MANEJO INTEGRADO DEL CULTIVO DE
ENFERMEDADES CAUSADAS POR BACTERIAS
Formation of hexagonal patterns as a result of convection is an example of a self- organised pattern formed as a result of physical processes. The interplay between reactivity and molecular motion driven by a concentration gradient (i.e. diffusion) can allow a plethora of analogous phenomena, such as spontaneous stationary patterns and propagating reaction-diffusion fronts to emerge in chemical209–211 and biolog-
ical212–215 systems alike. As early as 1896, Leisegang observed216 that mixing of inorganic salts—silver nitrate and potassium dichromate—results in the production of
Chapter 5 periodic bands in a layer of gelatinous media. Whilst the striped pattern (Figure 5.2) could not be explained by Liesegang at the time, the phenomena of spatial structure formation in precipitation reactions has now been established217 to arise as a result of the interplay between reaction, diffusion and precipitation. In this particular example, concentric rings are produced when salts react to form insoluble silver dichromate.
or gas phase22and have been found on a micrometer scale in
water-swollen polymer films.23 Certain geological formations
that display spatial rhythm may have formed by the diffusion of salts through a gellike matrix (cf. Chapter 13 in ref 1f), and some biological or biomedical structures are likely to be caused by a similar mechanism (e.g., gall stones,24urinary calculi, or
mucus around cancer cells).
In most systematic experiments, the two reactants are placed in a narrow test tube, which confines the direction of diffusion to essentially one spatial coordinate, resulting in a 1D structure (Figure 1A). The frequently observed regular and reproducible bands of deposited precipitate have been described in terms of a simple spacing and time law20,25that relates the location x
n of thenth band to its time of formationtnaccording to xn ≈
tn1/2and describes successive locations by the quotientxn/xn+1
)p, withpdenoting a constant spacing coefficient. The width of the bands also increases with increasing distance from the interface,26 and a corresponding power law has been formu-
lated.27
There are, however, numerous reports on different pattern characteristics: these include the revert spacing reported in 192828and further investigated in refs 3 and 29. Here the spacing
decreases with increasing distance from the junction of the solution. Some studies have shown secondary structures, such
as the separation of one band, once formed, into several closely adjacent thinner bands.30,31Quite frequently, under apparently
the same experimental conditions, “curiosities”32are observed
such as radially aligned gaps31or zigzag-shaped dislocations
of the ring structure in a Petri dish,33spirals or helices instead
of a set of rings or bands,34,35or “Saturn rings” in a test tube.6
A remarkable variety of structures have been reported recently in ref 36 with a phase diagram showing transitions between regular and irregular shapes of precipitation bands; the axes of this diagram are the initial concentrations of the interdiffusing reagents.
Macroscopic periodic structures may also arise from an electrolyte solution in the absence of any fields such as gradients of concentration or temperature or the gravitational field. For instance, if a solution of lead iodide is prepared at elevated temperatures, a gel-forming substance is added to the solution, and the solution is allowed to cool slowly; then a uniform nucleation of colloidal lead iodide is induced at sufficient supercooling, and precipitation occurs inhomogeneously in irregular patterns with length scales ranging, in general, from 0.5 to 10 mm.10,11,37In ref 38, examples are reported that can
be roughly characterized as a 2D network, a wavelike structure reminiscent of Liesegang rings, and a speckled pattern.
To explore the full range of structure formation from initially large concentration gradients (the classical Liesegang case) to low and vanishing initial gradients, we have made a systematic study of Liesegang ring formation and the dependence of their number, locations, and widths on∆andS.26,38For definitions
of S and∆ see Section II. The range ofS investigated goes from 17 to 1013, and that of|∆|from 0 to 7 M. As either∆is
decreased to zero or S reaches a lower critical limit, band formation becomes increasingly stochastic in the formation and placement of the rings. There is a continuity in structure formation from zero to increasing concentration gradients that has led to the suggestion that the origin of pattern formation is the same in all cases.12
During the past decade, a number of authors have studied the effects of externally imposed fields on the onset and evolution of precipitation bands. For this purpose, electric fields39and light40have been used. Other recent work focuses
on details of the morphological characteristics using microden- sitometric and microscopic techniques to elucidate the properties of a moving nucleation front that establishes a subsequent turbidity zone in which rings of silver chromate/dichromate subsequently form.41,42
Many theories have been proposed for a mechanistic explana- tion of patterned precipitation. Stern summarizes in his review43
of the Liesegang phenomenon the theoretical approaches developed until 1954, and Henisch presents a thorough overview of publications that have appeared on that issue until 1991.1e
In addition to the theories based on Ostwald’s supersaturation hypothesis mentioned above (OP model4) and a number of
approaches that turned out not to be supported by experimental observations, an early suggestion for a postnucleation scenario was made in the “coagulation theory”,44 which assumes that
precipitation bands arise by coagulation (or flocculation) of the colloid if certain critical electrolyte concentrations are exceeded. This theory alone cannot convincingly explain the creation of large clear regions between bands on long time scales. From the early 1970s onward, the role of competitive and autocatalytic particle growth in structure formation in the postnucleation phase involving instabilities was discussed in the literature (TI model4)
to explain the rich morphologies observed through the years in preciptation patterns in systems with and without macroscopic
Figure 1. (A) Parallel bands of precipitated lead iodide in agar gel (1%) with initially 240 mM KI in the upper and 9 mM Pb(NO3)2in
the lower portion of a tube. Scale bar: 1 cm. (B) Ring-shaped periodic precipitation of silver dichromate in a thin layer of gelatin gel, as reported by R. E. Liesegang in one of his earliest photographs taken in 1896 (from ref 17).
Figure 5.2 Typical Liesegang rings formed as a result of addition of a small drop of silver nitrate to a gel media containing potassium dichromate. Interplay of reaction and diffusion gives rise to periodic striped bands of insoluble silver dichromate. Figure reprinted with permission from Ref. 217. Copyright 2003 American Chemical Society.
In 1952, Turing proposed218a mathematical reaction-diffusion model, explaining the spontaneous evolution of spatially heterogeneous patterns in systems comprised of components that are both reacting and diffusing. Turing showed that a theoretical chemical system, formed by an activator and an inhibitor, though initially exhibiting no spatiotemporal phenomena, can evolve towards instability and ultimately, towards formation of spatial patterns. In the system described by Turing, the emergence of patterns was dependent on the diffusion of the inhibitor being greater than that of the activator. Turing suggested that in a scenario where such a reaction-diffusion system was built of the so-called morphogensa, cells could recognise the pre-pattern of morphogens
formed and would respond in a manner that leads to formation of spatial structure within a tissue. Turing reasoned that this process underlies the chemical basis of morphogenesis and is the mechanism behind many of the patterns observed in the animal kingdom. At the time of its conception, Turing’s theory was received with scepticism—there was little evidence for the existence of morphogenic compounds. Nevertheless, since then, theoretical simulation studies employing Turing’s reaction-diffusion model have been shown219 to replicate the patterns observed in nature extremely well (Figure 5.3).
One of the oldest and the most controversial examples of spatiotemporal phenomena reported is known as the Belousov-Zhabotinsky (BZ) oscillating reaction. In 1958, Belousov, a Soviet chemist, was investigating a reaction mixture containing cerium as a catalyst for the oxidation of citric acid by bromate that was meant to mimic
aMorphogen is a signalling molecule that diffuses from the tissue of its origin within an organism and its concentration gradient affects cells exposed to it.
Chapter 5
A
B
C
Initial condition
Six stable states
Case VI (Turing pattern) Case V
Uniform, stationary
Oscillatory cases with extremely short
wavelength
Oscillatory cases with finite wavelength
Stationary waves with finite wavelength (Turing pattern) Uniform, oscillating Stationary waves with
extremely short wavelength Both morphogens
diffuse and react with each other
I II III
IV V VI
Fig. 2.Schematic drawing showing the mathematical analysis of the RD system and the patterns generated by the simulation. (A) Six stable states toward which the two-factor RD system can converge. (B) Two-dimensional patterns generated by the Turing model. These patterns were made by an identical equation with slightly different parameter values. These simulations were calculated by the software provided as supporting online material. (C)
Reproduction of biological patterns created by modified RD mechanisms. With modification, the RD mechanism can generate more complex patterns such as those seen in the real organism. Simulation images are courtesy of H. Meinhardt [sea shell pattern (5)] and A. R. Sandersen [fish pattern (13)]. Photos of actual seashells are from Bishougai-HP (http://shell.kwansei.ac.jp/~shell/). Images of popper fish are courtesy of Massimo Boyer (www.edge-of-reef.com).
24 SEPTEMBER 2010 VOL 329 SCIENCE www.sciencemag.org
1618 A
B
C
Initial condition
Six stable states
Case VI (Turing pattern) Case V
Uniform, stationary
Oscillatory cases with extremely short
wavelength
Oscillatory cases with finite wavelength
Stationary waves with finite wavelength (Turing pattern) Uniform, oscillating Stationary waves with
extremely short wavelength Both morphogens
diffuse and react with each other
I II III
IV V VI
Fig. 2.Schematic drawing showing the mathematical analysis of the RD system and the patterns generated by the simulation. (A) Six stable states toward which the two-factor RD system can converge. (B) Two-dimensional patterns generated by the Turing model. These patterns were made by an identical equation with slightly different parameter values. These simulations were calculated by the software provided as supporting online material. (C)
Reproduction of biological patterns created by modified RD mechanisms. With modification, the RD mechanism can generate more complex patterns such as those seen in the real organism. Simulation images are courtesy of H. Meinhardt [sea shell pattern (5)] and A. R. Sandersen [fish pattern (13)]. Photos of actual seashells are from Bishougai-HP (http://shell.kwansei.ac.jp/~shell/). Images of popper fish are courtesy of Massimo Boyer (www.edge-of-reef.com).
24 SEPTEMBER 2010 VOL 329 SCIENCE www.sciencemag.org
1618 REVIEW
(a) (b)
Figure 5.3 Turing pattern observed in a(a)seashell and in(b)a fish. Patterns shown on the left-hand side are those observed in real organisms, whereas images on the right-hand side illustrate patterns generated through a simulation of a reaction-diffusion system. Figure adapted from Ref. 219. Reprinted with permission from AAAS.
features of glycolysis. Intrestingly, Belousov found220 that the mixture of reactants he
was investigating kept periodically changing colour from clear to yellow. The results reported by Belousov were widely rejected by his contemporaries, on the grounds that the findings contradict the second law of thermodynamics—a law stating that the total entropy in the universe, or in simplified terms the total disorder, must increase over time. Belousov’s discovery gained acceptance only after Zhabotinsky demonstrated221,222the oscillating nature of the reaction by employing malonic acid instead of citric acid, which improved the visualisation dramatically—changing periodically between red and blue (Figure 5.4). A crucial development in the understanding of the non-linear dynamic behaviour underlying the BZ system came with the proposal223,224of a mechanism for
the BZ reaction by Field, K¨or¨os and Noyes.
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doi: 10.1387/ijdb.072484vv
V. K. Vanag, I. R. Epstein, Int. J. Dev. Biol. 2009, 53, 673–681. Fig 1.
Figure 5.4 A series of images illustrating the spatiotemporal patterns exhibited by a Belousov- Zhabotinsky oscillating system over time. Figure taken from Ref. 210
Numerous examples209,225–229 of oscillations and wave phenomena have been
demonstrated in inorganic systems since the first report of the BZ system. Reports of organic systems, exhibiting such behaviour, however, are much more scarce—the literature reveals a single example of a propagating reaction-diffusion front based on small organic molecule, observed230 in the autoxidation of benzaldehyde. The lack of
examples exploiting autocatalysis based on organic systems is in stark contrast to the numerous examples of spatiotemporal dynamic phenomena, ranging from the black
and white stripes on a zebra, to BZ-like waves formed231–233 by the Dyctiostelium
discoideum(Figure 5.5), present in nature.
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to copyright restrictions.
J. J. Tyson, J. D. Murray, Development1989, 106, 421–
426. Fig 1.
Figure 5.5 Oscillating waves formed by aggregating cells ofDyctiostelium discoideumin a starvation state as a result of low cAMP concentration. Figure adapted from Ref. 231
Cells of this particular species of slime mould form spiral patterns in a response to starvation-induced chemical wave of cyclic adenosine 3,5-monophosphate (cAMP) propagating through aDyctiostelium discoideumcolony. The oscillating pattern stems from cAMP binding to the cell receptors, resulting in their transient desensitisation, producing a travelling wave that is strictly one-directional in character.
More recently, several examples of reaction-diffusion fronts exploiting RNA234,235
and DNA236,237have been reported. In 1989, McCaskill and co-workers reported234,235
the first example of a reaction-diffusion front observed in a system based on replicating ribonucleic acid (RNA). The propagating RNA fronts were initiated by addition of preformed molecules of RNA at a particular location in an essentially two-dimensional capillary tube reactor, containing a solution of RNA polymerase, nucleotide building blocks and buffer (Figure 5.6). Interestingly, McCaskill and co-workers were able to demonstrate that the fronts can emerge stochastically and are capable of evolving as they progress in space over time.
Distance
T
ime
Figure 5.6 A plot showing the distance travelled over time by a spontaneous propagating RNA wave front initiated by addition of two single RNA molecules (white dots). The white arrow indicates a change in the velocity of the RNA wave. Concentration of RNA is represented using a colour scale, where black denotes the lowest and orange the highest concentration. Figure adapted with permission from Ref. 235. Copyright 1993 National Academy of Sciences, USA.
Until 2013, this study was the only example of its kind. Rondelez and co-workers reported236an example of travelling concentration waves (Figure 5.7a) in a biochemical
network exhibiting predator-prey type of oscillations. The network presented in this study is an extension of their previous work238on DNA-based predator-prey systems, employing carefully designed DNA oligonucleotide-based molecules (Figure 5.7c), connected by a shared encoding sequence. Namely, the predator-prey network was constructed from three components: preyN, predator P, and grassG—the template required for growth of the prey (Figure 5.7bandFigure 5.7c).
Ti m e Prey Predator (c) (a) N (b) G 2 N N + P 2 P N, P (1) (2) (3) Predator
Grass (G) Prey (N) Predator (P)
Spatio-temporal waves Reaction
network
Figure 5.7 (a)A biochemical network exhibiting predator-prey type of oscillations, connected by a com- mon DNA oligonucleotide sequence capable of forming a propagating reaction-diffusion front.(b)Three reactions at the core of the predator-prey network: (1) autocatalytic growth of prey on the grass template, (2) autocatalytic growth of predator, with a consumption of prey, and (3) decay of predator/prey.(c)Structure of grass (G), prey (N), and predator (P). Complementary DNA sequences are highlighted in the same colour, whereas dark and light shade represent regions that can and cannot be destroyed through the action of an exonuclease. Figure adapted with permission from Ref. 236. Copyright 2013 American Chemical Society.
The reactivity within this network is controlled by three purified enzymes: a poly- meraseb, a nicking enzymec and an exonucleased—in the absence of these three en-
zymes, the replication reactions in the network would stall. The prey utilises the grass for its formation, and in turn, the predator consumes the prey component in order to form another molecule of itself. Both predator and prey can decay through the action of the exonuclease enzyme. By examining the molecular network in reaction-diffusion media—within the environment of an unstirred, 8 mm wide and 200 µm deep circular reactor, Rondelezet al. were able to demonstrate DNA-based travelling prey-predator reaction-diffusion fronts. Recently, this work has been elaborated237 to a more gen-
bPolymerase is a general term used to denote an enzyme that synthesises a DNA strand using nucleotides as building blocks.
cA nicking enzyme produces a break in one strand of a double-stranded DNA, leading to its unwinding. dAn exonuclease cleaves (i.e. removes through hydrolysis) one nucleotide at a time from a polynu- cleotide chain.
eral method for achieving control over the reaction and diffusion parameters of DNA components employed in programmable reaction-diffusion networks.