A model of cell-wall dynamics during sporulation in Bacillus subtilis
AA model of cell-wall dynamics during sporulation in
Bacillus subtilis
Li-Wei Yap , ‡ and Robert G. Endres , , ∗ Department of Life Sciences, Imperial College, London, United Kingdom. Centre for Integrative Systems Biology and Bioinformatics, Imperial College, London, United Kingdom.
To survive starvation,
Bacillus subtilis forms durable spores. After asymmetric cell division, theseptum grows around the forespore in a process called engulfment, but the mechanism of forcegeneration is unknown. Here, we derived a novel biophysical model for the dynamics of cell-wallremodeling during engulfment based on a balancing of dissipative, active, and mechanical forces.By plotting phase diagrams, we predict that sporulation is promoted by a line tension from theattachment of the septum to the outer cell wall, as well as by an imbalance in turgor pressures inthe mother-cell and forespore compartments. We also predict that significant mother-cell growthhinders engulfment. Hence, relatively simple physical principles may guide this complex biologicalprocess.
INTRODUCTION
Bacillus subtilis is a rod-shaped bacterium with athick (30-40 nm) outer cell wall made of peptidogly-can (PG) polymers for withstanding high ( ∼ . FIG. 1: Schematic representation of morphologicalchanges that occur during sporulation in
Bacillus sub-tilis . The cell wall is depicted in green, cell membranesin yellow, and DNA in blue. During DNA replication,septation is initiated near one of the poles (a), and thedaughter DNA strand is pumped into the forespore (b).Then, the septum grows around the forespore (c), allowingthe mother-cell membrane to engulf it (d).
GENERAL EQUATION FOR CELL-WALLDYNAMICS
Similar to the original framework [1], the active andmechanical forces in our model are given by thederivatives of the total energy E with respect to N shape degrees of freedom. Similar to Ref. 1, we fo-cused on the cell wall, but in the ESI † we also con-sidered the role of the membrane [14] (Figs. S1a-b inESI † ). These shape degrees of freedom are specifiedby the generalized coordinates q i ( i = 1 , ..., N ) andtheir respective velocities ˙ q i : η i V i ˙ q i ( q i ) = − ∂E∂q i , ∀ i (1)with viscosity constant η i and volume V i = h · A i overwhich dissipation of q i occurs. Here, h is the thicknessof the cell wall assumed to be constant and A i is thesurface area of dissipation.The left-hand side of Eq. 1 describes the dissipa-tive force, whose corresponding energy represents thework done to the medium when the cell shape deformsat a rate ˙ q i , arising from the insertion of newly syn-thesized PG strands into the cell wall [15–17]. Theright-hand side describes the sum of active and me-chanical forces, both of which, when integrated, rep-resent the work done to the cell wall when the cellshape deforms by q i . The active forces arise fromdistributed macromolecules that convert chemical en-ergy into mechanical work; these forces include the a r X i v : . [ q - b i o . S C ] O c t chemical potential for PG synthesis, as well as theline tension caused by the active remodeling of theseptum, which provides room for the mother-cell cy-toplasm to entropically expand and hence engulf theforespore [18]. The mechanical forces arise from theinternal turgor pressure and the opposing surface ten-sion of the elastic cell wall that act to increase anddecrease cell volume, respectively. Further includedin the mechanical force is the bending stiffness, re-ducing the degree of bending away from zero or anypreferred curvature. We neglected the energy of inter-action with cytoskeletal filaments due to lack of con-crete experimental evidence. (The FtsZ-initiated sep-tum is our initial condition in the models, and thereare no established roles of FtsZ and MreB during en-gulfment.) Assuming no external forces or thermalnoise, the dissipative energy is balanced by the activeand mechanical energies [13]. We implemented botha minimal model for engulfment and a more realisticmodel, which accounts for mother-cell and foresporegrowth. MINIMAL MODEL OF ENGULFMENT
B. subtilis is modeled as a cylindrical cell with twohemispherical poles. Here, we use a minimal modelfor the purpose of gaining intuition. In this mini-mal model, we assumed for simplicity that the sep-tum is already curved from the start (Fig. 2), al-though in reality the septum is initially flat [6]. Sincethe shape of the septum is fixed, the forespore is al-ways spherical. Therefore, the radius r and length L of the central cylindrical region of the mother cell,as well as the different turgor pressures and volumesof the mother cell ( p m , and V m = πr L ) and fores-pore ( p s , and V s = 4 πr /
3) are constant. The onlyshape degree of freedom is angle θ of engulfment (Fig.2), and the surface area over which dissipation oc-curs during increase in θ is A θ = 2 πr sin θ . Sur-face area A s = 2 πr (1 + sin θ ) of the forespore cellwall and distance r s = r · cos θ between the leadingedge of the engulfing membrane and the longitudi-nal axis are both functions of θ . The Helfrich bend-ing energy of the septum E bends = k s [2 πr (2 /r ) +2 πr sin θ (2 /r − /R ) ] / θ . Con-versely, surface area A m = 4 πr +2 πrL of the mother-cell wall and bending energy of the mother-cell wall E bendm = k m [2 πrL (1 /r − /R ) +4 πr (2 /r − /R ) ] / θ and hence are constant. Thebending energies are described in terms of circumfer-ential bending rigidity ( k m for mother cell, k s for fore-spore) and preferred radius R of the cell-wall cross-section.The sum of active and mechanical energies is givenby: E = − p m V m − p s V s + ( γ − ε )( A m + A s ) + 2 πr s f + E bendm + E bends , (2)with surface tension γ , chemical potential ε for PGremodeling, line tension f , mother-cell and foresporeturgor pressures p m and p s , and cell-wall and sep-tum bending energies E bendm and E bends . Similar toRef. 1, we used a constant surface tension as r is FIG. 2: Minimal model of engulfment. The only shapedegree of freedom is angle θ of engulfment. The red dotsrepresent the leading edge of the engulfing membrane, and θ increases over time to maximally π/
2. Surface area ofthe forespore cell wall, A s , as well as distance betweenthe leading edge and the longitudinal axis, r s , change inresponse to θ . All other parameters including cell radius r , cell length L , pressure p , volume V , and the surfacearea A m of the mother-cell wall are constant. either fixed or strongly constrained by MreB so thatdifferent functional forms of γ would not have mucheffect. The line tension may represent the energy costfor remodeling the attachment of the septum to theouter cell wall by the SpoIID/M/P complex [10, 18],or originate from the membrane, which has to bendbackwards onto itself. This expression for E was sub-stituted into Eq. 1 for cell-wall dynamics with q = θ .Since we are interested in the partial derivative of E with respect to θ , the terms of Eq. (2) that are eitherconstant or not functions of θ can be ignored: E ≈ δ · A s + 2 πf · r s + E bends , (3)with δ = γ − ε . Using Eq. (3) in Eq. (1), we obtained: dθdt = µ θ · θ r · sin θ · (cid:104) πrf · sin θ − δ · πr · cos θ − k s πr (cid:16) r − R (cid:17) · cos θ (cid:105) , (4)where µ θ = 1 / (2 πhη θ ) is the mobility coefficient ofengulfment. To make the various parameters dimen-sionless, surface tension was rescaled as ˜ γ = γ/ ( pR ),chemical potential as ˜ ε = ε/ ( pR ), line tension as ˜ f = f / ( pR ), and circumferential bending rigidity of themother-cell and forespore as ˜ k m = k m / ( pR ) = 3 . k s = k s / ( pR ) = 0 .
18 (see Table S1 in ESI † )respectively, with p = 1 . R = 0 . µ m [1] and ˜ δ = ˜ γ − ˜ ε . Since r is fixed at the samevalue as R (see Table S1 in ESI † ), the last term − k s πr (cid:16) r − R (cid:17) · cos θ in Eq. (4) effectively can-cels out. Moreover, as the septum and forespore cellwall are initially assumed to be a single PG layer [18], k s (cid:28) k m , so E bends has a minor contribution to theengulfment dynamics (Fig. S1f in ESI † ). As ˜ f and ˜ δ are not well-constrained by experiments, we scannedthrough these parameters.We first investigated the conditions of ˜ δ and ˜ f thatfavor engulfment. For engulfment to occur, energymust be released into the environment, i.e. ∂E/∂θ <
0, allowing θ to increase from 0 to the maximum π/ δ <
0, i.e. when the chemical potential for PG re-modeling is greater than the surface tension (Fig. 3a).Thus, there is competition between ˜ γ and ˜ ε , where˜ ε favors engulfment, whereas ˜ γ represents an energypenalty for engulfment (both parameters are assumedpositive). The more negative ˜ δ is, the easier it is toovercome the energy barrier for engulfment. Engulf-ment is favored when ˜ f > f is multiplied by r s = r cos θ , and r s decreases as θ increases and engulfment proceeds. In fact, if ˜ f is suf-ficiently large, engulfment may occur even for positive˜ δ (Fig. 3c). It may seem strange that engulfment isfavored by a positive line tension ˜ f , because a tensionrepresents an energy penalty. However, engulfmentforces are about changes in energy, and engulfmentreduces the penalty from the line tension due to de-creasing radius r s . Hence, Figs. 3a-b show that en-gulfment is driven by both growth (˜ δ <
0) and linetension ( ˜ f > † , we also varied r , andfound that deviation from the preferred radius R haslimited effect on the plot for θ ( t ) (Fig. S1c in ESI † ).We analytically verified the steady-state angles ofengulfment θ ∗ (Figs. 3a-b) using Eq. 4. The lowersteady state ( θ ∗ = 0) is in fact the initial θ ( t = 0),which increases over time towards the upper steadystate ( θ ∗ = tan − [( r ˜ δ + 2 r ˜ k s (1 /r − /R ) ) / ˜ f ]). Al-though engulfment is completed when θ = π/
2, theupper steady state might be greater than π/
2. Infact, tan ( π/
2) is undefined, so only in the absence ofline tension ( ˜ f = 0) will the steady state be exactly π/
2. This implies excess energy in the cell wall whenengulfment is complete. The larger the difference be-tween π/ π/ δ and ˜ f in the ensemble plots (Figs. 3a-b) todetermine stability at both steady states θ ∗ , using˙ δθ (cid:39) g (cid:48) ( θ ∗ ) δθ . If g (cid:48) ( θ ∗ ) >
0, the perturbation ( δθ )grows exponentially, indicating unstable equilibrium,whereas if g (cid:48) ( θ ∗ ) < δθ dampens out, indicating sta-ble equilibrium [20]. Note that g (cid:48) ( θ ∗ = 0) = 0, whichindicates a need for energy consumption at t = 0. Wefound that there is instability at small θ (near θ ∗ )and stability at θ ∗ , which explains the increase in θ over time. To analyze the stability of the lower steadystate θ ∗ = 0 further, we take the limit of θ (cid:38) dθ/dt = − πµ θ ˜ δ · θ , where engulfmentproceeds for ˜ δ <
0, i.e. ˜ ε > ˜ γ . Hence, synthesis isrequired to get engulfment started.The mother cell synthesizes considerable amountsof membrane, also required for compartment-specificexpression of transcription factors [14, 21]. To studythe effect of membrane synthesis on the cell wall, weextended the minimal model in the ESI † to includethe membrane surface areas (Fig. S1b in ESI † ). As-suming that the chemical potentials for synthesizing FIG. 3: Ensemble plots and phase diagram for minimalmodel. (a) Plot of θ ( t ) when difference ˜ δ = ˜ γ − ˜ ε betweensurface tension and chemical potential is varied, whilstline tension ˜ f = 0 .
04 is kept constant. The solid linesrepresent engulfment up to π/ π/ δ <
0. (b) Plot of θ ( t ) when ˜ f is varied, whilst ˜ δ = − . f >
0. (c) Phase diagram in the (˜ δ , ˜ f ) plane.For all other parameters, see Table S1 in ESI † . the cell wall and membrane are the same, the value of˜ δ in the ordinate axis of the phase diagram (Fig. 3c)is effectively reduced, so that it is easier for cells tobe in Region II in which engulfment occurs. Whilstthis seems counter-intuitive as additional membranesynthesis is required (a cost), we assumed that thereis sufficient energy available to drive membrane syn-thesis. This implies that engulfment actually relievesthis drive or ‘pressure’. REALISTIC MODEL OF ENGULFMENT
In reality the septum is initially flat, so that theforespore is hemispherical prior to engulfment [6, 7].Over time, the septum becomes increasingly curved asthe forespore expands into the mother cell to form ahemispheroid joined to the initial hemisphere. Hence,there might be competition between engulfment andforespore expansion for limited resources during star-
FIG. 4: Realistic model of engulfment. There are twomain shape degrees of freedom; angle θ of engulfment andexpansion l of the forespore into the mother cell. Withmother cell growth during engulfment, there are two moreshape degrees of freedom; cell radius r and cell length L .The red dots represent the leading edge of the engulfingmembrane, and θ increases over time to maximally π/ V m and V s ) and cell-wall surface areas ( A m and A s ), as well as distance be-tween the leading edge and the longitudinal axis, r s , arefunctions of one or more shape degrees of freedom. vation.With mother-cell growth, there are four shape de-grees of freedom: angle θ of engulfment, expansion l ofthe forespore into the mother cell along its longitudi-nal axis, as well as the mother-cell radius r and length L (Fig. 4). The mother cell and forespore volumes are V m = πr + πr L − πr l and V s = πr ( l + r ), re-spectively. The forespore cell wall surface area is A s = πr [1 + l · sin − ( (cid:112) − r /l ) / ( r (cid:112) − r /l ) + 2 sin θ ].The mother cell wall surface area A m , distance r s be-tween leading edge of engulfing membrane and lon-gitudinal axis, the surface area A θ over which dis-sipation occurs during increase in θ , as well as thecell-wall bending energy E bendm , are the same as inthe minimal model. Since k s (cid:28) k m , E bends hasa minor contribution to the engulfment dynamics,we neglect E bends in order to remain having analyt-ical expressions for the energies (further explainedin Discussion and Conclusions). The surface areaover which dissipation occurs during increase in l is A l = πr [1 + l · sin − ( (cid:112) − r /l ) / ( r (cid:112) − r /l )].The surface area over which dissipation occurs dur-ing increase in r is A r = A m + A s . The surface areaover which dissipation occurs during increase in L is A L = 2 πrL . With Eq. (2) for the sum of activeand mechanical energies, we derived dl/dt , dr/dt , and dL/dt (see ESI † for the complete formulae).We initially assumed for simplicity that no re-sources are diverted to mother-cell growth due to star-vation, so r and L are constant. This is consistentwith previously published time-lapse microscopy data[7, 18], showing that the cell volume remains constantthroughout engulfment. (Later, the mother cell is al- FIG. 5: Phase diagrams in (˜ δ, ˜ f ) plane for realistic model.(a) Effect of ∆ p in units of MPa for µ r = µ L = 0 (nogrowth of mother cell). As the mother-cell pressure in-creases relative to the forespore pressure, ∆ p and RegionIV increase, whilst Region III decreases. The lines in thephase diagram were determined by contour plots, whichshowed whether engulfment or expansion are complete forhundreds of thousands of combinations of parameters, in-cluding ˜ δ , ˜ f , and ∆ p . The contour plots were then super-imposed to obtain regions that show whether engulfmentis completed before expansion. (b) Effect of µ r and µ L atfixed ∆ p = − . † . lowed to grow during sporulation, so that r and L increase simultaneously with θ and l .) To make thevarious parameters dimensionless, surface tension andcircumferential bending rigidity were rescaled as inthe minimal model. The initial radius r is set to R [1], whilst the initial length L was set to 3 . µ m,which is the average experimentally measured value[22].We investigated the conditions of turgor pressurethat favor an increase in θ ( t ) and l ( t ). Phase di-agrams were plotted for varying values of the pres-sure difference between the mother cell and forespore∆ p = p m − p s (Fig. 5a). The value of ˜ f was set to0 .
2, as in the original framework [1], and we chose˜ δ = − . p . Region I represents theabsence of both engulfment and forespore expansion.Region II represents the situation of incomplete en-gulfment (terminated at L/
2) and excessive foresporeexpansion. Region III represents the situation whereforespore expansion is completed before engulfment.Finally, Region IV represents the situation where fore-
FIG. 6: Comparison of model predictions with time-lapsemicroscopy data ( n is number of individual cells observed).(a) Time trace of experimentally measured engulfment [18](black symbols) and results from numerical calculation us-ing realistic model (blue solid line). 50% engulfment is as-sumed to occur as soon as septum formation is completeand prior to septum remodeling. The vertical dashed linerepresents the start of engulfment at 0 . h . Parameters: µ θ = 1 . m J − h − , ˜ δ = − .
5, ˜ f = 0 .
2, and ∆ p = − . † . (b)Time trace of experimentally measured forespore surfacearea A s [18] (black symbols) and results from numericalcalculation using realistic model (blue solid line). Param-eters: µ θ = 1 . m J − h − , r = 0 . µm , ˜ δ = − . f = 0 .
2, and ∆ p = − . † . spore engulfment is completed without major fores-pore expansion.To understand which region in Fig. 5a might bephysiologically relevant, we compared the model withtime-lapse microscopy data [18]. For ˜ δ <
0, ˜ f > p < A s (Figs. 6a-b, see cap-tion as well as Table S1 in ESI † for extracted param-eter values). The experimental data [18] only showslittle increase in l once engulfment is complete, soRegion III represents the ideal set of parameter val-ues for engulfment and forespore expansion. As ∆ p decreases, Region III increases, whilst Region IV de-creases (transition from dashed to solid lines). Thisconfirms a previous hypothesis that the forespore ex-pands because of higher osmolarity and in turn higherturgor pressure than in the mother cell [23]. Fur-ther support comes from the ensemble plot of l ( t )for − . ≤ ∆ p ≤ . † ).The dashed lines in the ensemble plot show foresporeexpansion if allowed to continue after engulfment iscomplete, and for ∆ p ≥ l ( t ) is predicted to increase sharply to the point of complete foresporeexpansion only after engulfment is complete. This,taken together with the experimental data [18], sug-gests that ∆ p < θ again shows insta-bility at small θ (near θ ∗ ) and stability at θ ∗ =tan − [( r ˜ δ + 2 r ˜ k s (1 /r − /R ) ) / ˜ f ], which explainsthe increase in θ over time. Linear stability analy-sis of l shows that small l (near l ∗ = 0) is unstable for∆ p < p > l ( t ) occurs earlier when ∆ p < l = L/ l , suggesting a need for energyfor forespore expansion. Perhaps forespore expansionis driven by the utilization of limited energy resourcesin the cytoplasm [24]. Furthermore, the instability at l = L/ r and L , we wondered how forespore shapeinfluences engulfment. For that purpose, we com-pared the engulfment dynamics of the realistic (withfixed r and L ) and the minimal model (see ESI † ). In-deed, as differing only by forespore shape, we foundthat both models yield nearly the same engulfmentdynamics for the same set of parameter values, e.g.the same plot for θ ( t ) is produced by the minimal andthe realistic model for ˜ δ = − . f = 0 . † ).Finally, we investigated whether engulfment and/orforespore expansion would still be favored if r ( t ) and L ( t ) are allowed to increase simultaneously with θ ( t )and l ( t ). Phase diagrams for different µ r and µ L wereplotted for a constant ∆ p in the (˜ δ, ˜ f ) plane, usingthe same values of µ θ and µ l as before (Fig. 5b).Here, µ r = 1 / ( πhη r ) and µ L = 1 / (2 πhη L ) are themobility coefficients of radial and longitudinal growth,respectively. Notably, if µ r = µ L = 0, then the phasediagram is the same as in Fig. 5a, since dr/dt = dL/dt = 0.For large values of µ r = 2 m J − h − and µ L =4 · − m J − h − , mother-cell growth is significant,leading to a significant increase in Region II and a sig-nificant decrease in Region III (Fig. 5b, dashed lines).Thus, there is a higher chance that engulfment is notcompleted. Hence, significant growth is detrimentalto sporulation, potentially because a large amountof chemical potential is required for PG remodeling.If this chemical potential is diverted to mother-cellgrowth, then there would not be enough resourcesfor engulfment or forespore expansion, in line withsporulation being a starvation response. However,for small values of µ r = 4 · − m J − h − and µ L = 4 · − m J − h − , mother-cell growth is lim-ited and leads to only a slight decrease in Region IIIand no increase in Region II (solid lines). Thus, the in-hibition of engulfment by growth is limited. Althoughrod-shaped bacteria like B. subtilis usually grow lon-gitudinally rather than radially [26], our model pre-dicts similar behavior of r ( θ ) and L ( θ ) as compared tothe original framework [1], especially for ∆ p = − . r ( θ ), whichincreases slightly and reaches a plateau (Figs. S3cand S4c in ESI † ), as well as from the plot for L ( θ ),which increases linearly (Figs. S3d and S4d in ESI † ),whilst engulfment is not yet complete. That we re-obtain similar behavior of r ( t ) and L ( t ) is remarkablegiven that the original framework does not accountfor sporulation. DISCUSSION AND CONCLUSIONS
Elucidating the dynamics of cell-wall remodeling iskey to understanding the conditions that favor en-gulfment during sporulation in
B. subtilis . With ourmodel, we showed that it is energetically possiblefor PG remodeling to drive membrane migration andforespore expansion. Thus, PG likely arose before theevolution of sporulation as the latter is a consequenceof PG biochemistry [27]. To promote sporulation,PG remodeling is surprisingly aided by a line ten-sion and a turgor-pressure imbalance in the mothercell and forespore compartments. Furthermore, sig-nificant mother-cell growth is detrimental to sporu-lation, as it diverts excessive energy resources fromengulfment. How could these predictions be tested?Pressure could be measured in the mother cell andforespore by AFM indentation and fluorescence mi-croscopy [28, 29]. Indeed, packing of charged DNAin the small forespore may induce osmotic swellingin line with Region III [30]. High-throughput imag-ing could be used to test if some cells grow despitesporulation.While predictive, our modeling is minimalistic anda number of simplifications were necessarily made.First, similar to earlier models [1, 15–17], membraneenergetics and dynamics were neglected. Indeed, asshown in Figs. S1a-b with the minimal model, mem-brane contributions are negligible (as long as there issufficient energy available to drive membrane synthe-sis). Second, the detailed shape of the forespore cellwall is unknown. However, by comparing the spheri-cal and spheroidal forespore shapes without mother-cell growth, the results from the ensemble plots for θ ( t ) are very similar without qualitative differences(Figs. S2a and S2c in ESI † ). Third, the cell-wallbending energy of the forespore E bends was neglectedin Section 3. As the septum and forespore are initiallyassumed to be a single PG layer [18], using a signif-icantly smaller bending stiffness for the forespore inthe minimal model shows that its contribution to en-gulfment is indeed minor, especially when r ( t ) doesnot grow by more than 5% (Fig. S1f in ESI † ). In-deed, r ( t ) increases by 5% in the case of significantgrowth (Fig. S3c in ESI † ) and by 3% in the caseof limited growth (Fig. S4c in ESI † ), justifying thismodel simplification.Another model simplification concerns how the cell-wall surface energy is described. In Eq. (2) we as-sumed a constant surface tension γ , with the area en-ergy given by γ · A , where A is the total cell-wall areaof mother cell and forespore. This models a plastic cell wall, determined by growth [31]. In contrast, elas-tic cell-wall deformations could be considered aroundthe minimum-free energy surface area determined byLaplace law, where pressure, surface tension, and ge-ometry are all connected. However, Laplace law doesnot lead to stable rod shapes [32], and in sporula-tion, the pressures in mother cell and forespore arelikely different, which would lead to different radiiin mother cell and forespore compartments. How-ever, this has not been observed [18]. Furthermore,rod-shaped bacteria in the original framework [1] aremodeled with a constant surface tension as well, lead-ing to the same growth law as more detailed modelswith elastic strain [16, 17]. Lastly, including a smallnon-linear elastic correction [33] γ · A to our minimalmodel ( γ /γ ≤ .
1) has only minor effects on engulf-ment (Figs. S1d-e in ESI † ). This is in line with previ-ous studies involving microfluidics [31], which showedthat B. subtilis cell-wall deformations during PG syn-thesis are usually plastic, whereas any elastic deforma-tions tend to be transient and hence have little rela-tion to growth. Taken together, our presented modelprovides insights while being minimal and conceptu-ally straightforward to interpret.We believe that our findings could aid the de-sign of whole-cell radiation-biosensors, insecticides,probiotics, vectors for the delivery of drugs, vaccineantigens, or immunomodulators [34]. Furthermore,some spore-forming bacteria constitute major healththreats, such as
Clostridium difficile and
Bacillusanthracis [18], and our model may provide insightsinto preventing spore formation e.g. by influencingmother-cell growth, turgor pressure, or line tension.Indeed, our findings are of great interest to soft mat-ter, because bacterial cell walls, like other biologicalsystems, exhibit properties rarely found in condensedmatter physics which are often caused by growth [31].Future work may incorporate the effects of stochastic-ity and molecular-scale defects for closer connectionwith molecular biology experiments [1, 18].
ACKNOWLEDGEMENTS
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