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A growing bacterial colony in two dimensions as an active nematic

Abstract

How a single bacterium becomes a colony of many thousand cells is important in biomedicine and food safety. Much is known about the molecular and genetic bases of this process, but less about the underlying physical mechanisms. Here we study the growth of single-layer micro-colonies of rod-shaped Escherichia coli bacteria confined to just under the surface of soft agarose by a glass slide. Analysing this system as a liquid crystal, we find that growth-induced activity fragments the colony into microdomains of well-defined size, whilst the associated flow orients it tangentially at the boundary. Topological defect pairs with charges $$\pm {\textstyle{1 \over 2}}$$ are produced at a constant rate, with the $$+ {\textstyle{1 \over 2}}$$ defects being propelled to the periphery. Theoretical modelling suggests that these phenomena have different physical origins from similar observations in other extensile active nematics, and a growing bacterial colony belongs to a new universality class, with features reminiscent of the expanding universe.

Introduction

Active matter occurs at many length-scales: from bird flocks1, through shaken grains2 and swimming bacteria3 and synthetic colloids4, to gels in which protein motors ‘walk’ on filamentous ‘rails’5. All contain agents consuming energy to drive locomotion. The field attracts physicists (as a ‘grand challenge’ in non-equilibrium statistical mechanics), theoretical biologists (as a paradigm uniting phenomena across disparate scales) and materials scientists (who see applications in, e.g., self-assembly and artificial wound healing).

Active matter systems may show ‘polar’ (2π-periodic, like an arrow) or ‘nematic’ (π-periodic, like a double arrow) orientational ordering. Within the part of the system under consideration, momentum is either conserved or not. A bulk system which conserves momentum is termed ‘wet’, while in a ‘dry’ system, such as a two-dimensional (2D) layer interacting strongly with a substrate, momentum is not conserved. The polar/nematic and wet/dry dichotomies make up the four classical universality classes of active matter6. Recent developments show that ‘wet’ and ‘dry’ are the limiting cases of a continuum of possible theories7, and that ‘scalar active matter’ exists with no orientational order8. In all these cases, the number of active agents is conserved at all times. Studies of biological morphogenesis as active matter have also, until recently9,10, focussed on developmental phases with constant average particle number by balancing cell birth and death. In the following, we refer to these systems generically as ‘number-conserving’ active matter.

Here we instead study a simple active matter system with non-conserved particle number: a two-dimensional layer of dividing Escherichia coli cells confined in agarose gel by a glass slide. It represents the early stage of bacterial colonisation in many contexts, and constitutes perhaps the simplest example of biological morphogenesis. Despite the absence of cell differentiation, a growing bacterial colony is governed by similar physical constraints to those controlling eukaryotic morphogenesis, including tumour growth11.

Cell-cell mechanical interactions constrain colony growth12,13,14,15,16,17,18,19. An elongating E. coli pushing against its neighbours in a colony has been modelled as an extensile dipole in a viscous liquid, i.e., a wet active nematic, in which viscous drag between the cells and bulk liquid controls colony morphology10. A dry continuum model has also been proposed to describe the chaotic behaviour of nematic domains in simulated colonies19. However, no theory of growing active nematics has yet been tested against quantitative experimental data.

We provide such data by analysing growing E. coli colonies as ‘living anisotropic fluids’. We observe phenomena resembling those seen in number-conserving active nematics, including ‘active anchoring’ at the boundary20 and the proliferation of topological defects21. However, the underlying mechanisms differ. We propose a theoretical framework in which our observations emerge from growth-driven Hubble-like expansive flows22 and friction between cells and their substrate17. Alluding to this qualitative broad analogy, henceforth we call this class of systems (dry) ‘Hubble active nematics’. [Clearly, this suggestive analogy comes with limitations. Most notably, within our quasi-2D colonies the amount of matter grows over time, where in the expanding universe it is the metric that changes in scale.]

Results

Morphology and growth rate

We begin by reporting results on the morphology of our growing microcolonies (see Methods for experimental details on growth conditions and image processing). Confocal imaging (Supplementary Fig. 1) shows that cells are embedded just underneath the agarose surface, confined on one side by the cover slip. Figure 1 shows snapshots from a typical colony, which are representative of the behaviour we observe under our experimental conditions (see Methods). A number of qualitative changes occur as cells lengthen and divide. At early times, the colony is elongated, Fig. 1a, b, as cells grow longitudinally along a common axis. Later, this 1D order breaks down, giving way to a pattern of nematic domains bordered by defects, Fig. 1c. During this stage, cells push outwards in all directions, and the colony becomes more isotropic, Fig. 1 (inset, see also Supplementary Note 1 and Supplementary Fig. 2). Under our conditions, the middle of the colony turns phase dark after ~5 h when it contains N 103 cells, Fig. 1d (dark spots), indicating buckling of the 2D layer into the third dimension17, and we stop our analysis. Typically, buckling occurs when the colony size 2R 50 μm. All colonies show the same trend, although quantitative details are subject to noise (Supplementary Fig. 2).

Cells in the middle of a large colony may grow slower due to nutrient limitation. Measurement of cell lengths as a function of time, Fig. 2a, shows that their growth rate is in fact spatially uniform to within experimental uncertainties, so that nutrient limitation is unimportant for our colonies (although the growth rate is time dependent, Supplementary Note 2 and Supplementary Fig. 2b). The near-uniform growth rate is also the reason why we do not observe fingering instabilities, which appear when growth rate is limited by nutrient16 or depends on local density23.

‘Hubble’ expansion

We found no coherent azimuthal movement of cells, but there is a coherent radial movement, whose speed increases linearly with distance from the centroid, Fig. 2b, reminiscent of the Hubble expansion of the universe. Consider the velocity field, v, of this growth-induced motion: we now write an equation for this quantity based on our experimental observations. Mass conservation requires that the cell density, ρ, satisfies

$$\partial _t\rho + \nabla \cdot (\rho {\bf{v}}) = D_t\rho + \rho \nabla \cdot {\bf{v}} = {\mathrm{\Lambda }}\rho ,$$
(1)

where Λ is the growth rate, and Dt = ∂t + v ·  is the material derivative. Assuming incompressibility (we relax this assumption below), i.e., Dtρ = 0, we find

$$\nabla \cdot {\bf{v}} \equiv \partial _iv_i = {\mathrm{\Lambda }},$$
(2)

where hereafter repeated suffices are summed. Since Λ is independent of space in our colonies, and there is no coherent azimuthal movement, Eq. (2) predicts the radial component to be vr = Λr/2 ≡ Hr, as observed, Fig. 2b (the factor 1/2 arises as the system is 2D).

A linear fit to the population-averaged data in Fig. 2b gives our ‘Hubble constant’ H = 0.007 min−1. Theoretically, H = 0.5Λ, while individual values of (H, Λ) obtained from each of the 32 colonies, Fig. 2c, are consistent with a linear scaling of H with Λ and are fitted by H = 0.4Λ (the correlation between H and Λ is r ~ 0.77, corresponding to a p-value p < 0.00001). The discrepancy from the theoretical prediction H = 0.5Λ may arise either from small systematic errors in the fitting (e.g., due to small drift close to colony centroids) or from the approximations in the theoretical argument, such as that of isotropic growth. The scatter in the data in Fig. 2c gives an indication of the extent of intrinsic noise in our population of colonies.

Note that the expansive ‘Hubble flow’ we observe is distinct from the spontaneous shear flow exhibited by number-conserving active nematics24,25.

Active anchoring

A striking feature of the larger colonies is that cells tend to be aligned tangentially at the periphery, Fig. 1d. We fitted a Bézier curve through the centroids of the outermost cells, and measured the angle ψ between the curve and each boundary cell. For a typical colony just before it buckled, most peripheral cells show ψ 30°, Fig. 3a. Such ‘active anchoring’ is also seen in simulated number-conserving active nematics20, where it originates from the ingress of topological defects following the buckling of the boundary between an active nematic and an isotropic continuum. We will see later that the mechanism in our case is different.

Continuum description

Turning to the bulk of our colonies, we first construct a continuum structural description based on our raw data, which consist of the position, length and orientation of each cell as functions of time, {ri(t), Li(t), νi(t)}. We start this coarse graining procedure by defining a function that ‘smears out’ individual rod-shaped bacteria of diameter W:

$$\begin{array}{*{20}{l}} {f^i({\bf{r}})} \hfill & = \hfill & {\frac{1}{2}\left( {{\mathrm{tanh}}\left[ {\frac{{{\boldsymbol{v}}.{\bf{\Delta r}}^{\bf{i}} \,+\, {\textstyle{1 \over 2}}L^i}}{\sigma }} \right] - {\mathrm{tanh}}\left[ {\frac{{{\boldsymbol{v}}.{\bf{\Delta r}}^i \,-\, {\textstyle{1 \over 2}}L^i}}{\sigma }} \right]} \right)} \hfill \\ {} \hfill & {} \hfill & { \times \frac{1}{2}\left( {{\mathrm{tanh}}\left[ {\frac{{\overline {\boldsymbol{v}} .{\bf{\Delta r}}^{\bf{i}} \,+\, {\textstyle{1 \over 2}}W}}{\sigma }} \right] - {\mathrm{tanh}}\left[ {\frac{{\overline {\boldsymbol{v}} .{\bf{\Delta r}}^i \,-\, {\textstyle{1 \over 2}}W}}{\sigma }} \right]} \right),} \hfill \end{array}$$
(3)

with Δri = r − ri, $$\overline {\boldsymbol{v}} ^i$$ the unit vector perpendicular to νi and σ a smoothing length. We use σ = W to probe ordering at the single-cell level (σ → 0 gives sharp Li × W rectangles). The integrated cell ‘mass’ is preserved as σ varies. [Note that the functional form of fi we use is required to account for the non-spherical, rod-like shape of bacteria.] The (number) density field is then

$$\rho ({\bf{r}}) = \mathop {\sum}\limits_{i = 1}^N {\kern 1pt} f^i({\bf{r}}).$$
(4)

To quantify orientational order, we assume that the two poles of a cell are indistinguishable (i.e., ±νi are equivalent), and construct a traceless and symmetric Q-tensor26 field

$$Q_{\alpha \beta }({\bf{r}}) = \mathop {\sum}\limits_{i = 1}^N {\kern 1pt} f^i({\bf{r}})\left( {2\nu _\alpha ^i\nu _\beta ^i - \delta _{\alpha \beta }} \right),$$
(5)
$$= S({\bf{r}})\left[ {2n_\alpha ({\bf{r}})n_\beta ({\bf{r}}) - \delta _{\alpha \beta }} \right],$$
(6)

where we have also defined the scalar order parameter

$$S({\bf{r}}) = \sqrt {Q_{xx}^2({\mathbf{r}}) + Q_{xy}^2({\bf{r}})} ,$$
(7)

and the director n(r), which is the unit eigenvector associated with the positive eigenvalue of Qαβ(r). Supplementary Fig. 3 (see Supplementary Note 4) shows Qαβ(r) of the snapshots in Fig. 1.

Global order

The global version of Eq. (7) quantifies the degree of orientational order averaged over an N-cell colony:

$$S(N) = \sqrt {\left\langle {Q_{xx}} \right\rangle ^2 \,+\, \left\langle {Q_{xy}} \right\rangle ^2} = \frac{1}{N}\sqrt {\mathop {\sum}\limits_{i,j} {\kern 1pt} {\mathrm{cos}}\left[ {2\left( {\theta ^i - \theta ^j} \right)} \right]} ,$$
(8)

where θi is the angle between the axis of cell i and the director (i.e., νi × n = cosθi), and 〈…〉 denotes averaging over a colony. This evaluates to unity when all cells are perfectly aligned, and to $$1{\mathrm{/}}\sqrt N$$ for total orientational disorder. The measured degree of global order, S(N), varies significantly between colonies, Fig. 3b, but the trend is clear. Global order is high for up to 3 generations (N 10), and then decreases. Significantly, $$\left\langle S \right\rangle = 1{\mathrm{/}}\sqrt N$$ forms the lower envelope of the vast majority of our data. Thus, while there is no sustained growth-induced global ordering14, substantial residual order persists up to the buckling point, when our observations stop.

Correlation lengths and domains

Underlying the persistence of residual global order is a significant degree of local orientational order in the form of domains of aligned cells: Fig. 1c, d. Such local order can be quantified by calculating two-point correlation functions of the Q-tensor, C||(r) and C(r) (Supplementary Note 5), which measure the azimuthally-averaged orientational correlation at a distance r from a typical cell parallel and perpendicular to its axis, respectively. These functions for the colony snapshots in Fig. 1 are shown in Supplementary Fig. 4. In the earliest stage of growth (N = 15), C||(r) and C(r) decay over ≈10 and 5 μm respectively, i.e., the whole colony is ordered (as is obvious from Fig. 1a). At intermediate times, parallel correlations grow faster than perpendicular correlations, before C||(r) and C(r) become more comparable at late times.

This trend is generic. We calculated C||(r) and C(r) for 32 colonies, and obtained correlation lengths defined by

$$l_c^{||, \bot } = \frac{1}{{C^{||, \bot }(0)}}{\int}_0^\infty {\kern 1pt} C^{||, \bot }(r){\mathrm{d}}r,$$
(9)

so that two cells each of length and width L 2 μm and W 1 μm placed side by side and end-on give $$l_c^{||} = L{\mathrm{/}}2 \approx 1$$ μm and $$l_c^ \bot = W{\mathrm{/}}2 \approx 0.5$$ μm respectively. At early times, $$l_c^{||}{\mathrm{/}}l_c^ \bot$$ grows (Supplementary Fig. 5), with $$l_c^{||}$$ growing faster and peaking at 5 μm, compared to a maximum $$l_c^ \bot$$ 3 μm, corresponding to domains of some tens of cells, Fig. 1b. Thereafter, both correlation lengths decrease to 2 μm, corresponding to domains of 10 cells. Similarly-sized ‘micro-domains’ have been seen in recent simulated 2D bacterial colonies at late times19.

To highlight the role of active growth in generating this pattern of behaviour, we performed Monte Carlo simulations27,28,29,30,31 of passive (non-growing) sphero-cylinders32 with lengths and widths taken from bacterial colonies (see Supplementary Note 6 and Supplementary Fig. 6). In the passive case, $$l_c^{||}{\mathrm{/}}l_c^ \bot \approx 1$$ throughout the entire history of the colony (Supplementary Fig. 5).

Proliferation of topological defects

The existence of domains implies topological defects, where the director field is ill-defined. Defects control the elasticity and dynamics of passive liquid crystals, and feature prominently in number-conserving active nematics6, where they arise from the self-sustaining shear flow arising from the stirring of the fluid by active particles.

The topological charge, or winding number, of a defect, is the accumulated change (in units of 2π) in the angle ϕ made by the director field relative to an arbitrary direction on following a closed loop, $${\cal C}$$, around the defect: $$m = \frac{1}{{2\pi }}{\oint}_{\cal C} \,d\phi$$. Defects are identified via a topological charge density20:

$$q = \frac{1}{{2\pi }}\left( {\partial _xQ_{xx}\partial _yQ_{xy} - \partial _xQ_{xy}\partial _yQ_{xx}} \right),$$
(10)

which can be computed directly from the Q-tensor field.

An example of spatial maps of q is shown in Supplementary Fig. 7 (see also Supplementary Note 7). We identified individual defects as local maxima and minima of q(r), and determined their polarity from the unit vector $$\hat p_{{\mathrm{d}},\alpha } = - \partial _\beta Q_{\alpha \beta }{\mathrm{/}}\left| {\partial _\beta Q_{\alpha \beta }} \right|$$ at the local maximum. As in conventional active nematics, defects in our colonies mostly have topological charge $$\pm {\textstyle{1 \over 2}}$$. As the colony grows, we see the formation of multiple defect pairs in the colony interior, Fig. 4a. While defects are created continuously, the total signed topological charge $${\int} {\kern 1pt} q{\mathrm{d}}^2{\bf{r}} = 0$$. However, the total number of defects, which we compute as the total unsigned topological charge, $${\int} \left| q \right|{\mathrm{d}}^2{\bf{r}}$$ scales near-linearly with N, Fig. 4b, suggesting that defect generation and annihilation is a bulk, not boundary, phenomenon, and that topological (unsigned) charge grows exponentially in time. Linear scaling with N is expected on general grounds if, as we observe, the colony fragments into micro-domain of well-defined size (10 cells).

Defect dynamics

Tracking the topological defects through time, Fig. 4a (see also Supplementary Fig. 7), shows that the orientation of the three-fold symmetric $$- {\textstyle{1 \over 2}}$$ defects does not significantly impact their dynamics21. Presumably, these are simply advected by the expansion of the colony, perhaps aided by mutual elastic interactions. In contrast, comet-like ‘head and tail’ $$+ {\textstyle{1 \over 2}}$$ defects show marked directed motion—the speed just prior to buckling is v+ ~ 0.05 μm s−1. [This value should be seen as an order-of-magnitude estimate as the measurement requires unambiguous tracking of defects which we could only do in few selected cases.] The distribution of $$\widehat {\bf{p}}_{\mathrm{d}} \cdot {\mathrm{\Delta }}\widehat {\bf{r}}_{{\mathrm{d}} - {\mathrm{com}}}$$, Fig. 4c, where $${\mathrm{\Delta }}\widehat {\bf{r}}_{{\mathrm{d}} - {\mathrm{com}}}$$ is the unit vector pointing from a colony’s centre of mass to the defect, shows that positive defects propel along the direction of $$\widehat {\bf{p}}_{\mathrm{d}}$$ (i.e., tail to head), reminiscent of their behaviour in a non-growing extensile active nematic21.

The spatial distribution of $$\pm {\textstyle{1 \over 2}}$$ defects, Fig. 4d, shows ‘charge separation’. There is an excess of $$- {\textstyle{1 \over 2}}$$ defects near the colony centre, while $$+ {\textstyle{1 \over 2}}$$ defects predominate at the periphery. Presumably, inwardly-directed $$+ {\textstyle{1 \over 2}}$$ defects have a high chance of annihilating $$- {\textstyle{1 \over 2}}$$ defects advected by colony expansion, while outward-directed $$+ {\textstyle{1 \over 2}}$$ defects rapidly propel to the colony edge, where advected $$- {\textstyle{1 \over 2}}$$ defects have not yet reached. Moreover, once a $$+ {\textstyle{1 \over 2}}$$ defect has propelled itself to the colony edge in a tail-to-head direction (Fig. 4c), the director field pattern near its ‘head’ matches the tangential anchoring, Fig. 3a, and is therefore energetically favoured. Thus, we find that $$+ {\textstyle{1 \over 2}}$$ defects are rarely ejected from the boundary into the colony. This contrasts with a number-conserving active nematic surrounded by an isotropic medium, where ejection of $$+ {\textstyle{1 \over 2}}$$ defects from a deeply-undulating boundary into the interior is prominent20.

Intriguingly, the region in which we observe an excess of $$- {\textstyle{1 \over 2}}$$ defects, r 0.2R (here R is the radius of the inscribed circle), is precisely where we see the onset of buckling into the third dimension. This is reminiscent of the physics of passive liquid crystals, where the director field can avoid the formation of a costly defect core by escaping into the third dimension. This observation also suggests that defects may provide a means to relax stress in the system, as proposed recently for eukaryotic colonies33.

Theoretical description and length scales

Our observations and analyses so far suggest that it should be fruitful to describe a growing 2D colony of E. coli cells theoretically as an active nematic, albeit in a different universality class than almost all current theories of this kind because of particle number non-conservation. Here, we first delineate a number of components that must feature in such a theory and point out their implications. We then perform numerical simulations of a set of equations which assumes a particularly simple constitutive relation for the active stress.

There are two important length scales in our system. The first one, la, is related to the onset of the ‘generic instability’ in active nematics.In wet systems, la is given by a balance between active and elastic stresses34, as $$l_a\sim \sqrt {K{\mathrm{/}}a}$$, where K is a Frank elastic constant and a is the active stress coefficient, defined so that the growth-dependent active stress scales as aQαβ. In a growing colony, K and a both depend on growth-mediated cell deformations, and should scale as the cell wall elastic modulus, G, so that a ~ G and K ~ GL2 for cells of length L. They should also scale similarly with cell density16,19, so that we may expect K/a to be a constant, and la ~ L. In our dry system, the functional dependence of la on parameters is more complicated (see below and35). However, as la is generally lower for the dry than for the wet limit7, we expect la ~ L should still hold for our colonies.

A growing cell pushes and deforms its neighbours, generating stresses that scale as G. The resulting cell movements are opposed by dissipative forces, which we take to be lubricated frictional sliding between cells and agarose or glass controlled by a frictional drag coefficient per unit length, γ, so that the drag force on a cell of length L sliding at speed v is ~γLv. Force balance in terms of the stress tensor σαβ then reads

$$\partial _\beta \sigma _{\alpha \beta } = \left( {\gamma L\rho } \right)v_\alpha \equiv \hat \gamma v_\alpha ,$$
(11)

where ρ is the cell density. Our ‘dry’ description in Eq. (11) is motivated by the fact that our colony is very close to the glass coverslip (Supplementary Fig. 1); in general this equation will hold above a hydrodynamic ‘screening length’ $$l_s\sim \sqrt {{\textstyle{\eta \over {\hat \gamma }}}}$$, with η the effective viscosity of the colony.

Growth stress propagates by diffusion; dimensional analysis gives the diffusivity $$\sim G{\mathrm{/}}\hat \gamma$$16. [More rigorously, we can show that, within our theory, when Λ = 0 both ρ and the stress evolve according to a diffusion equation where the diffusion coefficient is $$\sim G{\mathrm{/}}\hat \gamma$$, given the simple constitutive relation for the stress considered below.] The extent of stress propagation during the lifetime of a single cell (birth to division) defines a stress propagation length

$$l_\sigma = \sqrt{\dfrac{G}{\hat \gamma }{\dfrac{1}{\mathrm{\Lambda }}}} .$$
(12)

Previous measurements give G ~ 106 Pa, albeit with large variations, while we estimate $$\hat \gamma \sim 10^{15}$$ (Pa s)m−2 using previous experimental results36 (Supplementary Note 8). Thus lσ ~ mm. This is an upper bound for the size of a colony in which internal growth stresses can relax into the agarose matrix. In practice, buckling into the third dimension will occur when the stress approaches the yield stress of 2% agarose, $$\sim 3 \times 10^3\,{\mathrm{Pa}} \ll G$$. With this revised stress estimate, lσ ~ 50 μm, similar to experiments. Intriguingly, v+Λ−1 ~ 50 μm as well, suggesting that defects may have a role in relaxing growth stresses.

Growth-induced alignment

To see how growth should induce orientational order, consider first a system of bacteria oriented along y (Qyy = 1, Qxx = −1, Qxy = Qyx = 0), Fig. 5a, with a single slightly-misaligned bacterium $${\cal B}$$ (centroid at y0) whose angle θ with the orthogonal (x) axis is <90°. The only velocity gradient component is ∂yvy = Λ. The fraction of $${\cal B}$$ at larger y moves with higher vy, and so must experience a higher force than the lower part; the net torque realigns this bacterium.

Quantitatively, the force giving rise to the velocity of a segment, dl, of $${\cal B}$$ located a distance l from the centroid must balance the frictional drag with the substrate at this position, γvy(y)dl. The total torque on $${\cal B}$$ is then

$$\tau = {\int}_{ - L/2}^{L/2} {\kern 1pt} \gamma v\left( {y_0 + l\,{\mathrm{sin}}\theta } \right)l\,{\mathrm{cos}}\theta {\mathrm{d}}l.$$
(13)

Setting τ = ζdθ/dt, where ζ is the rotational drag on a single cell due to its neighbours, and expanding to first order gives

$$\begin{array}{*{20}{l}} {\partial _t\theta = \dfrac{\gamma L^3} {24\zeta }\left( {\partial _yv_y} \right){\mathrm{sin}}\,2\theta ,\left[ {{\mathrm{1D}}} \right]} \hfill \end{array},$$
(14)

where ∂yvy = Λ > 0, Eq. (14) has a stable equilibrium at θ = 90°, i.e., growth drives alignment.

This quasi-1D analysis correctly predicts a high degree of alignment in our early-stage colonies, Fig. 1a, b. This persists only for a few generations, because the active instability length scale la ~ L. When the colony size 2R increases to the point when $$2R{\mathrm{/}}l_a \gg 1$$, the extensile stresses due to growth create bend distortions, as in number-conserving active nematics. The colony now grows in two dimensions.

For simplicity, consider the simplest case of circularly symmetric expansion in 2D, for which the cell velocity field is

$${\bf{v}} = g(r){\bf{r}},$$
(15)

with g(r) a local growth rate. Substituting Eq. (15) into nematohydrodynamic equations describing flow-orientation coupling (see Supplementary Note 9) yields the analogue of Eq. (14) for 2D:

$$\partial _t\theta = \frac{{\xi g\prime r}}{{4S}}{\mathrm{sin}}\,2(\theta - \phi ),\left[ {2{\mathrm{D}}} \right]$$
(16)

where ϕ denotes the azimuthal angle in polar coordinates and ξ > 0 is a flow-aligning parameter.

Equation (16) predicts that growth-induced alignment depends on the sign of g′. For compressible flows, g′ > 0, there is a stable steady state (∂tθ = 0) with θ = ϕ, giving a radially-ordered colony with a central +1 vortex defect. This is the analogue of the growth-induced alignment predicted by Eq. (14) for 1D.

We do not observe such alignment, because experimentally, we find incompressibility, i.e., g Λ/2 and g′ = 0, Fig. 2a. In this case, the steady state of Eq. (16) does not require any particular relation between θ and ϕ. The colony breaks into domains with a length scale controlled by la. The number of defects scales as the number of domains. There are ~10 ≈ 32 cells per defect, Fig. 4b, consistent with la ~ L. Our colonies are effectively incompressible because their sizes (2R 50 μm) are always < lσ, so that growth-induced stresses relax quickly.

At the edge of a colony, g Λ/2 (inside) transitions to g = 0 (outside), so that g′ < 0. Equation (16) predicts a stable steady state with θ = ϕ + π/2, i.e., tangential alignment, again as observed. Qualitatively, the origins of this effect is easy to understand. The presence and absence of growth-generated forces inside and outside the colony respectively, Fig. 5b, means that a cell not aligned tangentially will be rotated back into such alignment, as previously suggested37.

Non-uniform active stress

In theories of number-conserving active nematics, the active stress is taken to be

$$\sigma _{\alpha \beta }^{\left( a \right)} = - aQ_{\alpha \beta },$$
(17)

where a is a positive or negative constant for extensile or contractile activity respectively6. It is tempting to model growing bacterial colonies in the same way with a Λ.

To see why this does not work for a Hubble active nematic, consider again a 1D colony in the y direction. Force balance, Eq. (11), and mass conservation, Eq. (1), give

$$\partial _{yy}\sigma _{yy}^{(a)} = \hat \gamma {\mathrm{\Lambda }}.\,\left[ {1{\mathrm{D}}} \right]$$
(18)

For a 1D colony occupying 0 < y < y0 with spatially-constant Λ (cf. Fig. 2a), Eq. (18) solves to14,15

$$\sigma _{yy} = - P_0 - {\textstyle{1 \over 2}}{\mathrm{\Lambda }}\hat \gamma \left( {y_0^2 - y^2} \right),$$
(19)

where P0 is the pressure exerted by the agarose at the boundaries. The active stress field is non-uniform, increasing from the periphery to the centre by $${\mathrm{\Delta }}\sigma \sim \hat \gamma {\mathrm{\Lambda }}R^2$$ for a colony of dimension 2R. Numerically, this inhomogeneity can be substantial (our estimated parameters give Δσ ~ 3 × 103 Pa for 2R ~ lσ). Physically, such non-uniformity necessarily arises because peripheral cells have to push at fewer cells sliding frictionally against the substrate as the colony expands17.

The coupling of mechanics and nematic order in 2D gives rise to new complexity. Inspired by Eq. (19), we may expect

$$\sigma _{\alpha \beta } = - p({\bf{r}})\delta _{\alpha \beta } - a({\bf{r}})Q_{\alpha \beta }.\,\left[ {2{\mathrm{D}}} \right]$$
(20)

A simple constitutive relation consistent with a dry compressible active nematic is p = G max[(ρ/ρ0 − 1), 0], and a = a0ρ, where ρ0 is the close packed density of undeformed cells. The equations resulting from substituting this or other constitutive relations into the laws of nematohydrodynamics are only tractable by numerical simulations (see below). Nevertheless, the same physics we appealed to in the 1D case will still give rise to a stress field increasing from periphery to centre.

In a sufficiently small colony, $$2R \ll l_\sigma$$, these non-uniform growth-induced stresses relax rapidly through the periphery into the agarose, so that the flow is effectively incompressible, Fig. 2b. As 2R → lσ, however, stress relaxation becomes progressively more incomplete, eventually leading to the buckling when the colony is not confined to 2 dimensions (as is the case in our experiments).

Active field theory simulations

To complete our theoretical description, we now report results from direct numerical simulations of a full model comprising Eqs. (1) and (11), together with the following evolution equation for the liquid crystal order parameter,

$$\begin{array}{*{20}{l}} {\dfrac{{\partial Q_{\alpha \beta }}}{{\partial t}} + v_\gamma \partial _\gamma Q_{\alpha \beta }} \hfill & = \hfill & {\hat K\partial _\gamma \partial _\gamma Q_{\alpha \beta } + \alpha (\rho )Q_{\alpha \beta }} \hfill \\ {} \hfill & {} \hfill & { - \beta Q_{\gamma \delta }^2Q_{\alpha \beta } + \xi \left( {u_{\alpha \beta } - u_{\gamma \gamma }\dfrac{{\delta _{\alpha \beta }}}{2}} \right)} \hfill \\ {} \hfill & {} \hfill & { + \omega _{\alpha \gamma }Q_{\gamma \beta } - Q_{\alpha \gamma }\omega _{\gamma \beta }.} \hfill \end{array}$$
(21)

In Eq. (21), we denoted $$\hat K = K{\mathrm{/\gamma }}_1$$, with γ1 the rotational viscosity of the liquid crystalline colony, whereas uαβ = (∂βvα + ∂αvβ)/2 and ωαβ = (∂βvα − ∂αvβ)/2. We also introduced α(ρ) = α0(ρ − ρc), with α0 > 0 and ρc ≡ ρ0/2 to describe liquid crystalline ordering à-la-Onsager for sufficiently large bacterial density—the term proportional to β is a saturation term, and we choose β = α0ρ0/2 which yields 2S2 = 1 in the middle of the colony. The active stress in Eq. (11) is given by Eq. (20), with p = G max[(ρ/ρ0 − 1), 0], and a = a0ρ. For other parameter values, see Fig. 6; for more details, see Methods and Supplementary Figs. 8 and 9.

Our simulations confirm all qualitative predictions of our theory, and also compares favourably with experiments, see Fig. 6 and Supplementary Fig. 8. First, as expected the radial velocity increases linearly with distance to the colony centre—interestingly, our simulations also typically show a slighly smaller ‘Hubble constant’ than expected, as in experiments (H < Λ/2, Figs. 2b and 6a). Second, simulations reproduce the tangential alignment at the colony boundaries (Fig. 6b and Supplementary Fig. 8). Third, when the active coefficient a0 is sufficiently large, the colony breaks up into domains of well-defined lengthscale, la, separated by defects (Fig. 6b and Supplementary Fig. 8). Fourth, the total topological charge increases near-linearly with colony area (proportional to number of bacteria, see Fig. 6c). Finally, the positive topological charges (mainly associated with +1/2 defects) are more peripheral than the negative ones (mainly associated with −1/2 defects, Fig. 6d).

An analysis of our data suggest that there is a substantial range of a0 for which $$l_a\sim a_0^{ - 1/2}$$ (Supplementary Fig. 9). This is in line with simulations of turbulent-like dynamics in number-conserving dry active nematics35. Comparison with the latter study and with the results of19, together with dimensional analysis, suggest that for large a0 a possible functional form for the domain size may be $$l_a\sim \sqrt {{\textstyle{{\hat K\hat \gamma } \over {a_0}}}} \sqrt {{\textstyle{{\hat K} \over {\alpha _0}}}}$$, which is modulated by the defect size $$\sqrt {{\textstyle{{\hat K} \over {\alpha _0}}}}$$. A more systematic study would be required to confirm this however. As a0 decreases, the domain size increases until for sufficiently small a0 the bending instability is no longer active and the colony resembles a nematic tactoidal droplet with tangential anchoring (Supplementary Fig. 8).

Discussion

We have described the structural evolution of a growing 2D colony of rod-shaped E. coli until the point it buckles into the third dimension using the language of liquid crystal physics. Our system displays features that appear similar to those that are familiar from number-conserving active extensile nematics, such as tangential anchoring at the periphery, Fig. 3a, and a proliferation of $$\pm {\textstyle{1 \over 2}}$$ defects, Fig. 4. However, the underlying physical mechanisms differ. Thus, e.g., while invaginations from the boundary is a dominant mechanism of defect generation in a number-conserving active nematic embedded in an isotropic medium, we find that defects proliferated from the bulk of our colonies. The expansive growth of cells in these colonies drives new physics; we have named this system a ‘Hubble’ active nematic.

We have also proposed a continuum theory of such systems. Within this framework, the physics is controlled by two length scales: la associated with the generic bending instability in dry active nematics, and lσ associated with the diffusive propagation of growth stress. For a colony with size $$2R \ll l_\sigma$$, growth stress relaxes rapidly, and the cell flow is effectively incompressible. The colony breaks up into domains of size ~la. A colony with 2R → lσ will accumulate stress and eventually buckle at the centre, where the accumulated stress is highest. In all cases, we expect tangential alignment at the periphery. All these predictions are in agreement with our experimental observations, and are confirmed by computer simulations of this model. More quantitatively, such simulations further reproduce our experimental finding that the total number of defects increases approximately linearly with colony area, and that positive charge defects are more peripheral than negative charge ones.

To enable even more quantitative comparison between experiments and simulations of our growing active nematic system, progress needs to be made on a number of experimental fronts. Perhaps most importantly, the nature of the friction between growing cells and their substrate, i.e. the physics underlying our parameter $$\hat \gamma$$, needs to be elucidated. There is also a need to understand the mechanics of bacterial growth into bulk agarose. It is known that the elastic properties of the surroundings have a complex effect on, e.g., the point at which a 2D colony buckles into the third dimension17. The detailed mechanisms for these effects are far from understood. The effect of bacterial growth rate also needs to be elucidated. Finally, it will be fascinating to develop methods for studying the growth of bacterial colonies in 3D38, for which there are very few data at present39. Arriving at a predictive understanding of bacterial colony growth will constitute an advance not only in fundamental active matter physics, but also in practical fields such as food safety40.

Methods

Experimental methods

Overnight cultures of E. coli K-12 strain MG1655 were grown in M9 medium at 37 °C and shaken at 200 rpm, harvested in exponential phase, and diluted 100-fold (to an optical density of ≈0.2–0.3 at 600 nm). A number of 1.5–2 μL droplets were then inoculated onto a thin layer of 2% M9 agarose filling a polymeric frame (Thermo Scientific Gene Frame AB-0630, ≈0.25 mm thick) stuck onto a standard microscope slide, and a cover slip was sealed on top. Growing colonies were observed at 37 °C in phase contrast mode using a Nikon Eclipse E-800 microscope with a Nikon Plan Apo λ 100×/1.45 Ph3 Oil objective. Images were taken (Q-imaging Retiga 2000R camera) at regular intervals (typically 1 min). Using a published protocol41 we obtained the time-dependent centre of mass coordinates ri(t), length Li(t), and orientational unit vector νi(t) of each cell i. The cell diameter, W ≈ 0.9 μm, varies little between cells or during growth. The length varies from ~2 μm just after division to ~4 μm just before. We report data from 32 colonies.

Image recording and processing

A schematic of the experimental set-up is shown in Supplementary Fig. 1a. For observations under phase contrast illumination intensity was minimised and exposure time varied from 20 to 100 ms. In a separate experiment, we performed confocal microscopy using fluorescent bacterial cells and agarose. For this case, the dye used for agarose staining was rhodamine B (0.02% w/v) (red channel); this was used to enhance contrast with the GFP-labelled cells (green channel) and the glass that is the black area in the images. Rhodamine B preferentially concentrates at the agarose surface. Sideways reconstruction of an image stack, Supplementary Fig. 1b, shows that cells are embedded just within the agarose beneath the glass side.

Analysis of phase contrast images was performed using ImageJ and Matlab. Initially, frames were cropped to the actual area covered by the colonies at the final stage (the field of view achievable is at least three times larger). The frames were then adjusted in brightness and contrast, and corrected for inevitable drifts (a few) in the agarose pad, especially toward the beginning of the imaging period. The latter was due to the agarose taking ~10 min to cool to room temperature, having been poured into the frame at its melting point. The shift correction was performed using ImageJ (macro StackReg42). Finally, the frames were segmented using the Matlab-based Schnitzcells software41 to extract the positions, orientations and lengths of the individual cells. In order to extract velocities, the trajectories were stitched together in MATLAB using the standard Crocker and Grier code43.

Active field theory simulations

Numerical simulations of Eqs. (1), (11), and (21) were performed by using finite difference methods to discretise Eqs. (1) and (21). For reasons of numerical stability, we have set the growth rate Λ to 0 when the density was locally larger than 2ρ0. Note also that in Eq. (21) we need v, rather than ρv which is the natural variable in Eq. (11): to avoid problems arising from division by a very small number we have set v = 0 when ρ was below a threshold (ρ0/2 worked well for our parameter set).

In wet active gels, the key control parameter to determine when active turbulence sets in is $${\textstyle{{aL_{\mathrm{s}}^2} \over K}}$$, where a is the active stress parameter (in our case equal to a0ρ), Ls is the system size, and K is the liquid crystalline elasticity6. Active turbulence sets in when this control parameter exceeds a critical value: therefore an infinite system always develops active turbulence. On the contrary, for dry active fluids the turbulent regime sets in when $${\textstyle{{a_0\rho } \over {\hat \gamma \hat K}}}$$ exceeds a threshold35. This holds for our growing system as well, and we find the threshold is ~2, as can be shown in Supplementary Figs. 8, 9, where we plot the density of points with low orientational order. This plot also shows that the density of low order regions, which we assume should scale as $$l_a^{ - 2}$$ scales linearly with a0 close to the transition to active turbulence, corresponding to a scaling of la ~ (a0 + C)−1/2 in this regime (this is consistent with the behaviour found in the ref. 35). More data are required to firmly prove this scaling, as the range of a0 we investigated is relatively narrow.

The fact that in our experiments there are microdomains within our colonies suggests we are in the regime where nematic order is disrupted and defects arise (active turbulence). Therefore a suitable regime to describe experiments within our active field theory requires $${\textstyle{{a_0\rho } \over {\hat \gamma \hat K}}} \ge 2$$.

Code availability

The Matlab codes used for the analysis and in-house code written for active field theory simulations are available from the corresponding author upon request.

Data availability

Data on bacterial colony growth are available from the corresponding author upon request. The tracked colony data used in this manuscript are also available at https://dx.doi.org/10.7488/ds/2444.

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Acknowledgements

We thank A. Dawson for help with microbiological lab work, and M. E. Cates, A. Doostmohammadi, M. C. Marchetti, J. Toner, B. Wacław and J. M. Yeomans for helpful discussions. The majority of the work was funded by EPSRC grant EP/J007404/1 and ERC grant AdG 340877 PHYSAPS. D.D. acknowledges funding from EPSRC and SUPA. J.S.L. acknowledges EU intra-European fellowship 623637 DyCoCoS FP7-PEOPLE-2013-IEF.

Author information

Authors

Contributions

W.C.K.P. and D.M. designed research. D.D.A., A.T.B. and A.F.M. performed experiments. M.L.B., F.D.C.F., J.S.L., D.M. performed simulations. All authors analysed the results and wrote the manuscript.

Corresponding author

Correspondence to D. Marenduzzo.

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The authors declare no competing interests.

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Dell’Arciprete, D., Blow, M.L., Brown, A.T. et al. A growing bacterial colony in two dimensions as an active nematic. Nat Commun 9, 4190 (2018). https://doi.org/10.1038/s41467-018-06370-3

• Accepted:

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# Glut Mixing 2d and 3d

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Hello everyone! I am currently using Glut in an application that required that I mix 2d and 3d together. Right now I can switch between the two, and resize my window using the code below. However the problem is that when I resize the window it stretches the 3d part. I also need the 2d layer to be in sync with the 3d layer at all times. How can I accomplish this? I need the 3d to stay relative to the 2d when I resize my window, and the window should not stretch the 3d. I have a feeling I have to use orthographic projection for the 3d as well, but I am not quite sure if that's what needs to be done. Thanks for any future insight, I hope I explained my situation well enough. (here's my code if it helps to see what I am using.)
void glEnable2D(){
GLint iViewport[4];

// Get a copy of the viewport
glGetIntegerv( GL_VIEWPORT, iViewport );

// Save a copy of the projection matrix so that we can restore it
// when it's time to do 3D rendering again.
glMatrixMode( GL_PROJECTION );
glPushMatrix();

// Set up the orthographic projection
glOrtho( iViewport[0], iViewport[0]+iViewport[2],
iViewport[1]+iViewport[3], iViewport[1], -1, 1 );
glMatrixMode( GL_MODELVIEW );
glPushMatrix();

// Make sure depth testing and lighting are disabled for 2D rendering until
// we are finished rendering in 2D
glPushAttrib( GL_DEPTH_BUFFER_BIT | GL_LIGHTING_BIT );
glDisable( GL_DEPTH_TEST );
glDisable( GL_LIGHTING );
}
void glDisable2D()
{
glPopAttrib();
glMatrixMode( GL_PROJECTION );
glPopMatrix();
glMatrixMode( GL_MODELVIEW );
glPopMatrix();
}
static void resize(int width, int height){ //change 800 to width and 600 to height
GLdouble ASPECTRATIO = (GLdouble)width/(GLdouble)height;
glViewport(0, 0, width, height);              /* Set the viewport */
glMatrixMode(GL_PROJECTION);                  /* Select the projection matrix */
glLoadIdentity();				/* Reset The Projection Matrix */

gluPerspective(45.0f,(GLfloat)ASPECTRATIO,0.1f,200.0f);  /* Calculate The Aspect Ratio Of The Window FIRST 1*/

glMatrixMode(GL_MODELVIEW);                   /* Switch back to the modelview matrix */

glClearAccum(0.0, 0.0, 0.0, 1.0);
glClear(GL_ACCUM_BUFFER_BIT);

}

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you would have to keep aspectratio constant. Example, set it to 1.0

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Thanks for your reply, I attempted to just change the aspect ratio in my resize function to...

GLdouble ASPECTRATIO = 1;

However I still received stretching on window resize. As I think about it more and more I think I need to have the 3d use orthogonal view.

It's going to be a game where buildings will be 3d models, and characters and other things will be 2d. Right now I can position 2d objects using the standard 0-800, 0-600 coordinate system. So I need a way to position 3d buildings relative to the 2d stuff easily, but at the same time orthogonal would remove the right angles wouldn't it? So the buildings would just be seen top down regardless of their distance away from the viewport?

Is anything like this possible?

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solubility of alkali metal nitrates

0 yorum

Lithium-ion is the smallest in size and we know smaller the size of ion, the more hydrated it is. Why are alkali metals not found in nature ? Comment on each of the following observations: (a) The mobilities of the alkali metal ions in aqueous solution are Li+ < Na+ < K+ < Rb+ < Cs+ (b) Lithium is the only alkali metal to form a nitride directly. Electronic configuration of. (a) Na (b) K (c) Rb (d) Cs, Which one of the following alkali metals gives hydrated salts ? Halides. As we move down the group reactive nature increases. If you have a doubt or question that is not available here or in any of the chapters, contact us. These are the detailed explanation of NCERT textbook questions. Act as a reducing agent but not as powerful as alkali metals. [ M = Be, Mg, Ca, Sr, Ba] (b) Carbonates- Alkaline earth metal carbonates decompose on heating gives carbon dioxide and oxide. (iv) Solubility of carbonates and sulphates decreases down the group. Start studying Solubility Rules. They produce the metal nitrite and oxygen, but no nitrogen dioxide: $2XNO_3 (s) \rightarrow 2XNO_2(s) + O_2 (g)$ Each of the nitrates from sodium to cesium decomposes in this way; the only difference is in the temperature required for the reaction to proceed. Mg helps in maintains normal blood circulation in our body. (b) alkali metals are prepared by electrolysis of their fused chlorides ? KnopmanMarks. All nitrates, acetates, hydrogen carbonates, and chlorates are… All Alkali metals and Ammonium compounds are soluble. Lithium-ion is the smallest in size and we know smaller the size of ion, the more hydrated it is. The carbonates and sulphates of alkaline earth metals are less soluble than corresponding alkali metals . (4) Chlorides, bromides, and iodides are soluble Alkali earth metals form solid carbonates. The solubility data at 1 bar or saturation pressure for sodium nitrate are reviewed. Lithium forms monoxide, sodium forms peroxide, the other metals form superoxides. (iii) sodium peroxide dissolves in water ? General electronic configuration of Group 1 elements is, 2. In s-block, there is a total of two groups (1&2). Alkali metal salts are soluble and nitrate salts are not. Down the group thermal stability of nitrates increases. All nitrates are soluble. (b) Because their oxides are themselves are a very strong reducing agent in nature. 2H2O. Question 10.25 What happens when (i) sodium metal is dropped in water ? Any substance that fails to reach 0.1 M is defined to be insoluble. Q9. Solubility in water is related to the ionic nature and size. Heating the nitrates. General electronic configuration of Group 2 elements is, NCERT Solutions for Class 11 Chemistry Chapter 10 The s-Block Elements - Exercise Questions. 3. That’s … Sodium reacts vigorously with oxygen and forming sodium peroxides. Data on some, such as cesium and rubidium fluoride, nitrate, carbonate, and sulfate, appear to be reported for the first time. Nitrates of other alkali metals on decomposition gives corresponding nitrite. 2LiNO3 → Li2O + 2NO2 + O2 2Ca (NO3)2→ 2CaO + 4NO2 + O2 Hence it is more soluble in methanol. Most nitrates tend to decompose on heating to give the metal oxide, brown fumes of nitrogen dioxide, and oxygen. All nitrates are soluble in water. Hence ion can easily polarise ions and making it unstable and because of that lattice energy of is not very high and so it is soluble in water. Solubility of sulphates: Sulphates formed by alkali metals are highly soluble and form alums very easily. : Some Li + are insoluble, with Li 3 PO 4 being the most common example. The IUPAC Solubility Data Series documents found on this site are web-versions of the volumes. Learn vocabulary, terms, and more with flashcards, games, and other study tools. How to prepare for KVPY 2021? Any substance that can form 0.1 M or more concentrated is soluble. They burn vigorously in oxygen forming oxides. 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Fused Chlorides group hydration enthalpy alkali metals that can form 0.1 M is defined to be insoluble value, lattic. Compatibility with they again react with hydrogen and form hydrides except for lithium nitrate on heating, Beryllium nitrate nitrite. Form solution in liquid ammonia and give the metal oxide, brown fumes of nitrogen,... Please be aware that these rules are to be applied in the same periods group ionic size.!, Mikhail Yurievich Skripkin, Kiyoshi Sawada, Stefka Tepavitcharova and gradation properties... Are insoluble metal dissolves in liquid ammonia Chlorides, bromides, and oxygen high covalent character.! As we know that down the column polarising power bicarbonates: bicarbonates of alkali metals not be by. Dissolves and form hydrides except for lithium nitrate, it is very.! Forms peroxide, the more hydrated it is used in paper, paints textile. Fails to reach 0.1 M is defined to be more useful than potassium electrolysis of their Chlorides... These three factors that should be the same periods and caustic soda melting...: slightly soluble very helpful site two parts ) IUPAC Project part 1 NaCl s. Is, 2 to Chemistry, Basic concepts - periodic table soap borax. Slightly soluble unlike other alkali metals are sparingly soluble and nitrate ions detected., rather than lithium used in the free supply of air between Li and Mg– lithium shows to... Binary, ternary, and iodides are soluble silver, lead, and lead acetate are moderately.! That is not high so it can be solubility of alkali metal nitrates from sodium chloride.In this process electrolysis their! Are 32 questions in the exercise peroxide dissolved in water. ( compare the solubility thermal... Life, this app would have played a huge role in laying the foundation of your decisions! Find colleges in other sites, this is because lithium-ion is the cheapest form of solubility of alkali metal nitrates halogens to form and! Orlova alkali metal is heated in the manufacture of dye stuffs Solutions will help you well! Is having least melting point rules a SUMMARY of SOLUBILITIES rule exceptions 1 ) lithium other... Our Chemistry experts less soluble than their corresponding carbonates you can access it from profile... The cheapest form of alkali metals are soluble in water. ( coloured solution sulphate is soluble in as. And Industry rules | SIE Exam Preparation solubility of alkali metal nitrates most stable the answers will... Are silvery white, water-soluble salts with relatively similar melting points or aqueous Solutions the individual ions are! Decrease down the group lead acetate are moderately soluble know smaller the size hydrated... Dry air due to the formation of a cation, greater is its hydration energy not! To use not be overcome by hydration energy critically evaluated halides with definite. Sulphate and magnesium do not form alums to Chemistry, Basic concepts - periodic table nitrates, and! That are insoluble, with Li 3 PO 4 being the most common.. Density which increases down the group size of the aqueous solution as hydrates, has. ) alkaline earth metal and the lead ( ii ) is an important material! Of the other alkali metals and NH 4 + ions are detected by precipitating them as insoluble magnesium is... ) is soluble in water it gives lithium oxides Beryllium sulphate and magnesium sulphate is soluble because cation is in. This nitrate in non-polar solvents allows the separation of cerium from other rare earths forms nitrite and perchlorate! Obtained in water also capable to form hydroxide and dihydrogen decreases and both the and. Sulphates of alkaline earth metals do so rule 6 as we mix metal. Of colour change primary material for manufacturing cement and is also not amorphous with other sulphates what happens (! Guidelines only let us discuss the various reactions that occur in the Solvay process heating forms nitrites with the of! Sulphate is soluble paints and textile industries are reviewed so, lattice energy can use. Metal hydroxides and Ba ( OH ) 2 → 2MO + 4 NO 2 O... The alkali metals and ammonium ion are soluble, watch as we know that down the group from Li Cs! Enthalpy and hydration enthalpy sparingly soluble and form alums very easily are mostly soluble, lead. Carbonates ( c ) sulphates forms nitrites with the evolution of solubility of alkali metal nitrates obtained by chemical reduction, we know down... Not amorphous with other sulphates of alkali metal ion ion less is mobility nature of lithium decompose on,... Alkali fluorides nature increases > Chlorides > bromides > iodides with moisture form! Solution method because after liberating the metals they again react with halogens at high temperature to form solid halides... Unit Conversion - Duration: 3:01:41 process electrolysis of their oxides which in turn with... As insoluble magnesium carbonate is used as a result, the other hand carbonates of alkali metals give nitrites..., some teachers would have a low density which increases down the group ionic size increases than other fluorides!, this app would have a solvation number of 8 or 9 lithium compounds are soluble carbonates ( )... 10.31 which one of the alkali metal nitrates and alkali metal nitrates are soluble and form alums please aware... The overall effect of these elements have a low density which increases down the group hydration enthalpy decreases both. Contains two electrons solubility of alkali metal nitrates lose and attain their nearest noble gas configuration of these factors! Elements in which the last electron enters the outermost s-subshell ( or s-orbital.... The melting and boiling points of the individual ions have very good size with.: except: alkali and alkaline earth metals are low indicating sulphates increases down the column become more soluble organic! And magnesium sulphate is soluble in water increases on moving down the group reactive nature increases is much less than... Nitrate, it has a high tendency to distort the electron cloud the! Particular alkali metal is heated in the Solvay process sulphates formed by alkali metals give corresponding nitrites except Beryllium! ) bromide is a good idea to memorize them, the chemical name is calcium oxide the! Size compatibility with in our body ) Na ( c ) sulphates size and therefore it high. We will study the s-block elements are divided into two groups salts such as iron or! And water- nitrates tend to decompose on heating, Beryllium nitrate forms nitrite and, other nitrates yield,. Two groups charge density and can be prepared by reacting the corresponding oxides, and. Compatibility of both the anion and cation have very good size compatibility with ) alkali metals and ion... Is a good reference to use Eysseltová and Violetta Timofeevna Orlova alkali metal salts are soluble in water (! Have played a huge role in laying the foundation of your career decisions i ) sodium metal in! That these rules are to be more useful than potassium whereas LiCl soluble not only in water..... Important laboratory reagent both in qualitative and quantitative analysis, the lattic energy of carbonates usually.! The low ionisation energy of carbonates usually increases question 10.23 why is LiF almost insoluble in water is to! Scanned by NIST into PDF files and digitized in their outermost shell insoluble rule.: except: alkali and alkaline earth metal carbonates is thermally the most example... First IE is higher than those elements of the cation and anion are very high and it is because is... Because of the following: ( i ) compounds are soluble, except when paired with nitrate JPCRD... Very easily coloured solution, Beryllium nitrate forms nitrite and potassium perchlorate considered! Ammonia and give the deep blue coloured solution formation of a cation, greater is its hydration energy not! Preapred by Solvay process... KVPY result 2021 - Check SA, SX & SB Streams here... Bromide is a good reference to use any substance that can form 0.1 M or more concentrated is because... Lattice energies decrease more rapidly than the hydration energies result 2021 - Check Science Olympiad result! Anion are very high occur in the Solvay process our body to decrease in hydration enthalpy water due... By more water molecules be prepared by electrolysis of the compounds of the ionisation. Heating to form halogen halides mercury ( i ) compounds are insoluble the.. No2 and O2 K and group since the lattice energies decrease more rapidly the... Least water-soluble alkali metal compounds that are insoluble size as compare to ion SA... Colleges in other sites, this is the charge density and high polarising power are considered slightly soluble less. That of calcium and magnesium sulphate is soluble in water. ( forms monoxide, sodium peroxide... 10.31 which one of the formation of their fused Chlorides heating form their respective oxides NO2 and O2 solubility LiF... With a definite crystal structure last electron enters the outermost electron of potassium is lager the!

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## Astronaut Michael Collins, Apollo 8, and the anterior cervical fusion that changed the history of human spaceflight

In 1961, President John F. Kennedy declared that the United States would send a man to the moon and safely bring him home before the end of the decade. Astronaut Michael Collins was one of those men. He flew to the moon on the historic flight of Apollo 11 while Neil Armstrong and Buzz Aldrin walked on its surface. However, this was not supposed to be the case.

Astronaut Collins was scheduled to fly on Apollo 8. While training, in 1968, he started developing symptoms of cervical myelopathy. He underwent evaluation at Wilford Hall Air Force Hospital in San Antonio and was noted to have a C5–6 disc herniation and posterior osteophyte on myelography. Air Force Lieutenant General (Dr.) Paul W. Myers performed an anterior cervical discectomy with placement of iliac bone graft. As a result, Astronaut James Lovell took his place on Apollo 8 flying the uncertain and daring first mission to the moon. This had a cascading effect on the rotation of astronauts, placing Michael Collins on the Apollo 11 flight that first landed men on the moon. It also placed Astronaut James Lovell in a rotation that exposed him to be the Commander of the fateful Apollo 13 flight.

Here, the authors chronicle the history of Astronaut Collins’ anterior cervical surgery and the impact of his procedure on the rotation of astronaut flight selection, and they review the pivotal historic nature of the Apollo 8 spaceflight. The authors further discuss the ongoing issue of cervical disc herniation among astronauts.

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## Impact of insurance precertification on neurosurgery practice and health care delivery

### OBJECTIVE

Insurance preauthorization is used as a third-party tool to reduce health care costs. Given the expansion of managed care, the impact of the insurance preauthorization process in delaying health care delivery warrants investigation through a diversified neurosurgery practice.

### METHODS

Data for 1985 patients were prospectively gathered over a 12-month period from July 1, 2014, until June 30, 2015. Information regarding attending, procedure, procedure type, insurance type, need for insurance approval, number of days for authorization, or insurance denial was obtained. Delay in authorization was defined as any wait period greater than 7 days. Some of the parameters were added retrospectively to enhance this study; hence, the total number of subjects may vary for different variables.

### RESULTS

The most common procedure was back surgery with instrumentation (28%). Most of the patients had commercial insurance (57%) while Medicaid was the least common (1%). Across all neurosurgery procedures, insurance authorization, on average, was delayed 9 days with commercial insurance, 10.7 days with Tricare insurance, 8.5 days with Medicare insurance, 11.5 days with Medicaid, and 14.4 days with workers' compensation. Two percent of all patients were denied insurance preauthorization without any statistical trend or association. Of the 1985 patients, 1045 (52.6%) patients had instrumentation procedures. Independent of insurance type, instrumentation procedures were more likely to have delays in authorization (p = 0.001). Independent of procedure type, patients with Tricare (military) insurance were more likely to have a delay in approval for surgery (p = 0.02). Predictably, Medicare insurance was protective against a delay in surgery (p = 0.001).

### CONCLUSIONS

Choice of insurance provider and instrumentation procedures were independent risk factors for a delay in insurance preauthorization. Neurosurgeons, not just policy makers, must take ownership to analyze, investigate, and interpret these data to deliver the best and most efficient care to our patients.

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## Microsurgical resectability, outcomes, and tumor control in meningiomas occupying the cavernous sinus

### OBJECTIVE

Cavernous sinus meningiomas (CSMs) represent a cohort of challenging skull base tumors. Proper management requires achieving a balance between optimal resection, restoration of cranial nerve (CN) function, and maintaining or improving quality of life. The objective of this study was to assess the pre-, intra-, and postoperative factors related to clinical and neurological outcomes, morbidity, mortality, and tumor control in patients with CSM.

### METHODS

A retrospective review of a single surgeon's experience with microsurgical removal of CSM in 65 patients between January 1996 and August 2013 was done. Sekhar's classification, modified Kobayashi grading, and the Karnofsky Performance Scale were used to define tumor extension, tumor removal, and clinical outcomes, respectively.

### RESULTS

Preoperative CN dysfunction was evident in 64.6% of patients. CN II deficits were most common. The greatest improvement was seen for CN V deficits, whereas CN II and CN IV deficits showed the smallest degree of recovery. Complete resection was achieved in 41.5% of cases and was not significantly associated with functional CN recovery. Internal carotid artery encasement significantly limited the complete microscopic resection of CSM (p < 0.0001). Overall, 18.5% of patients showed symptomatic recurrence after their initial surgery (mean follow-up 60.8 months [range 3–199 months]). The use of adjuvant stereotactic radiosurgery (SRS) after microsurgery independently decreased the recurrence rate (p = 0.009; OR 0.036; 95% CI 0.003–0.430).

### CONCLUSIONS

Modified Kobayashi tumor resection (Grades I–IIIB) was possible in 41.5% of patients. CN recovery and tumor control were independent of extent of tumor removal. The combination of resection and adjuvant SRS can achieve excellent tumor control. Furthermore, the use of adjuvant SRS independently decreases the recurrence rates of CSM.

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## A comparison of lumboperitoneal and ventriculoperitoneal shunting for idiopathic intracranial hypertension: an analysis of economic impact and complications using the Nationwide Inpatient Sample

### Object

Complications following lumboperitoneal (LP) shunting have been reported in 18% to 85% of cases. The need for multiple revision surgeries, development of iatrogenic Chiari malformation, and frequent wound complications have prompted many to abandon this procedure altogether for the treatment of idiopathic benign intracranial hypertension (pseudotumor cerebri), in favor of ventriculoperitoneal (VP) shunting. A direct comparison of the complication rates and health care charges between first-choice LP versus VP shunting is presented.

### Methods

The Nationwide Inpatient Sample database was queried for all patients with the diagnosis of benign intracranial hypertension (International Classification of Diseases, Ninth Revision, code 348.2) from 2005 to 2009. These data were stratified by operative intervention, with demographic and hospitalization charge data generated for each.

### Results

A weighted sample of 4480 patients was identified as having the diagnosis of idiopathic intracranial hypertension (IIH), with 2505 undergoing first-time VP shunt placement and 1754 undergoing initial LP shunt placement. Revision surgery occurred in 3.9% of admissions (n = 98) for VP shunts and in 7.0% of admissions (n = 123) for LP shunts (p < 0.0001). Ventriculoperitoneal shunts were placed at teaching institutions in 83.8% of cases, compared with only 77.3% of first-time LP shunts (p < 0.0001). Mean hospital length of stay (LOS) significantly differed between primary VP (3 days) and primary LP shunt procedures (4 days, p < 0.0001). The summed charges for the revisions of 92 VP shunts ($3,453,956) and those of the 6 VP shunt removals ($272,484) totaled $3,726,352 over 5 years for the study population. The summed charges for revision of 70 LP shunts ($2,229,430) and those of the 53 LP shunt removals ($3,125,569) totaled$5,408,679 over 5 years for the study population.

### Conclusions

The presented results appear to call into question the selection of LP shunt placement as primary treatment for IIH, as this procedure is associated with a significantly greater likelihood of need for shunt revision, increased LOS, and greater overall charges to the health care system.

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## Perioperative complications in patients undergoing open transforaminal lumbar interbody fusion as a revision surgery

### Object

Transforaminal lumbar interbody fusion (TLIF) has been increasingly used to treat degenerative spine disease, including that in patients in whom earlier decompressive procedures have failed. Reexploration in these cases is always challenging and is thought to pose a higher risk of complications. To the best of the authors' knowledge, there are no current studies specifically analyzing the effects of previous lumbar decompressive surgeries on the complication rates of open TLIF.

### Methods

The authors performed a retrospective study of surgeries performed by a single surgeon. A total of 187 consecutive patients, in whom the senior author (A.N.) had performed open TLIF between January 2007 and January 2011, met the inclusion criteria. The patients were divided into two groups (primary and revision TLIF) for the comparison of perioperative complications.

### Results

Overall, the average age of the patients was 49.7 years (range 18–80 years). Of the 187 patients, 73 patients had no history of lumbar surgery and 114 were undergoing revision surgery. Fifty-four patients (28.9%) had a documented complication intraoperatively or postoperatively. There was no difference in the rate on perioperative complications between the two groups (overall, medical, wound related, inadvertent dural tears [DTs], or neural injury). Patients who had undergone more than one previous lumbar surgery were, however, more likely to have suffered from DTs (p = 0.054) and neural injuries (p = 0.007) compared with the rest.

### Conclusions

In the hands of an experienced surgeon, revision open TLIF does not necessarily increase the risk of perioperative complications compared with primary TLIF. Two or more previous lumbar decompressive procedures, however, increase the risk of inadvertent DTs and neural injury.

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## Do cystic vestibular schwannomas have worse surgical outcomes? Systematic analysis of the literature

### Object

The goal of this study was to perform a systematic quantitative comparison of the surgical outcomes between cystic vestibular schwannomas (CVSs) and solid vestibular schwannomas (SVSs).

### Methods

A review of English-language literature published between 1990 and 2011 was performed using various search engines including PubMed, Google Scholar, and the Cochrane database. Only studies that reported surgical results of CVSs in comparison with SVSs were included in the analysis. The primary end point of this study was surgical outcomes, defined by the following: 1) facial nerve outcomes at latest follow-up; 2) mortality rates; or 3) non–facial nerve complication index. Secondary end points included extent of resection and brainstem adherence.

### Results

Nine studies comprising 428 CVSs and 1287 SVSs were included in the study. The mean age of patients undergoing surgery was 48.3 ± 6.75 and 47.1 ± 9 years for CVSs and SVSs, respectively (p = 0.8). The mean tumor diameter for CVSs was 3.9 ± 0.84 cm and that for SVSs was 3.7 ± 1.2 cm (p = 0.7). There was no significant difference in the extent of resection among CVSs and SVSs (81.2% vs 80.7%, p = 0.87) Facial nerve outcomes were significantly better in the cohort of patients with SVSs than in those with CVSs (52.1% vs 39%, p = 0.0001). The perioperative mortality rates for CVSs and SVSs were not significantly different (3% and 3.8%, respectively; p = 0.6). No significant difference was noted between the cumulative non–facial nerve complication rate (including mortality) among patients with CVSs and SVSs (24.5% and 25.6%, respectively; p = 0.75)

### Conclusions

Facial nerve outcomes are worse in patients undergoing resection for CVSs than in patients undergoing resection for SVSs. There were no significant differences in the extent of resection or postoperative morbidity and mortality rates between the cohorts of patients with vestibular schwannomas.

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## The impact of comorbidities, regional trends, and hospital factors on discharge dispositions and hospital costs after acoustic neuroma microsurgery: a United States nationwide inpatient data sample study (2005–2009)

### Object

Hospitalization cost and patient outcome after acoustic neuroma surgery depend on several factors. There is a paucity of data regarding the relationship between demographic features such as age, sex, race, insurance status, and patient outcome. Apart from demographic factors, there are several hospital-related factors and regional issues that can affect outcomes and hospital costs. To the authors' knowledge, no study has investigated the issue of regional disparity across the country in terms of cost of hospitalization and discharge disposition.

### Methods

The authors analyzed the Nationwide Inpatient Sample (NIS) database over the years 2005–2009. Several variables were analyzed from the database, including patient demographics, comorbidities, and surgical complications. Hospital variables, such as bedsize, rural/urban location, teaching status, federal or private ownership, and the region, were also examined. Patient outcome and increased hospitalization costs were the dependent variables studied.

### Results

A total of 2589 admissions from 242 hospitals were analyzed from the NIS data over the years 2005–2009. The mean age was 48.99 ± 13.861 years (± SD), and 304 (11.7%) of the patients were older than 65 years. The cumulative cost incurred by the hospitals from 2005 to 2009 was $948.77 million. The mean expenditure per admission was$76,365.09 ± $58,039.93. The mean total charges per admission rose from$59,633.00 in 2005 to \$97,370.00 in 2009. The factors that predicted most significantly with other than routine (OTR) disposition outcome were age older than 65 years (OR 2.22, 95% CI 1.411–3.518; p < 0.001), aspiration pneumonia (OR 16.085, 95% CI 4.974–52.016; p < 0.001), and meningitis (OR 11.299, 95% CI 3.126–40.840; p < 0.001). When compared with patients with Medicare and Medicaid, patients with private insurance had a protective effect against OTR disposition outcome. Higher comorbidities predicted independently for OTR disposition outcome (OR 1.409, 95% CI 1.072–1.852; p = 0.014). The West region predicted negatively for OTR disposition outcome. Large hospitals were independently associated with higher hospital charges (OR 4.269, 95% CI 3.106–5.867; p < 0.001). The West region had significantly higher (p < 0.001) mean hospital charges than the other regions. Patient factors such as meningitis and aspiration pneumonia were strong independent predictors of increased hospital charges (p < 0.001). Higher comorbidities (OR 1.297, 95% CI 1.036–1.624; p = 0.023) and presence of neurofibromatosis Type 2 (OR 2.341, 95% CI 1.479–3.707; p < 0.001) were associated with higher hospital charges.

### Conclusions

The authors' study shows that several factors can affect patient outcome and hospital charges for patients who have undergone acoustic neuroma surgery. Factors such as younger age, higher ZIP code income, less comorbidity, private insurance, elective surgery, and the West region predicted for better disposition outcome. However, the West region, higher comorbidities, and weekend admissions were associated with higher hospitalization costs.