splinelike interpolation in particle tracking microrheology
TRANSCRIPT
PHYSICAL REVIEW E 86, 011501 (2012)
Splinelike interpolation in particle tracking microrheology
Timo Maier, Heike Boehm, and Tamas Haraszti*
Max Planck Institute for Intelligent Systems, Heisenbergstrasse 3, 70569 Stuttgart, Germany andBiophysical Chemistry Group, University of Heidelberg, Im Neuenheimer Feld 253, 69120 Heidelberg, Germany
(Received 24 August 2011; revised manuscript received 22 May 2012; published 5 July 2012)
Converting time dependent creep compliance to frequency dependent complex shear modulus is an importantstep in analyzing the results of particle tracking microrheology. Fitting a function to the whole time range andtransforming it to calculate the shear modulus is one way of solving this problem. However, the creep complianceof many samples, such as gels of biopolymers, shows different trends under different time regimes. Fitting inthese regimes segmentwise results in a function which usually cannot be transformed in a closed analytical form.In general, unlike for beta and cubic splines, also the continuity of the first derivative cannot be ensured. In thispaper, we present a method for using segmentwise fitting and numerical conversion, discussing interpolation forimproving the transition between the fitted ranges, and propose a dynamic sampling technique to control theaccuracy of the resultant complex shear modulus.
DOI: 10.1103/PhysRevE.86.011501 PACS number(s): 83.85.Ns, 83.60.Bc, 83.80.Lz
I. INTRODUCTION
Rheology characterizes the mechanical deformation andflow behavior of materials under dynamic stress. It is ableto derive characteristic properties of the sample down tothe molecular level by relating measured data to materialmodels [1–5]. There are many rheological techniques to testa sample, but only a small number of ways to probe softbiological samples available only in limited, microscopicamounts (such as within living cells) [3–5]. Particle trackingthermal, or passive, microrheology is a linear, nondestructiveway of analyzing the rheological properties of such softsamples [3–10]. Though embedding tracers does alter the localstructure and characteristics, using various tracer sizes andvarying the surface chemistry of the tracer, one can reconstructthe general properties of the sample [11,12].
This experimental tool has played a valuable role incharacterizing biological samples, such as protein gels (e.g.,actin, collagen, the extracellular matrix) and the intracellularenvironment [3–5,13,14]. The underlying physical principle isthe generalized Stokes-Einstein relation [15–17], connectingdiffusion characteristics and the thermal motion of the tracerparticle to the mechanical properties of the medium. Thisconnection relates the mean squared displacement (MSD) ofthe probe particle to the creep compliance of the sample.Assuming a spherical particle with radius a, traced in ND
dimensions at a temperature T , a linear relation [Eq. (1)]connects the MSD to the creep compliance [4,18]. In this work,the motion of the particle is observed in two dimensions, whichis common for video microscopy; therefore we set ND = 2.
J (t) = 3πa
NDkBT〈�r2(t)〉, (1)
G∗(ω) = 1
iωJ (ω), i = √−1. (2)
If available for all time values, the creep compliance [J (t)]completely describes the rheological properties of a sample.On the other hand, in active microrheology and macroscopic
experiments, it is also common to measure the frequencydependent complex shear modulus [G∗(ω)], applying oscil-latory shear stress. [G∗(ω) is complex, G∗ = G′ + iG′′; itsreal part G′(ω) is the storage modulus and its imaginarypart G′′(ω) the loss modulus.] In order to compare theresults from the different methods, it is common to convertthe creep compliance to the complex shear modulus usingEq. (2). This conversion contains the Fourier transform ofthe creep compliance [J (ω)] and may prove problematic forexperimental data. Various methods have been proposed inthe past to circumvent such difficulties [2,4–9,16–24]. Recentexamples include the “direct conversion method” by Evanset al. [21] and the method of Mason et al. [16], which is basedon the fitting of power law functions locally along the MSD.
However, both of these methods are strongly affected by theexperimental noise and the bandwidth of the data. In order toelucidate the extent to which they are affected, in the followingsections we first summarize the methods of Evans et al. andMason et al. and compare them to simple classical rheologicalmodels, such as the Maxwell liquid and the Kelvin-Voigt elas-tic body. Then we construct the general case of segmentwisefitting, and design a general, numerical conversion methodinvestigating the effect of increased bandwidth to improve theconversion accuracy. Finally we provide a dynamic samplingmethod, and present its application using an example of realdata obtained on elastic actin networks.
II. SUMMARY OF THE TWO CONVERSION METHODS
A. Transforming via linear interpolation
The “direct conversion method” proposed by Evans et al.[21] is based on the simplest assumptions about the data(solid line). Assuming no prior knowledge and consideringonly the discrete, measured points of {tk,Jk} for k = 1 . . . N ,we approximate the intervals between the data points usinga linear interpolation (Fig. 1). The time range of the datacan be extended using linear extrapolation to the short andlong time ranges. Extrapolating to the zero time value requiressome considerations though, because this strongly affects thehigh frequency part of G∗(ω). Instead of repeating the original
011501-11539-3755/2012/86(1)/011501(7) ©2012 American Physical Society
TIMO MAIER, HEIKE BOEHM, AND TAMAS HARASZTI PHYSICAL REVIEW E 86, 011501 (2012)
FIG. 1. (Color online) A schematic demonstrating how linearinterpolation (4) describes the experimental data. The data after the(k − 1)th data point is described by a linear equation Akt + Jk−1. Astep function H (t − tk−1) cuts the line to zero for lower t values(dashed lines). To correct the slope of the data after the next datapoint tk , the current slope Ak has to be subtracted, and the next Ak+1
one added. The constant Jk−1 is automatically formed by the sum ofthe previous segments.
arguments of Evans et al. [using Dirac-delta functions asthe second derivative of J (t)] here, we have developed analternative, more direct and transparent way of calculation.
The linear interpolation has the form Akt + Jk−1 for tk−1 �t < tk , where Ak is the difference quotient of the kth datapoint and its previous neighbor, defined in Eq. (3) for allk = 1 . . . N . Using the step function H (t) (which is 0 forall negative values and 1 for all positive values and zero),one can describe the changing directions by adding the nextAk component and subtracting the previous trend describedby Ak−1 (see Fig. 1). Using this method, we can constructJ (t) as a sum of linear functions in the form of Eq. (4). Thesum contains N experimental data points corresponding toN time domains, each beginning at time points tk . In orderto include all possible values from t0 = 0, one can completethe sum using a J0 value, obtained from linear extrapolation,or predefined according to a priori knowledge of the sample.The slope of the creep compliance is zero before t0, resulting inA0 = 0. For long time values, beyond the measured data, linearextrapolation provides the slope of AN+1 (at tN+1 = ∞), whichis the reciprocal of the long time viscosity (η = 1/AN+1) [21].
Ak = Jk − Jk−1
tk − tk−1, where 0 < k � N,
(3)
A0 = 0, AN+1 = 1
η,
J (t) = J0H (t) +N∑
k=0
(Ak+1 − Ak)H (t − tk). (4)
Equation (4) can be readily Fourier transformed and usingEq. (2), we obtain the frequency dependent complex shearmodulus G∗(ω) Eq. (5):
G∗(ω) = iω
iωJ0 + ∑Nk=0 (Ak+1 − Ak)e−iωtk
. (5)
The form of the resulting G∗(ω) looks simpler than theform in Eq. (9) in Ref. [21], but it can be proven to be
equivalent (by reorganizing the sum according to the Ak
coefficients, and splitting up the first and last terms of thesum). We present it in this form as it more clearly showsthe limitations of this conversion method. The complex shearmodulus in Eq. (5) contains the discrete Fourier transform ofthe Ak+1 − Ak terms. Consequently G∗(ω) is very sensitiveto sudden changes between the data points, i.e., experimentalnoise. Such jumps would produce a high absolute value of theAk+1 − Ak term, which in turn adds a strong oscillation to thesum in the denominator of Eq. (5) with a periodicity of 2π/tk .
B. Transform using power law interpolation
To summarize the transformation method developed byMason et al. [17], we consider the creep compliance in theform of a power function. The physical background lies in thefact that the tracer particles often show subdiffusion behavior inbiological gels, cells, or in various polymer gels. Subdiffusionis usually characterized by a power law time dependence withan exponent 0 < α � 1 [Eq. (6)]:
〈�r2(t)〉 = Dtα, J (t) = J0tα. (6)
Considering the simplest case, that a single exponent describesthe diffusion for all time values, J (t) can be readily Fouriertransformed and therefore the G∗(ω) can be calculated [Eq. (7)](the proportionality constant J0 is related to the diffusioncoefficient through Eq. (1), which results in J0 = πaD/(kBT )for three dimensional diffusion, in agreement with Ref. [17]):
G∗(ω) = eiπα/2 ωα
J0�(1 + α)= eiπα/2
J (t = 1/ω)�(1 + α)
= eiπα/2 kBT
πa〈�r2(t = 1/ω)〉�(1 + α(t = 1/ω)). (7)
A unique property of the power function is the directconnection between the value of the original function at timet = 1/ω and the value of its Fourier transformed counterpartat circular frequency ω as is apparent in the middle part ofEq. (7). This relation was generalized by Mason et al. in theirconversion method, fitting the experimental MSD data with alocal power law function, and calculating the correspondingG∗(ω) at ω = 1/t directly from the last part of Eq. (7) [16,17].
Applying this transform, one assumes that the t = 1/ω ↔ω symmetry and the last part of Eq. (7) are valid for the specificform of J (t). This is a very unique property, and generally afunction contains various Fourier components in the wholet = 0 − ∞ range; therefore the transformation can only bedone accurately using the whole data set. Dasgupta et al. havedeveloped an empirical correction term to Eq. (7) assumingthat the major error arises from the limited bandwidth of thediscrete sampling [19]. Their results present an improvementfor the case when G∗(ω) is the linear combination of powerfunctions, but does not improve the general case, when thet = 1/ω ↔ ω symmetry between Eqs. (6) and (7) is broken.
III. TESTING THE METHODS OF EVANS AND MASON
From their definition [Eqs. (4) and (6)] it is transparentthat the Mason method perfectly converts creep compliances,which follow a power law (single exponent for all time values),and the “direct conversion” method should work perfectly for
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a creep compliance following a linear equation, characteristicto an ideal Maxwell fluid. In these cases the conversion erroris determined only by the machine precision and round-offerrors from the algorithm used.
A Maxwell fluid is characterized by a creep compliance inthe form of Eq. (8):
J (t) = H (t)
(1
E+ t
η
). (8)
Here H (t) is again the step function, indicating that thedeformation was applied from t = 0, until that the creepcompliance was zero. The complex shear modulus can becalculated as before, from Eqs. (2) and (8):
G∗(ω) = iωEη
iωη + E. (9)
Using a Young modulus of E = 100 Pa and a dynamicviscosity of η = 2.5 Pa s, characteristic to a viscoelastic gel, wegenerated a numerical data set for the test. Jk = J (tk) valueswere calculated using 0 < t � 100 s in dt = 0.01 s steps(resulting in N = 10 000 data points) via Eq. (10). Then eitherEq. (5) or (7) was applied and compared to Eq. (9) (Fig. 2).
The results indicate a good match between the theoreticalG∗(ω) and the values obtained using the conversion of Evans,while the Mason method results in some deviation at highfrequencies. This deviation depends on the Young modulusused, since with a negligible 1/E part Eq. (8) would be apower law with an exponent of α = 1, perfectly fitting for thismethod as well.
To further investigate the accuracy of the two methods,we chose a function which deviates from both the linear andpower law, e.g., a Kelvin-Voigt elastic solid, where the creepcompliance is described by Eq. (10) and the complex shear
10-5
10-4
10-3
10-2
10-1
10
101
102
0.01 0.1 1 10 100 1000
G’,
G"
(Pa)
ω (1/s)
G’
G"
Mason Evans
FIG. 2. (Color online) Calculating the complex shear modulusof a Maxwell fluid from the creep compliance using the methods ofMason (lower frequency range on the left; G′ is denoted by [blue] ∗symbols, G′′ by [pink] �) and Evans (G′ by [red] +, G′′ by [green]×). The storage modulus (G′) values are the lower and steeper data,the loss modulus (G′′) are on the top part of the figure. The theoreticalvalues predicted by Eq. (9) are plotted with a solid black line. Thoughin the log-log representation it is not apparent, the Mason methodshows a relative error up to 40%, while the Evans method has arelative error of about 10−8.
0.01
0.1
1
10
100
1000
0.01 0.1 1 10 100 1000
G’,
G"(
Pa)
ω(1/s)
10-14
10-12
10-10
10-8
10-6
10-4
10-2
10
102
104
0.01 0.1 1 10 100
G’,
G"
(Pa)
ω(1/s)
FIG. 3. (Color online) Testing the conversion methods using aKelvin-Voigt creep compliance [Eq. (10), E = 100 Pa and η =2.5 Pa s]. The (+) symbols (red) depict the storage modulus, andthe (×) symbols (blue) the loss modulus. The solid and dashedlines are the theoretical values. Results from the “direct conversion”method are presented in the top panel, showing excellent match at lowfrequencies, but the storage modulus deviates strongly (up to 100%relative error) above about 30 s−1. In the bottom panel the resultsfrom the Mason method are plotted. It shows a relative error for thestorage modulus up to 50% at high frequencies, and up to 100% forthe loss modulus at low frequencies (below about 10 s−1).
modulus by Eq. (11). We chose again E = 100 Pa and η =2.5 Pa s as for the Maxwell fluid above, characteristic to a softviscoelastic gel, and generated the numerical data the sameway as in the previous test.
J (t) = H (t)
E[1 − e−(E/η)t ] (10)
G∗(ω) = E + iωη (11)
As is apparent in Fig. 3, both methods show strong deviationfrom theoretical predictions. Interestingly, the Mason methodprovides a slightly better match at high frequencies (50%instead of 100% relative error), but misses the loss modulusat low frequencies (below about 10 s−1). It is also importantto note that the Mason method always has a lower bandwidth,since it is defined using ω = 1/t . In our example this resultsin a maximal circular frequency 100 s−1 comparing to theNyquist maximum of π/dt ≈ 314 s−1.
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TIMO MAIER, HEIKE BOEHM, AND TAMAS HARASZTI PHYSICAL REVIEW E 86, 011501 (2012)
0.01
0.1
1
10
100
0.01 0.1 1 10 100 1000
G’(P
a)
ω(1/s)
FIG. 4. (Color online) Comparison of the storage modulus calcu-lated with the method of Evans, using the same settings as in Fig. 3(+ [red]) and 10× oversampled (× [blue]). The relative error of theoversampled data set is <1%.
The errors of the Mason method arise from two sources:First, the local data fragments may not fit well to a powerfunction in the form of Eq. (6), and second, the t = 1/ω ↔ω symmetry is broken between the underlying functionsEqs. (10) and (11).
In the case of the method of Evans et al., the source of errormay be best related to the limited sampling of the fast changingpart of the creep compliance. Therefore, one can alter the datasampling to elucidate how much of the error is really related tothe limited bandwidth of the test. Increasing the data sampling,but using the same circular frequency range 0 < ω � ωmax
(ωmax = π/dt = 314 s−1 for the Evans and ωmax = 1/dt =100 s−1 in the case of the Mason method) for G∗(ω) improvesthe accuracy (Fig. 4). The frequency range where the methodsprovide an accurate fit is increased, and for a 10× oversamplingthe Evans method gives for both the storage and loss modulian error lower than 1%. However, such improvement cannotbe observed for the Mason method (data not shown).
IV. SEGMENTWISE FITTING
While one can easily manipulate the sampling rate andbandwidth of model data, it is frequently beyond the limitationsof the given experimental equipment. A well known alternativeway of data treatment is to fit the creep compliance usinga single theoretical model equation in the whole data rangeand then use the resulting fit to approximate and interpret thecomplex shear modulus. However, for flexible and semiflexiblepolymers, such as many biological samples, the theory predictsthat the complex shear modulus follows a power function onlylocally [2,25,26], and one can expect similar behavior for thecreep compliance [17]. Thus, we can obtain proper fits onlyon segments of the data, and have to face the problem ofconverting such a segmentwise fitted function to the frequencydependent complex shear modulus.
Using an analogous but alternative argument to derivingEq. (4), we can construct a general formula for fitting a setof functions fk(t) to segments tk � t < tk+1, i = 1 . . . M ofthe experimental creep compliance J (t). Here M < N is the
number of functions for N data points, and f0 is the constantat t = 0:
J (t) = f0H (t) +M∑
k=0
[fk+1(t) − fk(t)]H (t − tk), (12)
J (ω) = f0
iω+
M∑k=0
[fk+1(ω) − fk(ω)] ∗(
eiωt
iω
). (13)
The Fourier transform of this creep compliance results inconvolutions of the form Eq. (13). That is, for using powerfunctions in the form of Eq. (6), this would result in a set offractional differentials [Eq. (14)], for which we do not have aclosed form solution. Alternatively, because we can control thesampling of the fitted functions fk(t) with arbitrary precision,numerical conversion methods can be employed to calculatethe appropriate G∗(ω).
J (ω) = f0
iω+
M∑k=0
Ak+1∂αk+1
∂ωαk+1
(eiωt
iω
)− Ak
∂αk
∂ωαk
(eiωt
iω
)
(14)
V. NUMERICAL CONVERSION WITHDYNAMIC SAMPLING
To convert Eq. (12) numerically, it is important to controlthe accuracy of the conversion. Considering the previouslydiscussed two numerical methods for this purpose, we focuson the “direct conversion” method. The error of this methodshowed a decrease with increased sampling, while the othermethod had a maximal error independent of the sampling.
The key question we have to address is how to definethe sampling of the fitted functions to obtain a result withcontrolled precision. From Eqs. (4) and (5) it is apparentthat the key source of the error lies in the accuracy of howAk approximates the local derivative of the underlying, fittedfk(t) function. For example, in the case of functions such asthe power function (6), this approximation can be very weak(e.g., as t → 0). For a known function fk(t), one can predict,in a first approximation, that the accuracy of Ak is proportionalto xc (the so called curvature scale) [Eq. (15)] [27]:
xc =√
fk(tk)
f ′′k (tk)
, f ′′k (tk) = ∂2f (t)
∂t2, (15)
h = εxc = ε
√fk(tk)
f ′′k (tk)
. (16)
Defining a minimum step size h as Eq. (16), where ε is anaccuracy parameter, the data sampling can be refined in eachmeasured time interval [tk,tk+1], by inserting new time pointsuntil h > (tk+1 − tk) is reached. The result is a dynamicallyrefined mesh depending on the local curvature scale of thefitting function.
Because the creep compliance is usually fast changingfor short time values, it is necessary to resample the fittedfunction in the (t0 = 0 s,t1) time interval as well, improvingthe evaluation of G∗(ω) for higher frequencies. Furthermoreh → 0 as t → 0 for many cases (such as the power function),so it is practical to define a new t ′1 first time point beforeutilizing Eq. (16) for all t ′1 � t .
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The required ε depends on the type of function and alsothe number of inserted points, since Eq. (5) contains the sumof all data. For example, our numerical tests indicate that theoverall error using power functions scales well with ε. Usinga new first time point of t1/1000 and a value of ε = 10−5
are required to reach subpercent accuracy. For Kelvin-Voigtmodel data t1/10 and ε � 10−3 are already sufficient to reachthe same precision.
A. Smoothing the transition between segments
Independent of the sampling, this method is very sensitive tojumps between the fitted regions. It follows from Eq. (5) that ifthe local slope at the meeting point of the two segments differsmuch, the local Ak+1 − Ak difference will have a high value.This will introduce an oscillation with a periodicity of 2π/tk ,which is an undesirable artifact. Assuming that both segmentsare fitted with a good accuracy, one can minimize or eliminatethis oscillatory term by replacing the local fitted values withan interpolated value generated as a linear combination of thetwo fitted functions within a chosen interpolation window ofNs points around the matching point k0 [Eq. (17)]:
Jj = wjfk+1(tj ) + (1 − wj )fk(tj ),(17)
wj = j − k0 + Ns/2
Ns
, j = 1 . . . Ns.
In the case of the power functions, this interpolation works wellwhen the exponent does not change too much (for example achange from an exponent of 0.3 to nearly zero), ideally lessthan 0.1.
B. The proposed algorithm
In summary, our proposed segmentwise fitting algorithmhas the following steps:
(i) Fit the experimental {tk,Jk} data with a power law oranother suitable function in M segments;
(ii) decrease t1 by a factor of 10 . . . 1000 (depending on thetype of the first fitted function) by inserting new time points inthe 0 . . . t1 region;
(iii) using a predefined ε = 10−5 . . . 10−3 value, insert in-terpolating time points between the data, such that the resultedtime steps always satisfy h = εxc � (tk+1 − tk);
(iv) calculate G∗(ω) using Eq. (5).
VI. EXAMPLE
As an example, below we present a measured data setobtained by imaging a polystyrene bead of 1.9 μm in diameter,embedded into a three dimensional actin gel. The gel wasprepared as discussed in the literature, and bundled using50 mM of magnesium chloride. The bead was selected suchthat it adhered to the network, resulting in a confined MSD(Fig. 5). The gel is expected to show an extended elasticplateau; because actin is a semiflexible polymer, the meshsize is much below the persistence length of actin (about15–17 μm) and the adhesion of the tracer particle hindersdiffusion.
10-4
10-3
0.001 0.01 0.1 1 10 100
MS
D (μ
m2 )
time (s)
FIG. 5. (Color online) Mean squared displacement measuredwith a tracer bead (diameter of about 1.9 μm) embedded into asemiflexible biopolymer (actin) network (+ symbols [red]). Thedouble-logarithmic presentation clearly shows two different powerlaw domains. The dashed straight lines show local power law fitsin separate time ranges, while the beginning was better describedusing a Kelvin-Voigt model. The full line (black) is the interpolatedand resampled data, indicating the extended dynamic range in the0.001–0.01 s interval.
A. Preparation and data recording
All chemicals were purchased from Sigma-Aldrich andused as is if not specified otherwise.
1. Actin gel
Globular actin (G-actin) was prepared from rabbit skeletalmuscle as described in Refs. [28,29], and mixed with abiotinylated one from TebuBio resulting in a 0.63 mg/ml(14.65 μM) 1 : 100 Biotin G-actin solution.
F-actin was polymerized from the biotinylated G-actin ina polymerization buffer. To achieve an end concentration of11 μM G-actin [30], 15 μl of the actin solution was mixedwith 2.0 μl of 10× F-buffer (resulting in a final concentrationof 2.0 mM TRIS, 2.0 mM MgCl2, 0.1 M KCl, 0.2 mMCaCl2, 0.2 mM DTT, 0.5 mM ATP, pH 7.4), 2.0 μl ofpurified 1.89 μm polystyrene (PS) beads from Polysciences(Cat. No. 19814, original concentration 2.64 wt./vol. %) and0.7 μl phalloidin-TRITC (from Sigma, dissolved in methanolat 0.1 μM concentration).
The experimental chambers were made by cutting a hole of3 mm in diameter out of a 0.7 mm thin polydimethylsiloxane(PDMS, Sylgard 184 from Dow Corning, Germany) layer.The PDMS film was placed on a glass slide and 9.5 μl of thefreshly prepared actin polymerization solution was added intothis chamber right after mixing with 0.5 μl of a 1 M Mg2+solution, resulting in an Mg2+ ion concentration of 50 mM.The chamber was then sealed with a second glass slide.
The beads were observed after 3–4 h at a distance of atleast 100 μm from the chamber surface, using a 60× LUMFlhigh distance objective (NA = 1.2) from Olympus at roomtemperature.
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2. Particle tracking microrheology
Microrheology was performed by using a modified Olym-pus IX71 microscope with a red LED illumination (Thorlabs,λ = 670 nm) and a Basler 602f CMOS firewire camera. Thecamera is capable of 100 frames/s recording speed.
Particle positions were analyzed from the microscopyimages using a MATLAB (version 1.10.0.499, Mathworks,Inc.) version of the tracking algorithm originally published byCrocker et al. [31], developed by D. Blair and E. Dufresne [32].
All further numerical calculations, including MSD calcula-tion and numerical conversion, were written in PYTHON [33],an interpreted programming language. The programs werebased on the scientific PYTHON package (including NUMPY,and the LEASTSQ algorithm for nonlinear fitting) [34] andfor visualization the MATPLOTLIB [35]. All graphs were thencreated using GNUPLOT version 4.4 [36].
The algorithms implementing Eqs. (5) and (7) were testedusing Maxwell liquid and power law models, respectively, toensure their correct operation.
B. Evaluation with segmentwise interpolation
A time series containing 30 000 positions of the tracers wasrecorded with about 100 frames/s, and the MSD calculatedwith time average of overlapping intervals for 1/4th of thewhole time interval [37]. Because of the crosslinking, the timeregime where the thermal motion of the individual filamentsdominates the MSD is shifted to very short time values, belowour experimental sampling time [38]. Still, the first pointsdeviate from a straight line in the double logarithmic plot, andfit well to a Kelvin-Voigt profile. The rest of the curve was
0.01
0.1
1
10
100
0.01 0.1 1 10 100 1000
G’,
G’’
(Pa)
ω (1/s)
FIG. 6. (Color online) Frequency dependent storage modulusG′(ω) and loss modulus G′′(ω) obtained using Eq. (5) on theexperimental data (+ and × symbols, respectively) and the fitted,interpolated, and smoothed data ([blue and green] solid and dashedlines between the symbols). The experimental data are presented asvalues averaged in logarithmically equidistant bins. The high noise ofthe experimental data is indicated by the error bars, which representan estimated ± standard deviation in the bin. The data also show abreakdown of the storage modulus at high frequencies, which is anartifact related to the finite sampling bandwidth. A slight remaining ofthe oscillatory noise caused by the transitions between the consecutivefit regimes is also visible in the figure.
fitted using power law curves into 2 intervals (the individualfits are indicated by dashed lines in Fig. 5), each containingmore than 500 data points.
The fitted MSD was oversampled using the dynamicsampling method described above, with inserting 10 datapoints in the 0–0.01 s interval, then adding new time pointsbetween the data points such to keep the step size below h
according to Eq. (16) with ε = 10−3. The resulting data setwas smoothed around the matching points of the fits in the±0.03 s time range using the linear interpolation of Eq. (17),then converted via Eqs. (1) and (5) numerically. The resultingfrequency dependent complex shear modulus G∗(ω) of theexperimental data and the fitted, smoothed, interpolated modelare presented in Fig. 6.
G∗(ω) is dominated by the elastic response of the gel,indicating that the test particle indeed attached to the filamentsand the diffusion is effectively blocked in the time range of theexperiment.
VII. CONCLUSION
In summary, we have presented a segmentwise fitting andconversion method for calculating the complex shear modulusfrom mean squared displacement data obtained in polymergels. To complement the direct conversion methods proposedin the literature, which do not rely on assumptions about thedata, but show elevated error, we proposed using segmentwisefitting matching the data to theoretical predictions and interpo-lating using the fitted functions. This method also eliminatesthe problem of using a single analytical function fitting thewhole time range, namely that the function may not describethe whole data range, and allows a flexible analysis of thetrends in the data segments.
Because the resulted fit converts to an analytical form[Eq. (12)], which may not have a closed form Fouriertransform, we proposed an algorithm calculating the complexshear modulus numerically with controlled accuracy.
Inserting extra sampling points between the original datavia interpolation, and adding extra points at shorter time valuesthan the first measured time, extends the dynamic range of thedata eliminating the aliasing effect of the discrete sampling ofthe fitting functions. In order to control the precision, we haveintroduced a dynamic sampling based on the accuracy of thenumerical derivative present in the conversion formula Eq. (5).
To remove the possible oscillatory artifacts arising fromjumps between the fitted segments, one can either include newfunctions in between, or apply a linear local smoothing ofthe transitional regions between the segments. The resultingalgorithm is not limited to power functions only, as we havedemonstrated by combining a better fitting Kelvin-Voigt modelwith two power law segments in an experimental data set.
ACKNOWLEDGMENTS
This work was supported by the Ministry of Science,Research, and the Arts of Baden-Wurttemberg (Az: 720.830-5-10a). The authors would like to thank Professor Dr. JoachimP. Spatz and the Max Planck Society for the generoussupport of this work, Christine Mollenhauer and Dr. MichaelBaermann for the actin purification, Dr. Christian Bohm forthe fruitful discussions, and Dr. Claire Cobley for assistance.
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[1] G. Harrison, G. V. Franks, V. Tirtaatmadja, and D. V. Boger,Korea-Aust. Rheol. J. 11, 197 (1999).
[2] J. W. Goodwin and R. W. Hughes, Rheology for Chemists: AnIntroduction, 2nd ed. (RSC Publishing, 2008).
[3] A. R. Bausch, W. Moller, and E. Sackmann, Biophys. J. 76, 573(1999).
[4] T. M. Squires and T. G. Mason, Annu. Rev. Fluid. Mech. 42,413 (2010).
[5] T. A. Waigh, Rep. Prog. Phys. 68, 685 (2005).[6] T. G. Mason, T. Gisler, and K. Kroy, J. Rheol. 44, 917 (2000).[7] H. Lee, J. M. Ferrer, F. Nakamura, M. J. Lang, and R. D. Kamm,
Acta Biomat. 6, 1207 (2010).[8] B. Schnurr, F. Gittes, F. C. MacKintosh, and C. F. Schmidt,
Macromolecules 30, 7781 (1997).[9] M. Tassieri, R. M. L. Evans, T. A. Waigh, J. Trinick, and
A. Aggeli, J. Rheol. 54, 117 (2010).[10] M. Claessens, R. Tharmann, K. Kroy, and A. Bausch, Nat. Phys.
2, 186 (2006).[11] Q. Lu and M. J. Solomon, Phys. Rev. E 66, 061504 (2002).[12] J. He and J. X. Tang, Phys. Rev. E 83, 041902 (2011).[13] S. S. Rogers, T. A. Waigh, and J. R. Lu, Biophys. J. 94, 3313
(2008).[14] P. A. Pullarkat, P. A. Fernandez, and A. Ott, Phys. Rep. 449, 29
(2007).[15] T. G. Mason and D. A. Weitz, Phys. Rev. Lett. 74, 1250 (1995).[16] T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C.
Kuo, Phys. Rev. Lett. 79, 3282 (1997).[17] T. G. Mason, Rheol. Acta 39, 371 (2000).[18] J. H. H. Dangaria and P. J. J. Butler, Am. J. Physiol-cell Ph. 293,
C1568 (2007).[19] B. R. Dasgupta, S.-Y. Tee, J. C. Crocker, B. J. Frisken, and D. A.
Weitz, Phys. Rev. E 65, 051505 (2002).[20] F. Cheong, S. Duarte, S.-H. Lee, and D. Grier, Rheol. Acta 48,
109 (2009).
[21] R. M. L. Evans, M. Tassieri, D. Auhl, and T. A. Waigh, Phys.Rev. E 80, 012501 (2009).
[22] R. M. Evans, British Soc. Rheol. Bull. 50, 76 (2009).[23] D. Preece, R. Warren, R. M. L. Evans, G. M. Gibson, M. J.
Padgett, J. M. Cooper, and M. Tassieri, J. Opt. 13, 044022(2011).
[24] M. Tassieri, G. M. Gibson, R. M. L. Evans, A. M. Yao,R. Warren, M. J. Padgett, and J. M. Cooper, Phys. Rev. E 81,026308 (2010).
[25] D. C. Morse, Macromolecules 31, 7044 (1998).[26] D. C. Morse, Macromolecules 31, 7030 (1998).[27] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,
Numerical Recipes in C++, 2nd ed. (Cambridge UniversityPress, 2002).
[28] J. D. Pardee and J. A. Spudich, Methods Enzymol. 85, 164(1982).
[29] S. MacLean-Fletcher and T. D. Pollard, Biochem. Biophys. Res.Commun. 96, 18 (1980).
[30] M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, andD. A. Weitz, Phys. Rev. Lett. 91, 158302 (2003).
[31] J. C. Crocker and D. G. Grier, J. Colloid Interface Sci. 179, 298(1996).
[32] See [http://physics.georgetown.edu/matlab].[33] See [http://www.python.org].[34] E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open Source
Scientific Tools for PYTHON, [http://www.scipy.org/].[35] J. D. Hunter, Comput. Sci. Eng. 9, 90 (2007).[36] T. Williams, C. Kelley, H.-B. Broker, E. A. Merrit,
et al., GNUPLOT 4.4: An Interactive Plotting Program,[http://sourceforge.net/projects/gnuplot].
[37] M. Saxton, Biophys. J. 72, 1744 (1997).[38] M. Keller, R. Tharmann, M. A. Dichtl, A. R. Bausch, and
E. Sackmann, Philos. Trans. R. Soc. London A 361, 699(2003).
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