Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution
Sergei Fedotov, Daniel Han, Andrey Yu. Zubarev, Mark Johnston, Victoria J Allan
VVariable-order fractional master equation and clustering ofparticles: non-uniform lysosome distribution
Sergei Fedotov , Daniel Han , Andrey Yu. Zubarev , Mark Johnston , and Victoria JAllan Department of Mathematics, University of Manchester, M13 9PL Faculty of Biology, Medicine and Health, School of Biological Sciences, University ofManchester, M13 9PL Ural Federal University, 620083 Ekaterinburg, RussiaJanuary 8, 2021
Abstract
In this paper, we formulate the space-dependent variable-order fractional master equation tomodel clustering of particles, organelles, inside living cells. We find its solution in the long time limitdescribing non-uniform distribution due to a space dependent fractional exponent. In the continuousspace limit, the solution of this fractional master equation is found to be exactly the same as the space-dependent variable-order fractional diffusion equation. In addition, we show that the clustering oflysosomes, an essential organelle for healthy functioning of mammalian cells, exhibit space-dependentfractional exponents. Furthermore, we demonstrate that the non-uniform distribution of lysosomesin living cells is accurately described by the asymptotic solution of the space-dependent variable-order fractional master equation. Finally, Monte Carlo simulations of the fractional master equationvalidate our analytical solution.
Anomalous transport has gained much interest due to its applications in physics, chemistry, biology[1, 2, 3, 4]. It has proven to be a powerful theory to characterise dynamics of biological processesusing the fractional exponent, µ for the mean squared displacement, (cid:104) x ( t ) (cid:105) ∝ t µ . When describing thebroad ensemble statistics of random walkers, a constant fractional exponent suffices. For cell biology[5, 6, 7, 8, 9], studies of tracer particles in mammalian cell cytoplasm [10], in cellulo vesicle transport[11, 12, 13] and loci in bacteria [14] were all shown to be well described through a constant fractionalexponent. Quantifying dynamic cellular processes has been a major success for anomalous transporttheory and much scientific work is still ongoing [15, 16, 17, 18, 19, 20]. However, current experimentalstudies [21, 22, 23] are finding evidence of heterogeneous anomalous transport in intracellular processeswhile the theory for heterogeneous anomalous transport (specifically when the fractional exponent µ isno longer a constant) remains largely neglected. In fact, it is given knowledge that the cellular cytoplasmis a vastly heterogeneous complex fluid [24].In particular, organelles responsible for cellular metabolism and degradation called lysosomes movepredominantly subdiffusively with heterogeneous fractional exponents that depend on their spatial po-sitioning [25, 21]. This implies that lysosome dynamics should be adequately described by a fractionaldiffusion equation with a space dependent fractional exponent, µ ( x ). More interestingly, lysosomesalso maintain a stable non-uniform spatial pattern clustered near the centrosome in the cell [25]. So achallenge for modelling was posed: What is the asymptotic distribution to the space-dependent variable-order fractional diffusion equation [26] and does the experimental distribution of lysosomes match thisasymptotic distribution?Recently, we found the asymptotic representation of the solution of the variable-order fractionaldiffusion equation analytically with an ultra-slow spatial clustering (aggregation) of subdiffusive particles[27]. This equation is ∂p ( x, t ) ∂t = ∂ ∂x (cid:104) D µ ( x ) D − µ ( x ) t p ( x, t ) (cid:105) (1)1 a r X i v : . [ q - b i o . S C ] J a n here p ( x, t ) is the PDF of a particle at position x and time t . The function p ( x, t ) can also be interpretedas the mean density of particles at x and t ; µ ( x ) is the space dependent anomalous exponent; D µ ( x ) = a / τ µ ( x )0 is the fractional diffusion coefficient with a time scale τ and length scale a ; and D − µ ( x ) t isthe Riemann-Liouville derivative. Since we are interested heterogeneous subdiffusion, µ ( x ) ∈ (0 , D − µ ( x ) t p ( x, t ) = 1Γ( µ ( x )) ∂∂t (cid:90) t p ( x, t (cid:48) )( t − t (cid:48) ) − µ ( x ) dt (cid:48) For a monotonically increasing fractional exponent with domain x ∈ [0 , L ], the asymptotic solution to(1) is [27] p ( x, t ) ∼ µ (cid:48) (cid:16) tτ (cid:17) ∆ µ ( x ) Γ(1 − ∆ µ ( x )) (cid:20) ln (cid:18) tτ (cid:19) − ψ (1 − ∆ µ ( x )) (cid:21) (2)where ∆ µ ( x ) = µ ( x ) − µ (0), µ (cid:48) = dµdx (0) (cid:54) = 0 and ψ ( x ) is the digamma function. In [27], we solved (1)directly using the Laplace transform in the long time limit. The asymptotic solution in Laplace spacecorresponding to (2) is s ˆ p ( s, t ) ∼ − ( τ s ) ∆ µ ( x ) µ (cid:48) ln( τ s ) . (3)The PDF in (2) describes a non-uniform distribution clustered about the point of minimum µ ( x ) locatedat x = 0. This clustering has no analogue in the classical advection-diffusion models.In this paper, we formulate the space-dependent variable-order fractional master equation. Thepurpose is to show that the solution converges to that of the space-dependent variable-order fractionaldiffusion equation presented in [27]. Furthermore, we demonstrate that the probability density function(PDF) for the asymptotic representation of the solution of the variable-order fractional diffusion equationmatches the observed density of lysosomes clustering near the centrosome in living cells [25]. In whatfollows, we will demonstrate that the same solution is obtained from the continuous approximation tothe discrete master equation that generates (1). In this section, our aims are to formulate a master equation for the subdiffusive movement of particlesin heterogeneous media and find the solution of this master equation. Consider the random movementof a particle in a domain [0 , L ] and divide it into N subintervals of length h = LN . The mean number ofparticles in the subinterval i , spanning [( i − h, ih ], is denoted as n i ( t ) for 1 ≤ i ≤ N . Every subinterval, i , is characterized by a fractional exponent, µ i . The non-homogeneous fractional master equation canbe written as [28, 29] dn i ( t ) dt = − I i ( t ) + I i − ( t )= 12 τ µ i − D − µ i − t n i − ( t ) + 12 τ µ i +1 D − µ i +1 t n i +1 ( t ) − τ µ i D − µ i t n i ( t ) (4)where the fluxes, I i ( t ) = 12 τ µ i D − µ i t n i ( t ) − τ µ i +1 D − µ i +1 t n i +1 ( t ) I i − ( t ) = 12 τ µ i − D − µ i − t n i − ( t ) − τ µ i D − µ i t n i ( t ) . (5)The flux of particles from subinterval i to i + 1 is I i ( t ) and from subinterval i − i is I i − ( t ). Aschematic of (4) showing the fluxes is seen in Figure 1Since we assume there is no external flux of particles entering our domain [0 .L ], the total mass isconserved: N (cid:88) i =1 n i ( t ) = n. (6)2igure 1: A diagram that shows the subinterval i with boundaries [( i − h, ih ] drawn with dashed lineslabelled with the corresponding fractional exponent µ i . The subintervals on either side have fractionalexponent µ i − and µ i +1 and every subinterval has width h . The small angled arrows represent particlesleaving each subinterval and the large round arrows show the flux of particles for subinterval i .For the boundary intevals, i = 1 and i = N in (4), we have dn i ( t ) dt = − I ( t ) , dn N ( t ) dt = I N − ( t ) . (7)Taking the Laplace transform of (4), we obtain s ˆ n i ( s ) − n i (0) = − ˆ I i ( s ) + ˆ I i − ( s ) (8)for 2 ≤ n ≤ N −
1, where Laplace transforms of (5) areˆ I i ( s ) = s ˆ n i ( s )2( τ s ) µ i − s ˆ n i +1 ( s )2( τ s ) µ i +1 , ˆ I i − ( s ) = s ˆ n i − ( s )2( τ s ) µ i − − s ˆ n i ( s )2( τ s ) µ i . (9)For i = 1 and i = N , (7) becomes s ˆ n ( s ) − n (0) = − ˆ I ( s ) s ˆ n N ( s ) − n N (0) = ˆ I N − ( s ) . (10)In the long time limit, τ s →
0, since ˆ I i − ( s ) ≈ n i ( s ) in terms of ˆ n i − ( s ) asˆ n i ( s ) (cid:39) ( τ s ) µ i − µ i − ˆ n i − ( s ) (11)Now we define µ i − µ i − = αh such that the difference between exponents at two neighboring sitestends to zero as N → ∞ . For µ ( x ) a linear function of x , α will be the gradient. Then (11) becomesˆ n i ( s ) (cid:39) ( τ s ) αh ˆ n i − ( s ) . (12)By summing (8) and (10) for all i and using the conservation of total mass from (6), we can obtain N (cid:88) i =1 s ˆ n i ( s ) = n. (13)Then using the recursive relation (12), (13) can be written as s ˆ n ( s ) (cid:104) τ s ) αh + ( τ s ) αh · · · + ( τ s ) ( N − αh (cid:105) = n. So, we can find the solution of ˆ n ( s ) in Laplace space as s ˆ n ( s ) = n (cid:80) N − k =0 ( τ s ) kαh = n − ( τ s ) αh − ( τ s ) Nαh . (14)3sing (12) together with (14), we find the solution of ˆ n i ( s ), for 2 ≤ i ≤ N , in Laplace space as s ˆ n i ( s ) = n ( τ s ) ( i − αh − ( τ s ) αh − ( τ s ) Nαh . (15)For the asymptotic limit τ s →
0, (14) and (15) become s ˆ n ( s ) (cid:39) n (1 − ( τ s ) αh ) ,s ˆ n i ( s ) (cid:39) n ( τ s ) ( i − αh (1 − ( τ s ) αh ) . (16)To verify that the master equation (4) does indeed correspond to the space-dependent variable-orderfractional diffusion equation in [27], we need to introduce the densityˆ p i ( s ) = ˆ n i ( s ) h , then s ˆ p i ( s ) = n ( τ s ) ( i − αh (1 − ( τ s ) αh ) h . (17)Then setting x = ih and using the well known formulalim h → ( τ s ) h − h = α ln( τ s ) , we obtain, in the continuous limit, s ˆ p ( x, s ) = lim h → s ˆ p i ( t ) = − αn ( τ s ) xα ln( τ s ) . (18)This is equivalent to the solution found in Ref. [27], shown as (3) in this paper. Monte Carlo simula-tions for the master equation (4) were performed and shown to have excellent correspondence with thisasymptotic solution as shown in Figure 3. The simulations were made by generating random residencetimes, T , for particles in box i drawn from the PDF ψ µ i ( τ ) = − ( ∂/∂τ ) E µ i [ − ( τ /τ ) µ i ] (details can befound in [30]) after which the particle jumps right or left with equal probability. The fractional exponent µ i was increasing linearly as i increased (full details can be found in [27]). In the next section, we showthat this asymptotic solution (18) and (2) corresponds to the distribution of lysosomes in living cells. Lysosomes are intracellular organelles that degrade macromolecules and regulate metabolism [31]. It iswell-established that under normal conditions, the majority of them are concentrated in the perinucleararea, although at any one time a fraction of them are undergoing active bi-directional movement towardsand away from the nucleus [25]. Figure 2 shows the non-uniform distribution of lysosomes (green).However, the exact mechanisms for how lysosomes maintain such a macroscopic spatial distributionremain unclear. The aim of this subsection is to show that lysosomal distributions in the cell can beexplained to a large extent by the anomalous mechanism detailed in this paper, since subdiffusion isthe most prevalent characteristic in lysosomal movement [25, 21]. Anomalous subdiffusion can occuras a result of non-uniform crowdedness [10] in the cytoplasm. Our hypothesis is that the non-uniformfractional exponent µ ( x ) can serve as a measure of crowdedness in the cytoplasm such that x is thedistance away from the nucleus and that lysosomes display a stable distribution across cells and timesdue to this non-uniform subdiffusion.We performed live-cell imaging experiments to analyze lysosome positions; experimental and analysismethods are detailed in Section 4. Figure 3a shows the empirical PDF (points) of finding a lysosome ata certain distance from the cell center from a sample of HeLaM cells and the asymptotic PDF (2) (line)for parameters µ (cid:48) = 0 . τ = 9 . × − s and t = 8 . × s. One can see that the predictioncorresponds well to the empirical PDF. Furthermore, we measured the fractional exponents directlyfrom the same HeLaM cells by calculating the time-averaged mean squared displacement (cid:104) r ( τ ) (cid:105) of eachtrajectory and fitting to a power law τ µ . Then from the lysosome population with 0 < µ < x = 0 to4igure 2: Non-uniform lysosome distribution in two HeLaM cells. Methanol fixed HeLaM cells werelabeled with antibodies to the lysosomal protein LAMP1 (Lysosomes, green) and DAPI to label DNA inthe nucleus (blue). Lysosomes are non-uniformly distributed with a large cluster around the perinuclearregion and fewer lysosomes throughout the rest of the cell. Scale bar shows 10 µ m.Figure 3: Data from live-cell imaging experiments of HeLaM cells with LysoBrite labelled lysosomes. (a)Normalized density of lysosomes, p ( x, t ), against the normalized displacement away from the cell centreinside 30 HeLaM cells. The experimental data (black dots) and fit of the asymptotic solution (blue solidline) show excellent correspondence with parameters µ (cid:48) = 0 . τ = 9 . × − and t = 8 . × .The Monte Carlo simulations (orange crosses) of N = 2 × particles with the same parameters asthe fit also show excellent correspondence with the analytical solution. (b) The plot of experimentallymeasured fractional exponent µ ( x ) against the normalized displacement away from the cell centre for 30HeLaM cells. x = 1. The fluctuations in µ ( x ) near x = 0 and x = 1 is due to low number of lysosomes found exactlyat the cell centre or the cell periphery.The anomalous mechanism presented in this paper is obviously not a complete theory to describethe non-uniform distribution of intracellular organelles. There are many other interactions and phe-nomena that occur in conjunction. Two primary additional phenomena that will affect this pattern isthe superdiffusion generated by motor protein transport of organelles [11, 12, 32] and the non-linearinteraction of subdiffusive organelles [33] such as the lysosome tethering to the endoplasmic reticulum5bserved in [25]. Furthermore, there are several other mechanisms, such as viscoelasticity and diffusionin labyrinthine environments, that lead to subdiffusive motion of organelles (see the excellent review [6]).Including these additional effects in future works should provide a more physical and accurate model oforganelle organization in the cell. However, (2) models the long time limit of lysosomal distributionswhere the effects of heterogeneous subdiffusion in the cell will dominate. HeLaM cells were maintained in Dulbecco’s Modified Eagle’s Medium (DMEM) - high glucose (Sigma-Aldrich, Dorset, UK) with 10% Fetal Bovine Serum (GE Healthcare, Buckinghamshire, UK) at 37 ◦ Cand 8% CO .For fixed images like Figure 2, HeLaM cells were seeded onto − ◦ C methanol for six minutes, rinsed in phosphate buffered saline (PBS)and labelled with mouse LAMP1 primary antibody (1/500 dilution) (Product Number: H4A3, Develop-mental Studies Hybridoma Bank, Iowa City, USA). After washing in PBS, the coverslips were incubatedwith donkey anti-mouse Alexa594 secondary antibody (1/800 dilution) (Jackson Immuno Research Lab-oratories Inc., West Grove, USA). Coverslips were left for 30 minutes at room temperature, washed oncewith PBS for 5 minutes and then stained with 4’,6-diamidino-2-phenylindole (DAPI) (0.1 µ g mL − inPBS) for 5 minutes, followed by a final wash with PBS. Coverslips were then mounted onto glass slidesusing ProLong ® Diamond mounting agent (Life Technologies, Paisley, UK). Cells were imaged by fluo-rescence microscopy using an Olympus BX Microscope, using a 60 × /1.4 objective, CoolSNAP EZ CCDcamera (Photometrics, Tucson, USA) and MetaMorph software (Molecular Devices, San Jose, USA).For live imaging, the HeLaM cells were first transfected with EGFP-C1 (Clontech Laboratories,Inc. CA, USA GenBank Accession µ L of jetPEI and 100 µ L of NaCl (150mM). Then they were stained withLysoBrite Red (AAT Bioquest), imaged using fluorescence microscopy and the lysosomes were trackedusing Imaris. The cells were grown in MEM (Sigma Life Science) and 10% FBS (HyClone) and incubatedfor 48 hr at 37 in 5% CO on 35 mm glass-bottomed dishes ( µ -Dish, Ibidi, Cat. No. 81150). LysoBritewas diluted 1 in 500 with Hank’s Balanced Salt solution (Sigma Life Science), then 0.5 mL of this solutionwas added to cells on a 35 mm dish containing 2 mL of growing media and incubated at 37 for at least1 hr. Then, cells were washed with sterile PBS and the media replaced with growing media.After at least 6 hr incubation at 37 ◦ C in 8% CO , the growing media was replaced with live-imagingmedia composed of Hank’s Balanced Salt solution (Sigma Life Science, Cat. No. H8264) with addedessential and non-essential amino acids, glutamine, penicillin/streptomycin, 25 mM HEPES (pH 7.0)and 10% FBS (HyClone). Live-cell imaging was performed on an inverted Olympus IX71 microscopewith an Olympus 100 × (cid:104) r ( mδτ ) (cid:105) = 1 N − m N − m (cid:88) i =1 (cid:0) [ x ( t i + mδτ ) − x ( t i )] + [ y ( t i + mδτ ) − y ( t i )] (cid:1) where x and y are the 2D co-ordinates obtained from tracking; and the video contains N frames separatedin increments of δt seconds. Then a power-law was fit to the time-averaged mean squared displacementsusing the in scipy.optimize package Python. 6 Conclusion
In this paper, we have formulated the space-dependent variable-order fractional master equation (4)and found its asymptotic solution (18) describing the clustering of particles due to a space dependentfractional exponent. In the continuous limit as the width of subintervals approaches zero, the solutionto this master equation converges to that in the space-dependent variable-order fractional diffusionequation found in [27]. This solution describing the non-uniform distribution of lysosomes inside the cellis confirmed by Monte Carlo simulations of the fractional master equation. We present a new mechanismfor lysosome clustering due to a non-uniform fractional exponent. This is a possible explanation for thenon-random, stable and yet distinct spatial distribution of lysosomes in the intracellular space foundbiologically [25]. The distribution of lysosome density in living cells matches the asymptotic probabilitydensity function (2) which is the solution of the space-dependent variable-order fractional diffusion andmaster equation, seen in Figure 3. Finally, lysosomes have been found to cluster dynamically based on ERspatial density and interact with a variety of other organelles dependent on position and particle density[25]. These dynamic interactions can be described in terms of non-linear subdiffusive fractional equationssimilar to [29, 33]. Our results give an alternative explanation for clustering and intracellular transportof gold nanoparticles within lysosomes in living cells [34]. It is important because gold nanoparticles haveproven to be promising radiosensitizers for improving proton therapy, since they enhance the radiationdamage to tumour cells [35, 36, 37].
SF and AYZ conceived and designed the study and drafted the manuscript. DH conceived and designedthe study, drafted the manuscript, carried out the experiments and performed data analysis. MJ andVJA carried out the experiments. All authors read and approved the manuscript.
SF and AYZ acknowledge financial support from RSF project 20-61-46013. DH acknowledges financialsupport from the Wellcome Trust Grant No. 215189/Z/19/Z. VJA acknowledges financial support fromthe EPSRC Grant No. EP/J019526/1.
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