Dynamics of self-propagating fronts of motile bacteria


Alison Kraigsley and Paul D. Ronney

Department of Aerospace and Mechanical Engineering

University of Southern California, Los Angeles, CA 90089-1453


Steven E. Finkel

Department of Biological Sciences

University of Southern California, Los Angeles, CA 90089-1340



      While much is known about self-propagating reaction-diffusion fronts that occur in many chemically reacting systems such as flames, polymerization processes and some aqueous reactions, this vast knowledge base has not previously been systematically applied to biological systems such as spreading motile bacteria.  The goals of this work are (1) to determine if such knowledge can be applied to biological systems and (2) if so, obtain a more quantitative and predictive understanding of biological systems that exhibit self-propagating fronts.  The analogy between spreading motile bacteria and other self-propagating fronts will be pursued with respect to the dynamical properties of such fronts.  Initial experiments using the E. coli bacterium do indeed show behavior analogous to reaction-diffusion systems.  We are attempting to conduct a comprehensive study of dynamical properties including propagation rates, front curvature effects, quenching limits, stability limits, buoyancy effects, effective diffusion coefficients and reproduction time scale.  This information will be used as inputs to existing analytical/numerical models of reaction-diffusion fronts and the validity of the proposed analogies determined.  The differences between conventional reaction-diffusion systems and microbiological systems will also be evaluated.  One key difference is that the bacteria evolve and respond to stress, a factor that will be studied by repeating the above testing with bacteria that have survived near-quenching conditions.  These data will then be used to obtain a better understanding of the survivability and adaptation of bacteria under adverse conditions.



      Self-propagating reaction fronts occur in many chemical and physical systems including flames, free-radical initiated polymerization processes and some aqueous reactions.  All of these systems are characterized by two key features: a reactive medium (for example a fuel-air mixture in the case of flames) and an autocatalyst that is a product of the reaction which also accelerates the reaction (for example thermal energy in the case of flames).  Self-propagation occurs when the autocatalyst diffuses into the reactive medium, initiating reaction and creating more autocatalyst.  This enables reaction-diffusion fronts to propagate at steady rates far from any initiation site.

            Two of the key characteristics of self-propagating reaction-diffusion fronts are the propagation rate (s) and the quenching limit.  The propagation rate is the speed at which the front advances into the reactive medium, which generally depends on the overall reaction time scale (t) and the diffusion coefficient (D) of the autocatalyst according to the relation s ≈ (D/t)1/2.  t in turn depends on the concentration and activity of the reactants.  The quenching limit is the minimum or maximum value of some parameter (for example, minimum reactant concentration or channel width) for which a steadily propagating front can exist.  The coupling that occurs between chemical reactions and diffusive and convective transport in flames [[1]], polymerization fronts [[2]] and aqueous reactions [[3]] is well understood quantitatively.

      Surprisingly, the analogous line of inquiry has not been applied to microbiological systems, even though the very first modeling of reactive-diffusive fronts, leading to the relation s ≈ (D/t)1/2, was conducted in 1937 in the context of the spread of infectious diseases (see [[4]]).  The purpose of our investigation is to assess the feasibility of doing so and from this develop a more quantitative understanding of such microbiological systems.  Most microbiological studies focus on individual bacterium and counting the behavior of individuals.  In this study we apply thermodynamic laws and study the aggregate behavior of a large number of individuals.  The test case we choose to focus on for this study is the very common and widely studied Escherichia coli bacterium, a motile bacterium that (like many others) swims using its tentacles or flagella in its nutrient media while frequently changing direction to seek regions of higher nutrient concentration.  Specifically the bacterium has two modes of behavior: “run” mode in which its flagella rotate to propel it in a more or less straight line, and “tumble” mode where the flagella cause it change orientation with little net motion [[5]].  The resulting motion is somewhat analogous to the random walk of molecules that leads to classical Fickian diffusion (although the mechanism is entirely different, since, unlike molecules the bacterium motion is not related to elastic collisions between bacterium).  Also, in a favorable nutrient gradient, the bacterium runs more than it tumbles, resulting in higher D, whereas in unfavorable gradients, it tumbles more, allowing it to seek out new directions and resulting in lower D; this also leads to a biased random walk that favors migration to regions of increased nutrient levels.

      These characteristics of motile bacteria can be exploited to model their behavior as reaction-diffusion fronts. The propagation speed (s) of E. coli we measured (see Preliminary Results section) in a 0.3% agar medium is about 4.5 mm/hr.  The reproduction time scale (t) of E. coli is about 20 min.  Since the propagation speed s ≈ (D/t)1/2, D ≈ s2t, thus D ≈ 1.5 x 10-5 cm2/sec.  This value is consistent with the value expected based on the kinetic theory of molecules which shows that D is proportional to a particle speed (c) multiplied by the particle mean free path (l), i.e., in the case of molecules the distance the particles move before colliding with each other, or, in the case of motile bacteria, changing direction.  l can be estimated as c multiplied by the time (t) the bacteria swim without changing direction.  For E. coli, an average swimming speed of 21 µm/s, with 21 changes in direction per 30 seconds has been measured.  This implies ≈ 1.4 s and l ≈ 3.0 x 10-3 cm and, thus the effective diffusivity D ≈ 6.3 x 10-6 cm2/s, which is of the same order of magnitude as the value of D inferred from D ≈ s2t. 

      We conclude that it is reasonable to characterize the spread of motile bacteria as reaction-diffusion fronts.  Many microbiological studies measure the response of the system to a gradient in temperature or nutrients that is imposed on the system; we intend to show that the bacteria can generate their own gradients and thus can be self-propagating.  Table 1 shows the proposed analogy between flames (a typical reaction-diffusion front) and microbiological systems.


Flame or molecular property

Microbiological equivalent


Concentration of bacteria



Heat diffusivity ≈ cl

Diffusivity of bacteria

Fuel diffusivity

Diffusivity of nutrient

Sound speed (c)

Swimming speed of bacterium in "run" mode

Mean free path (l)

c multipled by average time to switch from run mode to tumble mode and back

Reaction timescale (t)

Reproduction time

Heat loss

Death (of individual bacterium)


Death (of all bacteria)


Table 1.  Proposed analogy between flames and microbiological fronts


A complicating factor in flames as well as the motile bacteria is that the effective diffusion coefficient changes with product concentration.  In the case of flames, the production of products leads to higher temperature that increases D in nearly all cases.  In the case of motile bacteria, it is well known that the bacteria spend less time in the "run" mode and more in the "tumble" mode when the nutrient concentration is high (why look around when the grass is already green?)  This means that the mean free path l will be lower and thus the effective diffusivity will be lower when the nutrient concentration is high.  In the case of a propagating front, the few bacteria at the leading edge of the front see high nutrient concentrations and thus have low D.  The higher concentrations of bacteria (the "product" of reaction) at the trailing edge of front will exist in a medium with lower nutrient concentration and thus will "run" more and "tumble" less, increasing l and thus D will increase.  Thus the analogy with flames is reasonably broad.


Preliminary results

      The above discussion encouraged us to conduct preliminary experiments to test the viability of our propositions.  These experiments were conducted at 37˚C in standard petri plates using a water-based nutrient medium consisting of (unless otherwise noted) 1% NaCl, 1% tryptone, 0.5% yeast and 0.1% agar.  The agar increases the viscosity of the medium and essentially prevents any bulk motion.  The medium was inoculated with bacteria at a point (usually at the center of the dish) and the resulting advance (if any) of the bacteria front was observed visually.  Several strains of E. coli were tested.  After an initial transient, all exhibited a linear increase of front radius with time.  An example is shown in Fig. 1.

      Figure 1 shows that the front advances more slowly initially when the front radius is small and front curvature high.  All other tests showed this same trend.  This behavior is observed in chemical fronts in which the autocatalytic product has a lower diffusivity than that of the reactant (i.e., in the case of flames, a high thermal diffusivity and a low fuel diffusivity) and the reasons are well understood [[6]].  While the diffusivity of the tryptone nutrient is unknown, by analogy with similar molecules a value of 10-7 cm2/s can be estimated, which is lower than the apparent diffusivity of E. coli as estimated above. Consequently, the behavior shown in Fig. 1 is consistent with the reaction-diffusion model of front propagation. Furthermore, flame theory shows that if the ratio of product to reactant diffusivities is too different from the value 1, either higher or lower, various types of front instabilities may result. Pattern formation in microbiological systems is well known [[7]], but reaction-diffusion theory has not been used to quantify and predict such patterns.  We will investigate such instabilities by varying the medium diffusivity through the agar concentration.  Moreover, we suspect that this choice of the run-tumble timing of E. coli is not accidental because it affects D; if D is too low or too high, instabilities result that may be unfavorable for survival.  We will pursue this notion by examining the effect of agar concentration on the speed and run-tumble timing of the bacteria under a microscope and determine if the bacterium adapts (either on the fly, or by evolutionary processes) its speed and run-tumble timing to modify its effective D to maintain stable fronts.



Figure 1.  Example experimental data on bacterial front radii as a function of time.



Figure 2. Measured effect of agar and solution depth on steady propagation speeds of bacterial fronts.



      The effect of agar concentration and the depth of the nutrient medium on the steady values of the propagation rate s is shown in Fig. 2.  At 0.5% or greater agar concentration (not shown), the medium was essentially solid and the bacteria spread only on the surface of the medium, not in the bulk.  Diffusion-like behavior of the bacteria is unlikely in this case since they cannot swim.  At 0.4% and lower agar concentration, the bacteria spread through the medium.  The highest propagation rates (s) were observed at the lowest agar concentrations (and thus lowest viscosity of the medium).  This is expected since at lower viscosity, the bacterium can swim faster and thus should have higher effective D, leading to higher s (since s is proportional to D1/2 as discussed above).

The depth of the solution was found to have almost no effect on s.  By taking cross-sections of the medium after partial propagation of the front, it was found that the fronts are mostly uniform in the vertical dimension.  Thus buoyancy flow is not affecting these fronts, otherwise an effect of the solution depth on s, or non-uniformity in the vertical direction, would be expected.  Since the tests were conducted in air, these observations also indicate that oxygen diffusion into the medium is not significant, otherwise growth near the surface of the nutrient solution would be different from that below the surface.  This last result shows that the bacteria are growing anaerobically.

      To determine whether "quenching" limits exist in bacterial fronts, an experiment was conducted in which the bacterial fronts attempted to propagate through narrow and wide channels with antibiotic walls.  Since antibiotics remove the autocatalytic product (the bacteria), the effect of the walls is analogous to heat losses to cold walls in flames.  For front propagation in narrow channels, it is well known that a quenching limit occurs due to heat losses to the channel walls when the Peclet number Pe = sw/D, where w is the channel width and D is the diffusivity, is less than a critical value.  Thus, in the presence of heat losses, fronts can propagate through wide channels (high Pe) but not narrow channels (low Pe).  Figure 3 shows that fronts can propagate through the wide channel but not the narrow channel, indicating a quenching limit.  The applicability of the quenching relationship Pe = sW/D = constant at the limit will be tested in this work.

The effect of tryptone nutrient concentration was also tested; it was found that at half the nominal value no spread would occur, whereas s was almost unchanged by doubling the tryptone concentration from its nominal value.  This may indicate a quenching limit even in the absence of antibiotic walls.  Analogous behavior occurs in flames due to radiative heat losses when the fuel concentration is too low.  We will investigate this wall-free limit and attempt to determine the loss mechanism responsible for suppressing front propagation at low nutrient levels.

      Titering of the medium to obtain bacteria counts consistently showed a significant peak in the bacteria concentration at the leading front, followed by a major drop behind the front and, for later times, a second peak far behind the first one.  This is consistent with the images shown in Fig. 3.  In future work we will use titering analysis to determine if the front thickness ≈ D/s as predicted theoretically.  Based on visual inspection of Fig. 3 the front thickness is about 1 mm, which is close to the prediction D/s = (1.5 x 10-3 mm2/sec)/(4.5 mm/hr) = 1.2 mm.  Results with a wild strain of bacteria showed a slightly faster initial transient and a shorter time and distance to reach steady a steady propagation rate as compared to genetically marked strains.




Figure 3.  Photographs of fronts of motile E. coli bacteria propagating through narrow (6 mm, left) and wide (35 mm, right) channels with Kanamycin antibiotic side walls (100 microliters of Kanamycin per side), taken 6.5 hours after inoculation.   Both cases:  2086 wild strain of E. coli, 0.1% agar, standard nutrient medium.


Current research program

            Based on the apparent success of the reaction-diffusion model in describing in these preliminary experimental results, we are pursuing the following research program:

  1. Determine the propagation rates of E. coli fronts as a function of nutrient concentration, medium motility (through changing the agar concentration), ambient temperature, ambient oxygen concentration (aerobic vs. anaerobic environment), and the thickness of the layer of nutrient medium.  These tests will be performed in the same manner as the preliminary experiments described above.  Additionally, to improve visualization of the fronts, we will use E. coli specific fluorescent dyes from Molecular Probes, Inc (Eugene, OR). to improve visualization of the fronts.   An ultraviolet light source (purchased) or Ar-ion laser will be used to excite the fluorescent dyes.
  2. Determine D based on the relation D ≈ s2t and compare with that expected based on the relation D = c2t, where c is the measured (using video microscopy) speed (c) and t mean time to change direction.  Standard video is adequate since the run-tumble time scale is typically 1 sec and the video framing rate is 30 per sec.
  3. Determine the quenching limits and limit Peclet numbers by allowing the fronts to propagate through channels of varying width (w) having antibiotic walls (Kanamycin or similar) and determine if Pe is constant at the limit, i.e., is the propagation rate (s) at the limit proportional to 1/w?
  4. Determine the quenching limit in the absence of walls (by decreasing nutrient concentration, for varying agar concentrations) and ascertain the loss mechanism (i.e. cell death mechanism) responsible for this limit.
  5. Determine the stability limits as a function of the ratio of product (bacterium) to reactant (nutrient) diffusivity by varying the agar concentration in the nutrient medium and nutrient concentration.  The fluorescent dyes will facilitate imaging of non-uniform fronts and pattern formation.
  6. Determine if a buoyancy effect on front propagation exists.  Since E. coli bacteria have a density slightly different from water, at low viscosity a buoyancy effect will occur.  Most investigators use a high-viscosity agar medium to prevent any fluid flow, but many real E. coli environments have water-like viscosity.  Buoyancy effects will be examined by testing front propagation in agar-free media in a vertical parallel plate apparatus (called a "Hele-Shaw" cell in fluid mechanics literature) and the front pattern and propagation rate will be observed.
  7. Model these results using our existing analytical/numerical models of reaction-diffusion fronts.


      Of course, there are also some differences between conventional reaction-diffusion systems and microbiological systems.  One key difference is that the bacteria evolve and respond to stress, a factor that will be studied by repeating the above testing with "experienced" bacteria that have survived near-quenching conditions.  Another difference, as discussed above, is the biased diffusion properties. This bias can be modeled using a nonlinear diffusion relation to replace Fick's Law.


Practical importance

            This work is a study of the dynamics of a new class of propagating fronts involving motile bacteria.  The study of front dynamics has many applications as discussed above.  These studies, however, are of particular interest to microbiologists because it relates to the mechanisms of long-term survival and evolution used by bacteria.  One of the primary factors influencing the rates of evolutionary change in these bacterial systems are the levels of environmental stress, as well as the transition from one environment to another.  The culture systems we are developing represent new environments in which these questions of evolution and survival have not been addressed.  This work provides the first attempt, to our knowledge, to characterize these effects biophysically.

            The study of E. coli growing under anaerobic conditions is of particular interest.  When E. coli (and all other enteric bacteria) grow in association with the human digestive tract, they grow under anaerobic conditions.  We wish to begin to study bacterial stress responses in the absence of oxygen to further expand our general understanding of how this bacterium adapts to life in a wide variety of environmental niches.  For example, it is not currently known if the mechanisms of DNA damage repair and mutation that Prof. Finkel has been studying (and which he has shown to be important for adaptation to novel environments) function under anaerobic conditions.  These studies will bridge the gap from basic research into more medically relevant areas.


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[1].  Williams, F. A., Combustion Theory, 2nd Ed., Benjamin-Cummins, 1985.

[2].  Pojman, J. A., Hyashenko, V. M., Khan, A. M., "Free-radical frontal polymerization: self-propagating reaction waves."  J. Chem. Soc., Faraday Trans. 92, 2825 (1996).

[3].  Epstein, I. R. Pojman, J. A. An introduction to nonlinear chemical dynamics, Oxford, 1998.

[4].  Winfree, A.T., The Geometry of Biological Time, Springer-Verlag, 1990; Murray, J.D., Mathematical Biology, Springer-Verlag, 1993.

[5].  Berg, H. C., "Motile Behavior of Bacteria" Phys. Today 53, 24 (2000).

[6].  Lewis, B., von Elbe, G., Combustion, Flames, and Explosions of Gases, 3rd ed., Academic Press, 1987.

[7].  Budrene E.O., Berg H. C., "Complex patterns formed by motile cells of E. coli," Nature 349, 630 (1991).