A mechanism for sarcomere breathing: volume change and advective flow within the myofilament lattice
Julie A Cass, C. Dave Williams, Tom C Irving, Eric Lauga, Sage Malingen, Tom L. Daniel, Simon N. Sponberg
AA mechanism for sarcomere breathing: volumechanges and advective flow within themyofilament lattice.
J.A. Cass a,b,1 , C. D. Williams a,b,1 , T. C. Irving c , T. L. Daniel b , and S. N. Sponberg d,2 a Allen Institute for Cell Science, Seattle, WA 98109, USA; b Department of Biology, University of Washington, Seattle, WA 98195, USA; c BioCAT and CSRRI, Department ofBiological Sciences, Illinois Institute of Technology, Chicago, IL 60616, USA; d School of Physics & School of Biological Sciences, Georgia Institute of Technology, Atlanta,Georgia 30332, USAThis manuscript was compiled on April 30, 2019
During muscle contraction, myosin motors anchored on thick fila-ments bind to and slide actin thin filaments. These motors rely onATP, supplied at the limits of diffusion from the sarcoplasm to theinterior of the lattice of actin and myosin. Classic sliding filamenttheory suggests that lattice spacing is constant. If so, then thelattice changes volume during contraction and could provide fluidmotion and hence assist in transport of molecules between the con-tractile lattice and surrounding intracellular space. If, however, thelattice is isovolumetric, it must expand when muscle shortens. Do-ing so would alter the binding dynamics of myosin which are sen-sitive to spacing. We first create a convective-diffusive flow modeland show that flow into and out of the sarcomere lattice would besignificant in the absence of lattice expansion. Convective trans-port coupled to diffusion has the potential to substantially enhancemetabolite exchange within the crowded sarcomere. Using time re-solved x-ray diffraction of contracting muscle we next show that thecontractile lattice is neither isovolumetric, nor constant in spacing.Instead lattice spacing is time-varying, depends on activation, andcan manifest a negative (auxetic) Poisson ratio. This unusual mate-rial behavior arises from the multiscale interaction of muscle axialstrain, lattice spacing and myosin binding. The resulting fluid flowin the sarcomere lattice is even greater than would be expected fromconstant lattice spacing conditions. Akin to “breathing,” convective-diffusive transport in sarcomeres is sufficient to promote metaboliteexchange, and may play a role in the regulation of contraction itself.
Muscle | Convection-Diffusion | X-ray Diffraction | Cellular Flows T he cellular environment is crowded and intracellular trans-port by diffusion alone can often be limiting (1, 2). Muscleforce production is one of the most energy demanding physi-ological processes in biology. Yet diffusion of molecules thatsupply energy in muscle is especially challenging because thedense lattice of contractile proteins typically has spacings ofonly 10’s of nanometers, and muscle cells are frequently large(100’s of µ m). Moreover these scales interact and muscle is amultiscale material; the collective action of many molecularmotors at the nanometer scales are arranged in a hierarchicalordered structure that influences the production of macro-scopic forces and strains (3, 4). While muscle cells are likelyisovolumetric on the time scales of contraction, the environ-ment within the cell is highly anisotropic. We explore if themuscle contraction itself has the potential to act as an in-tracellular pumping mechanism, assisting diffusive transportthrough flow-mediated advection of cellular metabolites. Flowmediated transport may be especially important for enablinghigh frequency and high power contractions, such as occurduring locomotion (3, 5–7), sound production (8, 9), and car- diac function (10, 11), where the strain is periodic and energydemands can be high.In muscle, contractile proteins form a regular latticecomposed of myosin-containing thick filaments and actin-containing thin filaments (12). The lattice is subtended byz-disks on either end, forming the sarcomere, the fundamentalunit of muscle contraction. Each muscle cell contains manysarcomeres that share the same intracellular fluid and requirethe exchange of metabolites with surrounding organelles. Topower the sliding of the thick filaments relative to the thinfilaments, each of billions of myosin motors requires energyderived from ATP hydrolysis. However, the dense packing ofthe myofilament lattice is presumed to limit the diffusion ofcritical energetic metabolites (e.g. creatine phosphate, ADP,and ATP) and regulatory molecules (e.g. Ca ) (1, 2). Inter-estingly, as the thin filaments slide during contraction, and thez-disks to which they attach are pulled toward the midline ofthe sarcomere, mass conservation demands that either 1) fluidwithin the sarcomere is squeezed out of the lattice into theintracellular spaces surrounding sarcomeres, or 2) the filamentlattice expands radially mitigating mass flux by conserving thelattice volume. It is also possible that the in vivo dynamics ofthe lattice follow neither of these limiting cases. Indeed, recentx-ray diffraction studies point to possible radial motions of the Significance Statement
Muscle operates at the limits of diffusion’s ability to provideenergy-supplying molecules to the molecular motors that gen-erate force. We show that pumping due to flow inside musclecells assists diffusion because the lattice of contractile proteinsundergoes significant volume changes, while the surroundingcell is isovolumetric. High speed x-ray diffraction of contractingmuscle shows that the volume change is even greater thanexpected from classical muscle contraction theory. Musclecontraction itself can promote the exchange of metaboliteswhen energetic requirements are high, a process akin to therole breathing plays in gas exchange. Such multiscale phe-nomenon likely play an underappreciated role in the energeticsand force production of nature’s most versatile actuator.
TLD and SNS designed experiments. TLD, TCI, and SNS conducted the experiments. CDWextracted data from imaging. CDW, JAC, and SNS analyzed experimental data. JAC and TLDperformed and analyzed the models. All contributed to the writing.The authors declare that they have no conflict of interests. JAC and CDW. contributed equally to this work. To whom correspondence should be addressed. E-mail:[email protected]
Cass & Williams et al. a r X i v : . [ q - b i o . S C ] A p r attice that might influence fluid exchange (13–16). Radialmotions during natural contractions may result in conditionsthat are neither constant volume nor constant spacing becauseboth passive forces and active cross-bridges may modulateradial filament motion (17, 18). The interaction between fluidexchange, radial lattice motions and substrate delivery remainsunresolved.To assess flow as a result of lattice volume change duringcyclic contractions, we first consider the classic case of a con-tracting sarcomere with constant lattice spacing (19). We askhow flows and volume exchanges influence substrate deliveryinto the densely packed space within the myofilament lattice(Fig 1A). We focus on insect flight muscle where short timescales associated with cyclic contractions demand rapid sub-strate delivery. With diffusion greatly reduced in the crowdedsubcellular environment of a sarcomere(2), mechanisms thatenhance substrate replenishment may have a profound im-pact on the energetics of muscle contraction. We first use acontracting model sarcomere to assess the consequences ofadvective transport of substrate delivery due to convective-diffusive coupling. We then use time-resolved x-ray diffractionto measure nanometer scale changes in the filament lattice ofintact muscle contracting under in vivo conditions. Results & Discussion
Flow assisted transport can enhance substrate delivery.
Toassess the implications of contraction on flow, we first numeri-cally solved the one-dimensional diffusion-convection equationat the axial sarcomere center. We did so for a cylindricalsarcomere 1.5 µ m in radius and 3 µ m long undergoing pe-riodic length changes of 0.15 µ m (5%) amplitude at 25 Hz,consistent with synchronous insect flight muscle with a highmetabolic power requirement (7, 16). The one-dimensionaldiffusion-convection equation is given by the expression: ∂c∂t = D ∇ c − ~u · ~ ∇ c [1]where c is the radially-dependent concentration of ATP and ~u is the intra-sarcomeric flow field. The diffusion coefficient, D , was chosen from published estimates of intra-sarcomeric D ATP to be 0 . × − cm s − (2). We computed the flowfield from conservation of mass and appropriate boundaryconditions for the half-sarcomere (symmetric flow at the m-line, flow matching z-disk motion, see Methods). The resultingflow is symmetric and reversible (Fig 1B and Fig 1C); withthe addition of porosity, the field remains similar in shape,but changes in magnitude. The resultant vector flow fieldis significant at the scale of the sarcomere and evident intwo time-lapse visualizations, provided in the SupplementalMaterials, (i) a 2-D slice through the cylinder interior and (ii)the 3-D flow through the cylinder surface (Supp. Videos 1 and2, respectively).Flow augments ATP availability in the model sarcomere(Fig 1D). With an initial substrate concentration of zero inthe sarcomere interior we find that diffusion interacts withconvection in a nonlinear manner, augmenting the delivery ofATP into the sarcomere over diffusion alone. The difference inATP delivery between these two models (diffusion-convectionand diffusion-only) oscillates as convection pumps ATP in andout during cyclical contraction, but the average difference over each cycle is always positive (Supp. Video 3). To demonstratethis quantitatively, we average the difference in the radially-summed concentrations from each of these models (Fig 1E).The flow-based advantage is always positive, contributing asmuch as 6 mM increases in ATP concentration.The advantage of sarcomeric flow would apply to any ap-propriately sized substrate with lower concentration inside thelattice than outside. Conversely molecules more concentratedwithin the lattice ( e.g. ADP) would benefit from flow assistedtransport out of the sarcomere. The advantage is greatestduring the earliest cycles of the simulation (Fig 1E). As timeelapses and the cylinder fills with ATP, the advantage affordedby convection diminishes. It is interesting to note that thisdecrease is not monotonic, resulting from the non-linearityof the diffusion-convection equation. The shape and timingof this local maximum depends on the diffusion coefficient(Supp. Fig S1). These simulations do not model the effectof reaction. Depletion of a substrate would cause the in vivo convection-assisted delivery to have a persistent advantage asATP is consumed within the muscle.
Lattice spacing change within muscle is neither constant norisovolumetric.
While constant lattice conditions can lead tosignificant intra-sarcomeric flows and increases in substratedelivery, it is reasonable to ask if radial motions of the latticeduring natural conditions reduce or augment such volumechanges. Indeed, it is possible for the radial spacing to changein such a way that would lead to a constant volume for thelattice: as the z-disks move inward, the radial spacing wouldincrease inversely with the square root of the sarcomere length,corresponding to a Poisson ratio of 0.5. Time-resolved x-ray diffraction allows measurement of the radial myofilamentlattice spacing under physiological conditions (12, 15, 20).Prior analyses have led to disparate interpretations concerningvolume changes of the sarcomere during contraction. Earlyevidence supported the constant lattice volume hypothesis(19,21), but more recent evidence suggests otherwise: in
Drosophila asynchronous flight muscle, the radial spacing of the latticedid not vary down to Angstrom resolution (22). In tetanicallyactivated frog muscle fibers, the radial spacing of the latticechanges during rapid length perturbations, but in a way thatyields measurable volume changes of the lattice(13, 14). Incontrast to the results from
Drosophila , time-resolved x-raystudies of a cardiac-like synchronous insect flight muscle in
Manduca sexta showed significant changes in the radial spacingof the lattice during cyclic contractions, with the temporalpattern of radial changes influenced by the timing of cross-bridge activation(16). However, the question of how changes inlattice-spacing map to sarcomere volume has not been resolvedfor in vivo muscle conditions.Here we combine controlled length and activation of musclewith simultaneous time-resolved x-ray diffraction to show thatthere are significant changes in both the radial spacing ofthe filament lattice and the volume of the lattice. We do sousing a model preparation (
M. sexta ) under physiologicallyrelevant length change and activation conditions. Like mostflying insects, hawkmoths power their flight by two dominantmuscle groups: the dorsolongitudinal muscles which drivedownstrokes of the wings and the dorsoventral muscles whichdrive upstrokes. In the specialized, asynchronous flight muscleof
Drosophila , contraction is de-coupled from neural activation.In contrast,
M. sexta powers wingstrokes with muscles that are
Cass & Williams et al. t/T=0.75t/T=4.75 A r z r z BC E
Fig. 1. Time-dependent flow field of the sarcomere. (A) Schematic of contracting and lengthening sarcomeres, with red and blue rods representing thick and thin filamentsrespectively. (B) Surface flow fields in normalized radial ( r/R ) and axial ( z/Z ) coordinates at maximum contraction (left) and maximum lengthening (right). (C) A cross-sectionalview of the internal flow field in normalized coordinates during maximum contraction (left) and maximum lengthening (right). (D) Left: the concentration of ATP ([ATP]) as afunction of r at the axial center of the sarcomere. As this concentration is time-dependent, we display [ATP] at two sample time points. At each of these times, two models areplotted: a diffusion model (blue) and diffusion-convection model (yellow) of ATP transport. The green region between these models at both times highlights the positive impactof convection on ATP delivery. (E) The difference (green region in D) between convection with diffusion and diffusion alone is averaged over each cycle, and this cycle-averagedconvective advantage is plotted for 100 cycles; as it is always positive, convection always augments the delivery of ATP in our model.Cass & Williams et al. ctivated synchronously via motor neuron control (23). Thissynchrony allows direct experimental control of the timing ofcross-bridge recruitment and muscle activation (16, 24). Thuswe sinusoidally oscillate dorsolongitudinal muscle at 25 Hz(a physiologically relevant frequency) under controlled timingof the phase of activation with simultaneous time-resolvedimaging from X-ray diffraction (see Methods). We measuredthe interfilament distance from the separation of the first x-ray diffraction peaks (termed d , Fig 2) with millisecondresolution to provide high-speed imaging of the radial motionsof the myofilament lattice during the contraction cycle.Our time-resolved lattice spacing reveals that M. sexta flightmuscle does not follow the predictions for constant volumenor for constant lattice spacing. Under typical locomotorconditions, muscle length changes and phase of activation,lattice spacing changes by approximately 3%. With a typical d spacing of 49 nm, this corresponds to cyclic fluctuationsin lattice spacing of 1.5 nm in the radial direction (Fig 3A).While 1.5 nm is comparable to the magnitude of spacingchange predicted by the isovolumetric case, the change isnot in phase with axial length change. As such, there areappreciable volume changes of the lattice.Rather than having a Poisson ratio of zero (constant spac-ing) or 0.5 (constant volume), the change in lattice spacingvaries over the cycle (Fig 3B). This means that the muscle’slattice has a time varying Poisson ratio: at some points inthe length change cycle the Poisson ratio is positive and atothers it is negative (behaving briefly as an auxetic material).Recent theoretical and experimental evidence suggests thateven such modest changes in lattice spacing can profoundlyinfluence tension development (25) and rates of cross-bridgeattachment or detachment (17, 18).These unusual dynamic material properties likely arise fromthe interaction between radial forces generated by cross-bridgesas they transiently bind to the thin filaments. We next testedthe idea that cross-bridge activity influences lattice spacingeven when the periodic axial strain remains the same. Thesynchronous downstroke flight muscle of M. sexta receivesonly a single motor impulse per wingstroke (23). We alteredthe timing of cross-bridge force development by changing thephase of this electrical stimulation, defined as the timing ofactivation during the cyclical contractions. Over all phasesof activation, the radial spacing of the filament lattice variesconsiderably throughout the length change cycle (Fig 3C). Insome instances, the peak-to-peak amplitude was as much as 3nm (twice that of activation phase of 0.5), while in others thelattice was more constrained (e.g. phase of 0.4). In naturalconditions phase is known to vary under neural control (24).Varying the activation of the myosin motors results in differentlattice dynamics and different time varying Poisson ratios evenunder the same longitudinal strain.
Change in lattice spacing during contraction enhances vol-ume change beyond constant lattice spacing predictions.
While constant lattice spacing necessarily leads to volumechanges of the sarcomere, the magnitude of such changescould be even more extreme because of the time varying lat-tice spacing: cross-bridges could pull the myofilaments radiallyinward as the sarcomere shortens. Using the known axial strainand the measured lattice spacing we reconstructed the timevarying lattice volume (Fig 3D). The observed dynamics show51 +/- 22 % (95% CI of mean) more volume change than what would occur for constant lattice spacing (comparing volumeratio to 1, all conditions: p < − , N = 85; in vivo condi-tions: p < − , N = 19 ; 35 ◦ C alone: p = 0 . , N = 7).Temperature can also have an effect on lattice spacing (16),but we varied temperature across the in vivo range of 25-35 ◦ C and found no significant effect on volume change (Kruskal-Wallis, p = 0 . , N = 85). The phase of activation didweakly modulate the volume change by ±
22% (Kruskal-Wallis, p = 0 . , N = 85), but at all phases volume change wasstill larger than expected for a constant lattice ( p < .
01 inall cases). Overall, periodic lattice spacing change due tocross-bridge activation enhances volume changes in the lattice.These results show that in muscle contracting under physio-logically relevant conditions, cross-bridge binding dynamicallyalters lattice volume. Moreover, cross-bridges can activelyrestrict radial motions, even under conditions of active muscleshortening, creating a transient auxetic behavior in whichthere is a reduction in both the radial and axial dimensions ofthe lattice. Like breathing, volume changes and the resultingintra-sarcomeric flows interact with diffusion to augment sub-strate exchange. Convective flow due to muscle contractionis a previously unrecognized mechanism that can influenceenergy delivery and the flux of regulatory molecules in thecrowded intracellular environment.
Materials and Methods
Numerical solution of the diffusion-convection equation.
We modela contracting sarcomere with constant lattice spacing and ask howwhat flows and volume changes are necessarily generated duringcyclic contractions. Provided the intracellular fluid is incompressible,the sliding filament model requires that fluid moves into and out ofthe myofilament lattice during contractions. To derive the flow fieldin a cylindrical model sarcomere, we analytically solved the time-dependent vector flow field under standard boundary conditions:no-shear on the interior and the fluid velocity matches the axialperiodic velocity of z-disk. The resulting flow field takes the form: ~u ( r, z ) = u r ( r, z )ˆ r + u z ( r, z )ˆ z [2]where the radial and axial flow fields are given below: u r ( r, z ) = − u z max rz max h − (cid:16) zz max (cid:17) i [3] u z ( z ) = u z max zz max h − (cid:16) zz max (cid:17) i [4]Here, u z max is the z-disk speed, and z max is the axial position ofthe z-disks.We developed a Python-based numerical PDE-solver to calculatethe concentration of ATP in the sarcomere as a function of timeand radial position (26). Using the values mentioned in the text(a cylinder 1.5 µ m in radius and 3 µ m long, undergoing periodiclength changes of 0.15 µ m amplitude at 25 Hz) we solved the PDEfor the initial condition of 0 internal ATP concentration at time 0with a boundary condition of 10 mM ATP at the cylinder radius.We calculated the concentration by solving the one-dimensionaldiffusion-convection equation (Eq 1), with the time-dependent flowfield described in the previous section (Eq 2-4) serving as ~u . Wediscretized this equation using a central-difference Euler finite-differencing scheme, and solved it along a radial grid at the axialcenter of the sarcomere cylinder. This equation was then sequentiallysolved in steps along a temporal grid; the resolution of the temporaland radial grids were coupled by a von Neumann stabilizationcriterion: ∆ t < (∆ r ) D , [5]
Cass & Williams et al.
10 t=1 t=0t=… … t=2 S force andlengthbeam line DLM DVM
M3M6 S S Fig. 2. Schematic of X-ray fiber diffraction procedure for measuring lattice spacing.
From bottom left: the lattice of contractile proteins, containing myosin (red) and actin(blue) filaments is shown, with the lattice spacing ( d ) marked. Zooming out, these parallel filaments are part of the dorsolongitudinal muscle (DLM, purple) which runsperpendicularly to the dorsoventral muscle (DVM, dark green) in the hawkmoth thorax (light green). Muscles from the thorax are placed in the synchrotron x-ray diffraction beamline (dotted red line). Muscle length is controlled and the force is measured by a force-feedback motor (light green). As the beam line passes through the sample, the resultingdiffraction pattern is recorded on a detector. This is repeated over the course of many contractions, resulting in a series of frames of the X-ray diffraction pattern. On eachpattern, two bright spots (marked S ) are used to calculate the d interfilament lattice spacing. Zooming farther out from the x-ray diffraction pattern, we see the diffractionlines: M3 (7.5 nm), M6 (14.3 nm), S (average of 40 nm) and S (reflection of S ).Cass & Williams et al. BA phase of activation (0-1) v o l u m e r a t i o D in vivo
25º C30º C35º C
Fig. 3. Example data measuring the radial lattice spacing. (A) Time-dependent difference between instantaneous and mean d spacings for muscles experiencingsinusoidal length changes at 25 Hz with amplitude of 0.4 mm (strain 4%) and an activation phase of 0.5 (yellow dot – onset of shortening). The observed data (blue with CI) differs from the predictions for constant lattice spacing (green) and constant volume (red). (B) The lattice spacing and predictions from A are plotted against themuscle length, illustrating the hysteresis in lattice spacing. (C) Whisker plots (mean, 50%, 90% quantiles) for lattice spacing as function of time within the 40 ms length changecycle. The extent and timing of lattice spacing changes depend on the phase of stimulation scaled to the range of 0 to 1 (compare red and blue). A phase of 0 corresponds toactivation occurring at the onset of lengthening which would be at the beginning of the strain cycle (time = 0 ms). (D) Volume ratio as a function of the phase of activation. Theactual volume change ( V a ) is a sum over the time intervals from C and taken as a ratio to the volume change predicted from the sliding filament model, ( V p ). Values greaterthan 1 indicate more volume change than would occur if the actin-myosin lattice had constant spacing. Colored data points correspond to different temperatures, black markersare means across all temperatures with 95% confidence intervals of the mean in black brackets. The in vivo condition of a phase of 0.5 is plotted separate from the modifiedphases which produce systematic variation in the mean volume change, but always result in a volume ratio greater than 1. Color bars at the top show which phases correspondto data in C. Cass & Williams et al. here ∆ t and ∆ r are the temporal and radial grid resolutions,and D is the diffusion coefficient. For our model, we prescribe:∆ t = 0 . (∆ r ) D . The result of this simulation is a numerical calculation of theATP concentration on a radial-temporal grid, during many cyclesof successive sarcomere contractions.
Preparation of specimens.
Hawkmoths (
M sexta ) were grown atthe University of Washington Insect Husbandry Facility. Mothswere cold anesthetized at 4 ◦ C with a thermo-electrically cooledstage after which wings, head, and legs were removed. The leftand right pair of dorsolongitudinal, downstroke muscles (DLMs)were isolated and together mounted in the apparatus as describedpreviously(16, 23). The anterior phragma region of the scutumwhere the DLMs originate was rigidly adhered to a custom brassmount shaped to the curvature of the thorax. The DLMs insertonto the second phragma, an invagination of the exoskeletalbetween the meso- and meta-thoracic segments. A second brassmount with two stainless steel prongs were inserted into thephragma to provide a rigid attachment. This mount connectedto a muscle ergometer (Aurora Scientific 305C) that controlledlength and measured force. After mounting, the ventral side ofthe thorax was removed immediately below the DLMs to severthe upstroke dorsoventral muscles and the steering muscles. A ∼ in vivo operating length, L op , which is 0 . L rest (23).Two bipolar tungsten-silver wire electrodes were inserted throughthe five subunits of each DLM. Stimulation amplitude (bipolarpotentials, 0.5 ms wide) was set by monitoring the isometric twitchresponse in the muscle and setting stimulation voltage to the twitchthreshold plus 1 V. Typically 3 V stimuli were used. Time-resolved small angle x-ray diffraction.
X-ray experimentsused the small angle instrument on the BioCAT undulator-basedbeamline 18-ID at the Advanced Photon Source (ArgonneNational Laboratory, Argonne, Illinois). The overall experimentalarrangement is shown in Fig 2B. The X-ray beam energy was 12 keV(wavelength 0.103 nm), with a specimen-to detector distance of 3.2m. Fiber diffraction patterns were recorded with a photon-countingPilatus 100k (Dectris Inc.) pixel array detector collecting images at125 Hz. A rapid shutter closed during the 4 ms detector readouttime and beam intensity was adjusted appropriately in the range10 − with aluminum attenuators to obtainadequate counting statistics in the X-ray pattern with minimalradiation damage to the specimen. In addition, the preparationwas oscillated in the beam ( ∼ − ) to reduce the X-ray doseon a given region of the muscle. For most trials, we recorded 100cycles and images were collected at the same time within each cycle.The first 10 cycles were not considered so that data were collectedentirely under steady periodic conditions. X-ray images wereprocessed using an automated machine vision algorithm(27) thatfits equatorial diffraction intensity peaks superimposed on diffusebackground scattering. The distance between the first diffractionpeaks, S , is related to the inter-filament lattice spacing, d , byBragg’s law. Sample to detector distance was calibrated usinga silver behenate scattering image. Lattice spacing, d , wasnormalized to its maximum value in each trial to account forpreparation to preparation variation. ACKNOWLEDGMENTS.
We thank Nicole George and Andrew Mountcastle for their illustration and Sage Malingen for her helpfulcomments. This project was supported by grant W911NF-14-1-0396from the Army Research Office to TLD, TCI, and SNS, NationalScience Foundation CAREER 1554790 to SNS, grant 9 P41GM103622 from the National Institute of General Medical Sciencesof the National Institutes of Health, and the Richard KomenEndowed Chair to TLD and NSF Physics of Living SystemsStudent Research Network grant 1205878. This research usedresources of the Advanced Photon Source, a U.S. Department ofEnergy (DOE) Office of Science User Facility operated for the DOEOffice of Science by Argonne National Laboratory under ContractNo. DE-AC02-06CH11357.
1. Kushmerick MJ, Podolsky RJ (1969) Ionic mobility in muscle cells.
Science
Biophys J
Physiology
Physiological reviews
Journal of Experimental Biology
Integrative and comparative biology
Journal of Experi-mental Biology
Journal of Experimental Biology
Annual Review of Physiology
Integrative and ComparativeBiology
Phys-iological Reviews
Nature’sversatile engine: Insect flight muscle inside and out , ed. Vigoreaux JO. (Springer, New York),pp. 197–213.13. Cecchi G, Bagni MA, Griffiths PJ, Ashley CC, Maeda Y (1990) Detection of radial crossbridgeforce by lattice spacing changes in intact single muscle fibers.
Science
Biophys J
Sci-ence
Science
PLoS Comput Biol
PLoS Comput Biol
Proc R Soc Lond B Biol Sci
Nature
J Mol Biol
Biophys J
Journal of Experimental Biology
Proceedings Of The Royal Society B-BiologicalSciences
Proc Biol Sci
A guide to NumPy. (Trelgol Publishing).27. Williams CD, Balazinska M, Daniel T (2016) Automated analysis of muscle x-ray diffractionimaging with mcmc. in
Biomedical data management and graph online querying , eds. WangF, et al. (Springer International Publishing, Switzerland), pp. 126–133.