2 2 −1 2 −2 −1 2 〈 〉 − 〈〉 〈〉 〈 〉 5 Variance measurement from raw single molecule data. Data processing. During an experiment, the positions of i) the magnetic bead tethered to the protein-construct-of- interest and ii) a nearby surface-bound reference (non-magnetic) bead are recorded, see Figs. S2A&B, respectively, for data acquired using the protein L monomer construct, centred around an unfolding event. The same low-frequency thermal drift dominates both signals, disguising the unfolding event.
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