DRAFT Bulk-surface coupling reconciles Min-protein pattern formation in vitro and in vivo Fridtjof Brauns a,c,1 , Grzegorz Pawlik b,1 , Jacob Halatek c,1 , Jacob Kerssemakers b , Erwin Frey a,2 , and Cees Dekker b,2 a Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany; b Department of Bionanoscience, Kavli Institute of Nanoscience Delft, Delft University of Technology, Van der Maasweg 9, 2629 HZ Delft, the Netherlands; c Biological Computation Group, Microsoft Research, Cambridge CB1 2FB, UK Abstract 1 Self-organisation of Min proteins is responsible for the spatial con- trol of cell division in Escherichia coli, and has been studied both in vivo and in vitro. Intriguingly, the protein patterns observed in these settings differ qualitatively and quantitatively. This puzzling dichotomy has not been resolved to date. Here, we experimentally show that the dynamics crucially depend on the bulk-to-surface ratio, which is vastly different between the cell geometry and traditional in vitro setups. We systematically control the bulk-to-surface ratio in vitro using laterally wide microchambers with a well-controlled bulk height. A theoretical analysis shows that in vitro patterns at low bulk height are driven by the same lateral oscillation mode as pole-to-pole oscillations in vivo. At larger bulk height, additional vertical oscilla- tion modes set in, marking the transition to a qualitatively different in vitro regime. Our work resolves the Min system’s in vivo/in vitro conundrum and provides important insights on the mechanisms un- derlying protein patterns in bulk-surface coupled systems. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 M in protein patterns determine the mid-cell plane for cell 1 division in many bacteria. They have been intensively 2 studied in E. coli where the Min system comprises three 3 proteins: MinC, MinD, and MinE (1–7). These Min proteins 4 alternately accumulate on either pole of the cylindrical cell 5 (8). These oscillations with a one-minute period result in time- 6 averaged Min protein gradients with a minimum concentration 7 at the center of the long cell axis, which localizes the FtsZ- 8 coordinated cell-division machinery to this point (8, 9). The 9 oscillating pattern is driven by cycling of proteins between 10 membrane-bound and cytosolic states, a process governed by 11 cooperative accumulation of MinD (driven by association with 12 ATP) on the membrane and MinD-ATP hydrolysis stimulated 13 by MinE followed by dissociation of both proteins to the 14 cytosol (1, 2, 10). MinC is not involved in the spatiotemporal 15 Min-system dynamics and acts only downstream of membrane- 16 bound MinD to inhibit with the FtsZ polymerization (10–13). 17 The Min system was discovered E. coli (14, 15), and subse- 18 quently purified and reconstituted in vitro on supported lipid 19 bilayers that mimic the cell membrane (16). This reconstitu- 20 tion provides a minimal system that enables precise control of 21 reaction parameters and geometrical constrains (16–27). This 22 enabled the study of the pattern-formation process and its 23 molecular mechanism in a well-controlled manner, and showed 24 the ability of the Min system to form a rich plethora of dynamic 25 patterns, predominantly travelling waves and spirals, but also 26 “mushrooms”, “snakes”, “amoebas”, “bursts” (16, 17, 28) as 27 well as quasi-static labyrinths, spots, and mesh-like patterns 28 (26, 27). The characteristic length scale (wavelength) of these 29 in vitro patterns (with the exception of the quasi-static ones) 30 is on the order of 50 μm — an order of magnitude larger than 31 the approximately 5 to 10 μm wavelength in vivo (8, 9). The 32 dichotomy between the disparate Min protein patterns found 33 in vivo and those found in vitro remained puzzling so far. In 34 particular, it raises the question how these two conditions are 35 related, and, more generally, how we can gain insights on in 36 vivo self-organization from in vitro studies with reconstituted 37 proteins. 38 The rich phenomenology of the Min system is remark- 39 able, given its molecular simplicity compared to other pattern- 40 forming systems, such as the Belousov–Zhabotinsky reaction 41 (29–32), or protein-pattern formation in eukaryotic systems 42 (see e.g. (33)). It suggests that a multitude of distinct pattern- 43 forming mechanisms are encoded in the simple Min-protein 44 interaction network. Disentangling and deciphering these 45 mechanisms and the underlying principles poses an ongoing 46 challenge. It is also an opportunity to gain a deeper under- 47 standing of fundamental principles underlying pattern forma- 48 tion in general. 49 Experimental studies of the Min-protein system were closely 50 accompanied by theoretical studies, making the Min-protein 51 system a paradigmatic model system for (protein-based) pat- 52 tern formation. Modelling and theory have elucidated various 53 aspects of the Min-protein dynamics in vivo (34–37) and in 54 vitro (16, 23, 26, 38, 39). A particular theoretical insight is 55 that the same protein interactions can drive pattern formation 56 through distinct mechanisms in different parameter regimes 57 (26, 37, 39). The most well-known mechanism for protein 58 pattern formation is the so-called “Turing” instability (40) 59 which is a lateral instability that arises due to the interplay 60 of lateral diffusion and local chemical reactions (in our case: 61 protein interactions at the membrane and nucleotide exchange 62 in the bulk). A qualitatively distinct mechanism that can 63 drive pattern formation are coupled local oscillators that form 64 an oscillatory medium (Ref. (41), ch. 4), akin to neurons which 65 can exhibit oscillations individually and yield a rich spatiotem- 66 poral phenomenology when coupled, see e.g. (42). The key 67 distinction to the Turing mechanism is that local (nonlinear) 68 oscillators in such systems are able to oscillate autonomously, 69 that is, independently of the lateral coupling to their neighbors 70 (see SI Box 1). 71 Local oscillations in the in vitro reconstituted Min system 72 were theoretically predicted (39), and experimentally observed 73 F.B., G.P., J.H., E.F., and C.D. designed research; G.P. and C.D. designed and carried out the experiments; F.B., J.H., and E.F. designed the theoretical models and performed the mathematical analyses; F.B., G.P., and J.K. analyzed data; and F.B., G.P., J.H., E.F., and C.D. wrote the paper. The authors declare no conflict of interest. 1 F.B., G.P., and J.H. contributed equally to this work. 2 To whom correspondence should be addressed. E-mail: [email protected] or [email protected]1–12 . 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DRAFT
Bulk-surface coupling reconciles Min-proteinpattern formation in vitro and in vivoFridtjof Braunsa,c,1, Grzegorz Pawlikb,1, Jacob Halatekc,1, Jacob Kerssemakersb, Erwin Freya,2, and Cees Dekkerb,2
aArnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37,D-80333 München, Germany; bDepartment of Bionanoscience, Kavli Institute of Nanoscience Delft, Delft University of Technology, Van der Maasweg 9, 2629 HZ Delft, theNetherlands; cBiological Computation Group, Microsoft Research, Cambridge CB1 2FB, UK
Abstract1
Self-organisation of Min proteins is responsible for the spatial con-trol of cell division in Escherichia coli, and has been studied bothin vivo and in vitro. Intriguingly, the protein patterns observed inthese settings differ qualitatively and quantitatively. This puzzlingdichotomy has not been resolved to date. Here, we experimentallyshow that the dynamics crucially depend on the bulk-to-surface ratio,which is vastly different between the cell geometry and traditional invitro setups. We systematically control the bulk-to-surface ratio invitro using laterally wide microchambers with a well-controlled bulkheight. A theoretical analysis shows that in vitro patterns at low bulkheight are driven by the same lateral oscillation mode as pole-to-poleoscillations in vivo. At larger bulk height, additional vertical oscilla-tion modes set in, marking the transition to a qualitatively differentin vitro regime. Our work resolves the Min system’s in vivo/in vitroconundrum and provides important insights on the mechanisms un-derlying protein patterns in bulk-surface coupled systems.
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M in protein patterns determine the mid-cell plane for cell1
division in many bacteria. They have been intensively2
studied in E. coli where the Min system comprises three3
proteins: MinC, MinD, and MinE (1–7). These Min proteins4
alternately accumulate on either pole of the cylindrical cell5
(8). These oscillations with a one-minute period result in time-6
averaged Min protein gradients with a minimum concentration7
at the center of the long cell axis, which localizes the FtsZ-8
coordinated cell-division machinery to this point (8, 9). The9
oscillating pattern is driven by cycling of proteins between10
membrane-bound and cytosolic states, a process governed by11
cooperative accumulation of MinD (driven by association with12
ATP) on the membrane and MinD-ATP hydrolysis stimulated13
by MinE followed by dissociation of both proteins to the14
cytosol (1, 2, 10). MinC is not involved in the spatiotemporal15
Min-system dynamics and acts only downstream of membrane-16
bound MinD to inhibit with the FtsZ polymerization (10–13).17
The Min system was discovered E. coli (14, 15), and subse-18
quently purified and reconstituted in vitro on supported lipid19
bilayers that mimic the cell membrane (16). This reconstitu-20
tion provides a minimal system that enables precise control of21
reaction parameters and geometrical constrains (16–27). This22
enabled the study of the pattern-formation process and its23
molecular mechanism in a well-controlled manner, and showed24
the ability of the Min system to form a rich plethora of dynamic25
patterns, predominantly travelling waves and spirals, but also26
well as quasi-static labyrinths, spots, and mesh-like patterns28
(26, 27). The characteristic length scale (wavelength) of these29
in vitro patterns (with the exception of the quasi-static ones) 30
is on the order of 50µm — an order of magnitude larger than 31
the approximately 5 to 10µm wavelength in vivo (8, 9). The 32
dichotomy between the disparate Min protein patterns found 33
in vivo and those found in vitro remained puzzling so far. In 34
particular, it raises the question how these two conditions are 35
related, and, more generally, how we can gain insights on in 36
vivo self-organization from in vitro studies with reconstituted 37
proteins. 38
The rich phenomenology of the Min system is remark- 39
able, given its molecular simplicity compared to other pattern- 40
forming systems, such as the Belousov–Zhabotinsky reaction 41
(29–32), or protein-pattern formation in eukaryotic systems 42
(see e.g. (33)). It suggests that a multitude of distinct pattern- 43
forming mechanisms are encoded in the simple Min-protein 44
interaction network. Disentangling and deciphering these 45
mechanisms and the underlying principles poses an ongoing 46
challenge. It is also an opportunity to gain a deeper under- 47
standing of fundamental principles underlying pattern forma- 48
tion in general. 49
Experimental studies of the Min-protein system were closely 50
accompanied by theoretical studies, making the Min-protein 51
system a paradigmatic model system for (protein-based) pat- 52
tern formation. Modelling and theory have elucidated various 53
aspects of the Min-protein dynamics in vivo (34–37) and in 54
vitro (16, 23, 26, 38, 39). A particular theoretical insight is 55
that the same protein interactions can drive pattern formation 56
through distinct mechanisms in different parameter regimes 57
(26, 37, 39). The most well-known mechanism for protein 58
pattern formation is the so-called “Turing” instability (40) 59
which is a lateral instability that arises due to the interplay 60
of lateral diffusion and local chemical reactions (in our case: 61
protein interactions at the membrane and nucleotide exchange 62
in the bulk). A qualitatively distinct mechanism that can 63
drive pattern formation are coupled local oscillators that form 64
an oscillatory medium (Ref. (41), ch. 4), akin to neurons which 65
can exhibit oscillations individually and yield a rich spatiotem- 66
poral phenomenology when coupled, see e.g. (42). The key 67
distinction to the Turing mechanism is that local (nonlinear) 68
oscillators in such systems are able to oscillate autonomously, 69
that is, independently of the lateral coupling to their neighbors 70
(see SI Box 1). 71
Local oscillations in the in vitro reconstituted Min system 72
were theoretically predicted (39), and experimentally observed 73
F.B., G.P., J.H., E.F., and C.D. designed research; G.P. and C.D. designed and carried out theexperiments; F.B., J.H., and E.F. designed the theoretical models and performed the mathematicalanalyses; F.B., G.P., and J.K. analyzed data; and F.B., G.P., J.H., E.F., and C.D. wrote the paper.
The authors declare no conflict of interest.
1 F.B., G.P., and J.H. contributed equally to this work.
.CC-BY 4.0 International licenseavailable under a(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made
The copyright holder for this preprintthis version posted March 3, 2020. ; https://doi.org/10.1101/2020.03.01.971952doi: bioRxiv preprint
in vesicles (25, 43) where they manifest as homogeneous “puls-74
ing” of the Min-protein density on the vesicle surface. In75
contrast, in cells such pulsing was never observed, indicating76
that there are no local oscillation in vivo. Instead, the pole-77
to-pole oscillations in cells are driven by a lateral (“Turing”)78
instability, based solely on the lateral redistribution of proteins79
(37). The ratio of cytosolic bulk-volume to membrane sur-80
face (short: bulk-surface ratio) of cells is considerably smaller81
compared to that of the spatial confinements (vesicles and82
microchambers) used in reconstitution studies. This suggests83
that the bulk-surface ratio is the key parameter that distin-84
guishes the in vivo regime from the in vitro regime — an85
important hypothesis that merits an in-depth study and exper-86
imental verification. Moreover, an analysis of the transition87
between these two qualitatively different regimes is expected88
to inform about the organizational principles underlying the89
system’s dynamics.90
The key experimental challenge is to systematically control91
the bulk-surface ratio in vitro without influencing the pattern92
formation process laterally along the membrane. In previous93
reports, various methods have been employed to enclose the94
bulk in all three spatial dimensions using microchambers, mi-95
crowells or vesicles (17, 19, 21, 22, 24, 43). In contrast to96
the classical in vitro setup on a large and planar membrane,97
patterns cannot evolve freely in such geometries because adap-98
tion to the confinement (“geometry sensing”) interferes with99
pattern formation (38, 44, 45). As an illustrative example,100
consider a traveling wave which is the typical unconfined in101
vitro phenomenon. When a traveling wave is confined to a com-102
partment of size comparable to its wavelength, the wave will103
be reflected back and forth between the opposite confinement104
walls and thereby visually resemble a standing wave like the105
in vivo pole-to-pole oscillation, despite the fact that the mech-106
anism underlying these dynamics may be very different (as we107
will show below). Hence, the in vivo and in vitro regimes can-108
not be straightforwardly distinguished in three-dimensionally109
confined geometries (9, 22).110
Here, we eliminate the interfering effects of geometric con-111
finement by using laterally large microchambers with flat sur-112
faces on top and bottom and well-controlled heights between 2113
to 60µm (see Fig. 1A). The microchambers’ height of directly114
controls the bulk-surface ratio, while the MinE patterns on115
the membranes can evolve freely in lateral directions along116
the surfaces.117
In experiments with reconstituted Min proteins in such118
microchambers, we observe a rich variety of patterns from119
standing wave chaos at low bulk heights (< 8 µm), to sus-120
tained large-scale oscillations at intermediate bulk heights121
(≈ 15 µm), to traveling waves at large bulk heights (> 20 µm).122
The mathematical analysis of the reaction–diffusion equations123
(homogeneous steady states and their linear stability analysis)124
in the microchamber geometry with planar, laterally uncon-125
fined surfaces enables us to identify the characteristic modes126
that drive pattern formation. From these modes, we predict127
a number of signature features of the distinct mechanisms128
— lateral and local oscillations — underlying pattern forma-129
tion, in particular the synchronization of patterns between130
the microchamber’s top and bottom surface. We verify these131
predictions experimentally and thereby provide evidence that132
indeed distinct lateral and local oscillations underlie pattern133
formation in the various parameter regimes. Importantly, we134
find that the patterns in microchambers with low bulk height 135
are driven by the same lateral oscillation mechanism as in vivo 136
pole-to-pole oscillations. In contrast, a combination of both 137
lateral and local oscillations govern pattern formation at the 138
large bulk heights that are typical in most traditional in vitro 139
setups. 140
Taken together, we find that the in vivo vs in vitro di- 141
chotomy can be excellently reconciled on the level of the 142
underlying mechanisms. Systematic variation of the bulk 143
height experimentally confirms our theoretical prediction that 144
the bulk-surface ratio is the key parameter that continuously 145
connects the in vivo and in vitro. 146
Results 147
Finite bulk heights lead to drastically different Min patterns. 148
To study the effect of the bulk-surface ratio on Min pattern 149
formation in vitro, we need to control this parameter with- 150
out imposing lateral spatial constraints that affect pattern 151
formation. To achieve this, we created a set of PDMS-based mi- 152
crofluidic chambers of large lateral dimensions (2 mm × 6 mm) 153
but with a low height in a range from 2 to 57 µm (Fig. 1A). 154
In these wide chambers, patterns can freely evolve in the lat- 155
eral direction while we study the effects of the bulk-surface 156
ratio which is controlled by the microchamber height. All 157
inner surfaces of the microchambers were covered with sup- 158
ported lipid bilayers composed of DOPG:DOPC (3:7) which 159
has been shown to mimic the natural E. coli membrane com- 160
position (20). Proteins were administered by rapid injection 161
of a solution containing 1 µM of MinE and and 1 µM of MinD 162
proteins, together with 5 µM ATP and an ATP-regeneration 163
system (22). 164
Figure 1C and Movie S1 show snapshots and kymographs 165
of the characteristic patterns observed in microchambers of 166
different heights. We clearly observe distinct Min patterns 167
that can be identified qualitatively by simple visual inspection: 168
standing wave chaos, “large-scale oscillations”, and traveling 169
waves. Next, we will qualitatively describe these different 170
pattern types in detail. Further below in the section Compe- 171
tition of different oscillation modes leads to multistability of 172
patterns, we will provide a detailed quantification of the pat- 173
tern characteristics (wavelength, frequencies, and propagation 174
velocities, based on auto-correlation analysis) in dependence 175
of bulk height and E-D ratio. There we will also show that 176
the different pattern types exist in overlapping regions of the 177
parameter space (bulk height, E-D ratio). 178
At low bulk height (2–6 µm in Fig. 1C) we observe inco- 179
herent wave fronts of the protein density propagating from 180
low density towards high density regions, thus continually 181
shifting these regions in a chaotic manner as can be seen in the 182
kymographs. We will refer to these patterns as standing wave 183
chaos. The chaotic character is also evidenced by the irregular 184
shapes and non-uniform propagation velocities of wave fronts 185
within the same pattern (see Fig. S1). Still, these patterns 186
clearly have a characteristic length scale. 187
At an intermediate bulk height (13 µm in Fig. 1C), we 188
observe patterns with large areas that have fairly homoge- 189
neous Min-protein density and temporally oscillate as a whole. 190
We refer to these patterns as large-scale oscillations. (We 191
will use this rather general term to subsume a wide range of 192
phenomena that are commonly found in oscillatory media.) 193
Phenomenologically similar oscillations have been observed as 194
2
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Fig. 1. Effect of a change of a bulk height on Min patterformation. A General concept of the experimental setup.MinD and MinE proteins were reconstituted in laterally largeand flat microchambers of different heights. All inner wallsof microchambers were covered with supported lipid bilay-ers made of DOPC:DOPG:TFCL (66 : 32.99 : 0.01 mol%)mimicking the E. coli membrane composition. Min pro-teins cycle between bulk and the membrane upon whichthey self-organize into dynamic spatial protein-density pat-terns. B Min-protein interaction scheme. MinD monomers(light-green hexagons) bind ATP resulting in dimerizationand cooperative accumulation on membrane (dark-greenhexagons). Next, MinE dimers bind to MinD, activatingits ATPase activity, detachment from the membrane, anddiffusion to the bulk where ADP is exchanged to ATP, andthe cycle repeats. C Influence of the bulk height on Min pat-tern formation. Snapshots show overlays of MinD channel(green) and MinE channel (red) 30 minutes after injection.Kymographs below were generated along the dashed redlines. In each microchamber, the concentrations of thereconstituted proteins are 1 µM MinE and 1 µM MinD (cor-responding to a 1:1 E-D ratio). (White scale bar: 50 µm.) DSnapshots and kymographs from numerical simulations ofthe reaction–diffusion model describing the skeleton Min-model in a three-dimensional box geometry with a mem-brane on top and bottom surfaces and reflective boundarieson the sides (see SI Materials and Methods for details).The colors are an overlay of MinD density (green) and MinEdensity (red) on the top membrane. Parameters (from leftto right): H = 2 µm, E/D = 0.8; H = 6 µm, E/D = 0.75; H =14 µm, E/D = 0.75; H = 20 µm, E/D = 0.725; H = 40 µm,E/D = 0.625. The remaining, fixed model-parameters aregiven in the SI Materials and Methods (Table S1).
an initial transients in some previous experiments (17, 46). In195
contrast, however, the oscillations that we observed at inter-196
mediate bulk heights persisted throughout the entire duration197
of the experiment (90 minutes).198
The lack of spatial coherence for large-scale oscillations is199
in stark contrast to the traveling waves we observed in the200
large height regime (57 µm in Fig. 1C). Traveling waves are201
characterized by high spatial coherence of the consecutive202
wave fronts that propagate together at the same velocity and203
with a well-controlled wavelength. Finally, the wave patterns204
found at 25µm shows phenomena indicative of defect-mediated205
turbulence: continual creation, annihilation, and movement of206
spiral defects (Movie S2). This behavior is commonly found in207
oscillatory media at the transition between spiral waves and208
Taken together, we find that the bulk height has a profound210
effect on the phenomenology of Min protein pattern forma-211
tion. Notably, the bulk-surface ratio at the lowest bulk height212
(2 µm) is of the same order of magnitude as in E. coli cells213
which have a diameter of about 0.5–1 µm. However, there is214
no obvious phenomenological correspondence between the in215
vivo system and the laterally unconfined in vitro system as216
the patterns found in these two settings differ qualitatively.217
Moreover, at low bulk heights, where the bulk-surface ratio218
is on the same order of magnitude as in vivo, the pattern219
wavelength in vitro is approximately 40 µm (see Quantification220
of experimentally observed patterns). This is much larger than221
the typical wavelength in vivo (approximately 5 to 10 µm), as222
found for stripe oscillations in filamentous cells (8) and large223
“sculpted” cells (9). We will further elaborate on the question224
of pattern wavelength in the discussion section. 225
A minimal model reproduces the salient, qualitative pattern 226
features. To explain the observed diversity of patterns found 227
in experiments, we performed numerical simulations and a the- 228
oretical analysis. We used a minimal model of the Min-protein 229
dynamics that is based on the known biochemical interactions 230
between MinD and MinE (Fig. 1B). This model encapsulates 231
the core features of the Min system and has successfully re- 232
produced and predicted experimental findings both in vivo 233
(36–38) and in vitro (23, 39). Finite-element simulations of 234
this model in the same geometry as the microchambers (lat- 235
erally wide cuboid with membrane on both top and bottom 236
surfaces, see Fig. S2) qualitatively reproduce the three pattern 237
types found in experiments the three regimes of bulk heights, 238
as shown in Fig. 1D and Movie S3. 239
At low bulk heights (0.5–5 µm), the model exhibits standing 240
wave chaos (incoherent fronts that chaotically shift high- and 241
low-density regions) in close qualitative resemblance of the 242
patterns found experimentally. At intermediate bulk heights 243
(5–15 µm), we find nearly homogeneous oscillations, meaning 244
large areas with a nearly homogeneous oscillator density that 245
are phase separated by phase defect lines where the oscilla- 246
tor phase jumps. Shallow gradients in the oscillation phase 247
lead to the impression of propagating fronts, with a velocity 248
inversely proportional to the phase gradient (sometimes called 249
“pseudo waves” or “phase waves” in the theoretical literature 250
(49, 50). In contrast to genuine traveling waves (sometimes 251
called “trigger waves”), phase waves are merely phase shifted 252
local oscillations. They do not require lateral material trans- 253
port (lateral mass redistribution) and the visual impression of 254
3
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(Movie S5). Dominance of vertical membrane-to-membrane 370
∗Mathematically, in the linear stability analysis, the two vertical transport modes — membrane-to-membrane and membrane-to-bulk — become equivalent at large bulk heights: the vertical modesare sinh(z) and cosh(z), which asymptotically approach exp(z) for large z. Since for linearstability, only the vertical bulk-gradient at the membrane surface matters, the stability properties ofthese modes become identical in the asymptotic limit H → ∞; see SI Materials and Methods.
4
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lateral instabilitym-to-m local osc. m-to-b local osc.
no instability
Fig. 2. Distinct lateral and local instabilities at different bulk heights. A Phase diagram for bulk height and E-D ratio showing three types of linear instabilities that exist inoverlapping regimes: lateral instability (green), local membrane-to-membrane instability (“m-to-m”, blue) and local membrane-to-bulk instability (“m-to-b”, orange). See Fig. S3for representative dispersion relations in the various regimes. The green dot-dashed line marks the commensurability condition for lateral instability that indicates the transitionfrom chaotic to coherent patterns at larger bulk heights (39). A representative example of a chaotic pattern is shown in Movie S4 for the parameter combination marked by thered star. Black dots mark the parameters used for the simulations shown in Fig. 1D. The red line indicates the parameter range in which adiabatic sweeps of the bulk heightwere performed to demonstrate hysteresis as a signature of multistability (see Fig. 4 below). Panels (B–D) illustrate the instabilities at different bulk heights. The top row showslaterally isolated compartments, to illustrate local vertical oscillations due to vertical bulk gradients. The bottom row illustrates the interplay of lateral and local oscillation modesin a laterally extended system. B At low bulk height, where the bulk height is too small for vertical concentration gradients to form (see Movie S5). Hence, a laterally isolatedcompartment does not exhibit any instabilities. In a laterally extended system as depicted in the bottom row, exchange mass of mass can drive a lateral instability (green arrows).The cartoon of an E. coli cell illustrates that this instability also underlies pattern formation in vivo. C For bulk heights above Hc, vertical concentration gradients lead to delaysin the vertical transport of proteins between top and bottom membrane (see Movie S6). Consequentially, vertical membrane-to-membrane oscillations (blue arrows) emerge thatdo not require lateral exchange of mass, i.e. they occur in a laterally isolated compartment. The cartoon of a E. coli cell illustrates that the m-to-m oscillations in vitro can alsobe pictured as equivalent to in vivo pole-to-pole oscillations, where the two cell-poles represent the top and bottom membrane of a local compartment of the in vitro system.D At bulk-heights larger than the penetration depth of vertical gradients, the top and bottom membrane effectively decouple (see Movies S7 and S8). In this regime, whichcorresponds to the classical in vitro regime, the bulk in-between the membranes acts as an effective protein reservoir that facilitates membrane-to-bulk oscillations.
oscillations leads to large-scale oscillations (Movie S6).† These371
large-scale oscillations clearly demarcate the transition from372
the in-vivo-like regime, where only lateral oscillations but no373
vertical oscillations exist, to the in vitro regime where vertical374
oscillations come into play. At large bulk height, the interplay375
of lateral oscillations and local membrane-to-bulk oscillations376
drives traveling waves (Movie S7) and standing wave chaos at377
low E-D ratios (see Movies S4 and 8 for simulations the full378
geometry (2+3D) and in slice geometry (1+2D), respectively).379
The large bulk-height regime was investigated in detail in a380
previous theoretical study (39). In particular, it was found381
that the transition from travelling waves standing wave chaos382
corresponds to a commensurability condition in the disper-383
sion relation, marked by a dot-dashed green line in the phase384
diagram (Fig. 1A).385
In passing, we note that the patterns we find in numerical386
simulations have large amplitude. It is no a priori clear whether387
the linear stability analysis of the homogeneous steady state388
is informative regarding such large amplitude patterns. In the389
SI Materials and Methods, we briefly describe how a recently390
developed theoretical framework called “local equilibria theory”391
can be used to characterize large amplitude patterns locally392
and regionally (39, 51). Centrally, this framework utilizes393
the fact that the Min-protein dynamics conserve the total394
amounts of MinD and MinE. Technical details and several395
concrete examples for local equilibria analysis of the patterns396
†We subsume several phenomena, including homogeneous oscillations, phase chaos, and defect-mediated turbulence (52–54), under the general term “large-scale oscillations.” A detailed quan-tification and distinction is beyond the scope of this work. Instead, we relied on the anti-phasesynchronization between top and bottom membrane that is characteristic for the membrane-to-membrane oscillation mode.
found in numerical simulations are provided in the SI and 397
tion modes in experiments. Recall that the vertical synchro- 400
nization of patterns on the opposite (top and bottom) mem- 401
branes is a a key signature that distinguishes the lateral os- 402
cillation mode at low bulk height, the vertical membrane- 403
to-membrane oscillation mode at intermediate bulk height, 404
and the vertical membrane-to-bulk oscillation mode at large 405
bulk height. Respectively, these modes correspond to strong 406
in-phase coupling, anti-phase coupling, and de-coupuling of 407
the dynamics on the two opposites membranes. In the follow- 408
ing, we will use these characteristics to infer the underlying 409
oscillation modes directly from the experimentally observed 410
patterns. 411
Interplanar synchronization of patterns can be visualized 412
by overlaying snapshots and kymographs where the protein 413
densities on the two membranes are shown in different colors 414
(Fig. 3A). Recall that at low heights, we found only lateral 415
instability (cf. Fig. 2B) where the bulk is uniform in the verti- 416
cal direction. This leads to strong in-phase synchronization 417
of the top and bottom membrane (Fig. 3B, left). In contrast, 418
the local instability driven by vertical membrane-to-membrane 419
transport for leads to anti-phase synchronization (Fig. 3B, 420
center; cf. Fig. 2C ). Finally, at large height, the large bulk 421
reservoir in between the two membranes decouples their pat- 422
terns (cf. Fig. 2D), thus removing any synchronization between 423
them (Fig. 3B, right). 424
To test these theoretical predictions, we imaged the Min 425
5
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Fig. 3. Min cross-talk between opposite membranes. A Patterns form on the membranes both on the top and bottom surface of the microchambers. Overlaying thesepatterns in different colors (blue and orange) reveals the synchronization between them. B Kymographs from numerical simulations showing perfect in-phase synchrony ofpatterns at low bulk height, anti-phase synchrony driven by local membrane-to-membrane oscillations at intermediate bulk height and de-synchronization at large bulk height,where two membranes effectively decouple. (E-D ratios from left to right: 0.75, 0.725, and 0.55). C Snapshots and kyomographs from simulataneous (< 0.1 s delay) imaging ofthe top (orange) and bottom (blue) membrane in microchambers of different heights (1 µM MinD, 1 µM MinE). In the overlay, areas where peak protein concentrations coincideare white. Areas of coinciding low protein density are black. Bars correspond to 50 µm. D Each field of view (FOV) was divided into a grid of cells for which the correlationanalysis was performed individually. Histograms show frequency distribution of correlations of individual cells in the grid measured for 30 timepoints in each FOV. Perfectin-phase correlation corresponds to a correlation value of 1 and perfect anti-phase to a value of –1 respectively; lack of correlation corresponds to a correlation measure 0. EExample of coexistence of in-phase and anti-phase synchrony within adjacent spatial regions. F Classification of top-bottom correlation as a function of bulk height, extractedfrom the histograms in panel D. Correlation values above 0.7 are classified as correlated, values less than –0.3 as anticorrelated.
6
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oscillations, and vertically anti-phase traveling/standing waves.486
Bulk height [ ]
C anti-phase TW/SWOSC
topbottom
MinD membrane density
in-phase SW
anti-phase TW/SW
OSC
Multistability
Patte
rn ty
pe
BC
A
4 208 12 16
OSCin-phase SWB
Fig. 4. Multistability and coexistence of different pattern types in numericalsimulations. A Using adiabatic parameter sweeps of the bulk height (along the redline, E/D = 0.725, in Fig. 2D; see SI for details) we demonstrate multistability of differentpattern types. A hallmark of multistability is hysteresis as shown in (B) for the transitionfrom in-phase standing waves (SW) to homogeneous oscillations (OSC); and in (C)for the transition to homogeneous oscillations to anti-phase traveling/standing waves(TW/SW). B Kymograph showing the transitions from in-phase SW to anti-phaseOSC as the bulk height is adiabatically increased from 12.16 µm to 12.88 µm. Upondecreasing the bulk height back to 12.16 µm, the homogeneous oscillations persist. Infact the transition back to in-phase SW takes place around H = 6 µm, see Fig. S7A. CKymograph showing the transitions from anti-phase OSC to anti-phase TW/SW as thebulk height is adiabatically increased from 17.84 µm to 18.20 µm. Upon decreasingthe bulk height back to 17.84 µm, the anti-phase TW/SW pattern persists. Similarlyas OSC, these patterns persist down to around H = 6 µm, see Fig. S7B.
In the multistable regime, the pattern exhibited by the sys- 487
tem depends on the initial condition (see Fig. S5). Moreover, 488
in simulations with large system size (lateral domain size of 489
500 µm), we find spatiotemporal intermittency at intermediate 490
bulk height (∼ 18 µm). This phenomenon can be pictured as 491
the coexistence of large-scale oscillations, traveling waves, and 492
standing waves in space where they continually transitioned 493
from one to another over time (Fig. S6A). In simulations in 494
full 3D geometry, we observe coexistence of in-phase synchro- 495
nized standing waves and large-scale anti-phase oscillations in 496
neighboring regions regions of the membranes (Fig. S6A and 497
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large scale oscillations segmented waves (”amoebas”)
0.5
0.75
1
2
3
bulk heigth [μm] bulk heigth [μm]
0.5
0.75
1
2
3
E-D
ratio
E-D
ratio
2 6 8 15 25 57 2 6 8 15 25 57
B
DC
Fig. 5. Experimental phase diagramshowing multistability. A–D showrepresentative snapshots to indicatewhere each of the four pattern types— traveling waves (TW), standing wavechaos (SWC), large scale oscillations(OSC) and segmented waves — wasobserved as a function of E-D ratio andbulk height (cf. Movies S11–S14). Tovary the E-D ratio, the concentration ofMinE was varied from 0.5 to 3 µM at aconstant MinD concentration of 1 µM.(Snapshots show overlays of MinDchannel (green) and MinE channel(red); field of view: 307 µm × 307 µm.)
Movie S10).498
To experimentally test the predicted multistability, we sys-499
tematically varied the bulk height (from 2 to 57 µm) and the500
E-D ratio (from 0.5 to 3), and repeated the experiment several501
times (N = 2 to 5) for each parameter combination. Figure 5502
and Movies 11–14 show an overview of the patterns found in503
this assay. Notably, at intermediate bulk heights, qualitatively504
different patterns were observed in repeated experiments with505
the same parameters, clearly indicating multistability (see the506
“phase diagram” in Fig. S8). This confirms our theoretical pre-507
diction from linear stability analysis and numerical simulations.508
For several parameter combinations (H = 6 µm, E/D = 2 and509
H = 13 µm, E/D = 2), we found threefold multistability510
(Movie S15).511
By contrast, at low bulk height (2 µm) we found only a512
single instance of (twofold) multistability (H = 2 µm, E/D =513
2), whereas at large bulk height (57µm) we did not observe514
any multistability at all. In agreement to these experimental515
findings, numerical simulations at small and large bulk heights516
do not show multistability of qualitatively different patterns.517
Quantification of experimentally observed patterns.518
Segmented waves (“amoebas”). In addition to the three pat-519
tern types presented in Figure 1, we also found a fourth type520
of patterns that resembles “segmented waves” (55, 56). These521
patterns are similar to a phenomenon previously observed in522
the in vitro Min system where it was called “amoeba pat-523
tern” (23, 28). The segmented waves consist of small separate524
“blobs” of MinD which, in contrast to standing waves, are525
not surrounded by MinE but instead feature a one-directional 526
MinE gradient, resulting in directional propagation of the 527
blobs (Movie S14). This type of pattern occurred mostly at 528
large E-D ratio (& 1) and for a broad range of bulk heights 529
(2–25 µm). Due to its incoherent and non-oscillatory charac- 530
ter, we did not characterize this pattern by autocorrelation 531
analysis. 532
In the vicinity to the traveling wave regime, the segmented 533
waves emerge due to the segmentation of the spiral wave front. 534
Such “segmented spirals” were previously observed and studied 535
in the BZ-AOT system (55, 56). This might provide hints 536
towards the mechanism underlying segmented wave formation. 537
Explaining this phenomenon likely requires an extension of the 538
minimal Min model, e.g. by the cytosolic switching dynamics 539
of MinE (23). 540
Traveling waves. As predicted by the model, traveling waves 541
are found mainly for large bulk heights (> 15 µm) and suf- 542
ficiently large E-D ratios (≥ 1). In addition, we found in 543
traveling waves down to 2 µm bulk height at an E-D ratio 544
of 2. We quantified the experimentally observed patterns us- 545
ing autocorrelation analysis (Fig. S9). Traveling waves exhibit 546
oscillation period that increase as a function of E-D ratio and 547
bulk height in the range 2.5 to 8 min. Their wavelength ranged 548
mostly from 35 to 55 µm and did not vary systematically across 549
the conditions we investigated. 550
Standing wave chaos. Standing wave chaos occurs in two dis- 551
tinct regions of the phase diagram. First, at low bulk heights, 552
where our theoretical analysis shows that only lateral instabil- 553
ity exists (cf. Figs. 2 and 3). Second, we found standing wave 554
8
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segmented waves patterns, and traveling waves. Clearly, a 642
classification of the dynamics based on the topology of the 643
protein-interaction network is not sufficient in such a situation. 644
Instead, our theoretical analysis shows that a classification is 645
possible in terms of the lateral and vertical oscillation modes 646
that can be identified by linear stability analysis. While the 647
classical linear stability analysis of a homogeneous steady state 648
is only valid for small amplitude patterns, local equilibria the- 649
ory enabled us to reveal how these instabilities drive patterns 650
far from the homogeneous steady state. Diffusive mass redis- 651
tribution between the compartments changes the equilibria 652
and their stability properties that serve as proxies for the 653
local dynamics. This principle made it possible to systemati- 654
cally identify distinct lateral and local instabilities as physical 655
mechanisms of (strongly nonlinear) pattern formation. 656
On the level of pattern-forming mechanisms, we showed 657
that the archetypical in vivo and in vitro patterns of the 658
Min system correspond to different instabilities that arise in 659
separate parameter regimes: lateral mass-transport oscillations 660
at low bulk heights (in vivo), and two additional modes of 661
vertical oscillations at large bulk heights (in vitro). Central to 662
these distinct oscillation modes are vertical bulk gradients that 663
couple top and bottom membrane. As a consequence of this 664
coupling, we observed characteristic in-phase synchronization 665
of patterns between both membranes low bulk heights and 666
anti-phase synchronization at intermediate bulk heights. At 667
large bulk heights, the top and bottom membrane decouple 668
and the patterns are no longer synchronized between them. 669
These findings serve as a clear signature of the underlying 670
instabilities in the experimentally observed patterns. Crucially, 671
this allowed us to characterize the observed patterns on this 672
mechanistic level and directly link the observation to the 673
theoretical analysis. Furthermore, synchronization of patterns 674
9
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across the bulk serves as an experimental proof of the role of675
such bulk gradients for pattern formation.676
A second important evidence for the distinct lateral and677
vertical oscillation modes are multistability and coexistence678
of different pattern types. Both in numerical simulations679
and in experiments, we found multistability in the transition680
regime in parameter space where multiple instabilities based on681
different transport modes coexist. This raises many interesting682
questions for future research which we will briefly discuss in683
the Outlook below.684
Reconciling pattern formation in vivo and in vitro. Taken to-685
gether, robust and qualitative features of the patterns — syn-686
chronization between top and bottom membrane, and multi-687
stability in the transition regime — serve as clear signatures688
of the underlying pattern-forming mechanisms in the experi-689
mental data. This enabled us to reconcile pattern formation690
in vivo and in vitro on a mechanistic level. Importantly these691
features are inherent to the vertical transport modes driving692
the pattern-forming instabilities. They can be intuitively un-693
derstood (cf. Fig. 2B–D) and are not sensitive to parameter694
changes or model variations. This establishes a strong con-695
nection between the experimental system, the minimal model,696
and the theoretical framework.697
Length-scale selection, pattern wavelength and front width. Ama-698
jor phenomenological feature of patterns that is typically dis-699
cussed in the context of the in vivo vs in vitro dichotomy is the700
pattern wavelength. From a theoretical standpoint, however,701
the principles underlying nonlinear wavelength selection of702
large-amplitude patterns are largely unknown. As of yet, only703
(quasi-) stationary patterns in one- and two-variable systems704
have been systematically studied (67–69). Importantly, the705
wavelength of strongly nonlinear patterns is not necessarily706
determined by the dominant linear instability of the homoge-707
neous steady state. Instead, the wavelength is subject to a708
subtle interplay of local reactions and lateral transport (many709
nonlinearly coupled modes) and therefore can depend sensibly,710
and non-trivially on parameters. For instance, a previous the-711
oretical study of the in vivo Min system showed that a subtle712
interplay of recruitment, cytosolic diffusion and nucleotide713
exchange can give rise to “canalized transfer” which plays an714
important role for wavelength selection in this system (37).715
Moreover, experimental studies on the reconstituted Min sys-716
tem showed that many “microscopic” details, like the ionic717
strength of the aqueous medium, temperature, and membrane718
charge can also affect the wavelength of Min patterns (20, 22).719
Matching the wavelengths found in simulations to those720
found in experiments would require fitting of parameters, and,721
potentially, model extensions (e.g. the switching of MinE (23)).722
Given the already large number of experimentally unknown723
reaction rates, such fitting would not be informative. Due to724
a lack of understanding of the underlying principles, fitting725
would also be a laborious and computationally expensive task,726
as one would need to “blindly” search large parameter ranges.727
Besides their wavelength, i.e. the distance between con-728
secutive wave nodes, patterns have a second characteristic729
length-scale — the width of fronts, also called “interfaces”730
or “domain walls”, that connect low-density to high-density731
regions. While linear stability analysis of the homogeneous732
steady state does not predict the wavelength of large-amplitude733
patterns, the front width is quantitatively linked to the under-734
lying lateral instability that creates and maintains them (51).735
Indeed, the front widths observed in experiments at low bulk 736
height are of the same order of magnitude as the length scale 737
of in vivo patterns (∼ 5 µm), which agrees with our finding 738
that both are driven by the same type of (regional) lateral 739
instability (see Fig. S10). 740
Understanding the principles of wavelength selection of 741
highly nonlinear patterns in general, and the Min-protein 742
patterns in particular, remains an open problem for future 743
research, both theoretically and experimentally. Such an un- 744
derstanding might ultimately answer why the Min-pattern 745
wavelengths are so different in vivo and in vitro. 746
Outlook 747
We found a large range of dramatically different phenomena 748
exhibited by the in vitro Min system in laterally extended, 749
vertically confined microchambers at different chamber heights 750
and average total densities, both in experiments and in simula- 751
tions. This leads to many interesting questions and connections 752
to other pattern-forming systems. In the following, we discuss 753
some of these promising avenues for future research. 754
Large-scale oscillations (lateral synchronization and defect-medi- 755
ated turbulence). At intermediate heights of the microcham- 756
bers, we observed a phenomenon that was not previously 757
observed in the Min system: temporal oscillations between 758
the opposite membranes with spatially nearly homogeneous 759
protein concentrations (termed “large-scale oscillations”). Be- 760
cause each pair of opposite membrane points together with the 761
bulk column in-between them constitutes a local oscillator, this 762
can be understood as lateral synchronization of coupled oscilla- 763
tors. We found stable lateral synchronization of membrane-to- 764
membrane oscillations at intermediate bulk heights. Towards 765
larger bulk heights, homogeneous membrane-to-membrane 766
oscillations become unstable, giving rise to defect-mediated 767
turbulence, spatiotemporal intermittency, and eventually trav- 768
elling wave patterns such as spirals. These phenomena are 769
generic for oscillatory media and have been studied theo- 770
retically (see e.g. (47, 52–54) and experimentally (see e.g. 771
(32, 48, 70, 71)). Interestingly, in contrast to membrane-to- 772
membrane oscillations, membrane-to-bulk oscillations were 773
never found to stably synchronize laterally, neither in experi- 774
ments nor in simulations. Investigating this different behavior 775
of the two vertical oscillation modes is an important question 776
for future research. 777
More generally, synchronization of coupled oscillators is a 778
topic of broad interest pervasive in nonlinear systems (72). 779
Applications include catalytic surface reactions (see e.g. (70)), 780
self-organization of motile cells (see e.g. (71)) cardiac calcium 781
oscillations (see e.g. (73)), neural tissues (see e.g. (74)), and 782
power grids (see e.g. (75, 76)). More specifically, for calcium os- 783
cillations in cardiac muscle tissue, the transition from laterally 784
synchronized (i.e. homogeneous) oscillations to spirals and tur- 785
bulent waves is thought to be a main reason for fibrillation (73). 786
The molecular simplicity and experimental accessibility make 787
the Min system a potential model system to further study 788
oscillatory media experimentally and theoretically. 789
Multistability and coexistence of distinct patterns. Another in- 790
triguing phenomenon found at intermediate bulk-heights is 791
multistability. Studying these phenomena in detail was be- 792
yond the scope of this work. What are the precise conditions 793
under which these phenomena can be found? Can hysteresis 794
10
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