Physica A 370 (2006) 793807

A theoretical analysis on self-organized formation

Biolms with a conspicuous hierarchical and fractal structure have been paid much attention in recent years. Available

Microbial biolms are encountered in a wide variety of applications, including traditional industrial

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www.elsevier.com/locate/physa

Abbreviations: 2D, two dimension; CA, cellular automaton; DLA, diffused-limited aggregation; IbM, individual-based model; Eq.,

equation0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.physa.2006.03.022

Corresponding author. Tel.: 86 22 27890550; fax: 86 22 87402076.E-mail address: lhchai@tju.edu.cn (L.H. Chai).processes, such as various biochemical operations, wastewater treatment, pipes for water supply and sewers,and various aqueous environments, etc. The importance of biolms in a wide variety of applications hasprovided motivation for numerous investigations on its mechanisms during the past several decades. Asubstantial number of efforts have been devoted to understanding and modeling the microbial colonies andbiolms. Many empirical correlations are now available in the literature [1]. Existing researches show thatmicrobial biolms are remarkably heterogeneous virtually in all parameters that can be measured accuratelyand reproducibly [24]. The emergence of these heterogeneities: structural, physiological, ecological, electrical,metabolic, etc. have not been well understood yet. Due to the multiplicity and complexity of variablesanalyses of biolm formation typically employed macroscopic differential equations, simulation approaches such as

cellular automaton (CA), diffusion limited aggregation (DLA), Monte Carlo simulation, and hybrid model, etc. This paper

proposed a new non-equilibrium statistical mechanics framework to analyze the interactions among cells and their

environment, and the self-organizing formation process of biolm was elaborated. The paper tried to make a renewed

attempt to illustrate the emergence of complexity of the biolm community and reveal the mechanism of producing a

macroscopic microbial biolm pattern from the microscopic microbial cells interaction. These studies may not only

provide a more reasonable physical description on microbial biolms, but also nds important engineering instructions.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Biolm; Microbial colony; Nonlinear interaction; Structure; Fractal

1. Introductionof microbial biolms

Li Ming Chen, Li He Chai

School of Environmental Science and Engineering, Tianjin University, Tianjin 300072, China

Received 17 October 2005; received in revised form 8 March 2006

Available online 27 April 2006

Abstract

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807794Nomenclature

b1b4 constantsfi functions introduced in Eq. (17)Fu, Fs uctuation forces introduced in Eqs. (15) and (16)h mass transfer coefcient, mg/m2 hJ; J ux, mg/m3 hl characteristic length, mp exponential coefcientinuencing the biolm systems and strong nonlinear features [3,4], a complete theory on the formation ofbiolms is still far from being created.In classical theories, the predictions of biolm formation and structure remained principally an empirical

art, and traditional modeling efforts typically used a linearized or discrete approach or differential equations[5]. For example, the physical phenomena were analyzed based on one-dimensional (1D) assumption, andthe mass transfer rate was obtained for a given biolm thickness by assuming the uniform density [6], i.e.,the cells had no effect on the formation of adjacent cells. Consequently, possibly important interactionsbetween cells were ignored. However, for practical biolm process, interactions did occur between adjacentcells [7,8]. Also, the traditional 1D model assumptions often conicted with observations of heterogeneousbiolm with channels, holes or cavities [3,4]. More severely, many available theories could not effectively

S function dened in Eq. (15)t time, sT function introduced in Eq. (16)u velocity, m/sWs, Wu potential functions of stable modes and unstable modesx driving forcex driving force vectordt boundary layer thickness

Greek Symbols

r probability densityr; r constantsZ constantsl damping coefcienta controlling parameterb the fractionU; U potential functionsz constantx normalized concentration difference variablem constantc constantO area unit

Subscripts

0 reference stateS bulks; s0; s00; s000 stable modesu; u0; u00; u000 unstable modes

associate microscopic mechanisms with macroscopic structure because it was based on uniform 1Dassumptions [5].The initial microbial cells on the substratum surface are randomly distributed. Cell immobilization and

biolm formation may roughly classify into four-step processes, which are integration of the physical,chemical, biological and hydrodynamic process, and these processes change the local aggregation and theother parameters that determine the stability of adjacent cells [9].The formation, pattern or structure of microbial colonies or biolms have been investigated by various

popular simulation methods, such as cellular automaton (CA) [1014] or partial CA [1518], Monte Carlo[1921], diffusion limited aggregation (DLA) [22], and individual-based model (IbM)-based Swarm system[2325], etc. However, these researches often pay little attention to clear microscopic mechanisms.The development of modern measurement techniques has provided much visualization of biolm

community and structure and then synergetic effects on understanding the biolm structure. Thus, the

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807 795fractal structure of the biolm had been observed and some fractal dimensions of biolms were measured[2630]. These researches have deepened the understanding of biolms, but can rarely provide a dynamicexplanation from microscopic mechanisms.As shown in Fig. 1, biolm system is an open complex system far from equilibrium with continuous exchanges

of the substrate and the product, which results in many nonlinear effects. These nonlinear effects include non-uniform cell distributions, growth and division of microbial cells, the formation and detachment of micro-clustersand macro-clusters, hydrodynamics, diffusion and conversion of substrate, etc. Of course, the nonlinear behaviorof biolm systems would be investigated if we could solve the controlling partial differential equations in a controlvolume of the two-phase system in the boundary layer adjacent to the biolmliquid surface [31]. However,problems rest with the fact that we have difculties in dealing with this kind of nonlinear partial differentialequations (though numerical computation can make some achievements). For another reason, cells are initiallydistributed stochastically on a substratum surface. The very special boundary conditions make it difcult toobtain exact theoretical solutions for controlling nonlinear partial differential equations. Furthermore, we oftenlack the typical values of characteristic parameters available [5]. In addition, the available simulations for biolmprocess, such as CA, Monte Carlo modeling usually lack a sound theoretical foundation. In a word, an alternativetheoretical framework is highly needed to describe the biolm process.In our opinion, to tackle the biolm process alternatively, we need to pay much attention to the microscopic

perspective. However, the microscopic mechanisms of biolm process are not very clear as yet. In the paper,new analytical frameworks for biolms were provided from non-equilibrium statistical mechanics, which weredifferent from partial differential equations and discrete or stochastic model, and a correspondingmathematical description of cell interactions was elaborated. Self-organization phenomena on biolm processwere correspondingly investigated. Industrial instructions based on theoretical results were nally discussed.

2. Non-equilibrium statistical descriptions on biolm formation

The equivalence of Lagranges analytical mechanics to Newtons framework for mechanics (partialdifferential equation for mechanics) gives us a hint that we can evade partial differential equation in an

Substratum

Flow Substrate

Product Fig. 1. The schematic drawing of a heterogeneous biolm system.

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807796alternative way when dealing with biolm systems. Statistical mechanics based on Lagrange optimization maybe an alternative way.For problems in thermal equilibrium of closed systems, entropy is maximized as an extreme. By dealing with

entropy, we can understand thermal equilibrium of closed systems from a microscopic view. For non-equilibrium open systems, ux may be chosen as a maximized property [32]. The classic theory on closedsystems is ruled by the Hamilton Principle, which is a minimal cost principle. However, Hamilton Principlecannot be applied to the open complex systems directly. Therefore, to deal with the open complex system, weneed to modify the Hamilton Principle to maximal ux principle (MFP), which is also a minimal costprinciple.The novel maximum ux principle may be described as follows: an open system far from equilibrium always

seeks an optimization process so that the obtained ux J from outside is maximal under given prices orconstrains [33]. Maximization of ux can be developed by Lagrange optimization [33,34]. Then let us discusshow Lagrange optimization can bring new understanding of biolm process from a microscopic nonlinearstatistical dynamics.Assuming the driving forces of subelements, i.e., the cause of receiving ux from liquid bulk, can be

expressed as x1; x2; ; xn, which are lumped as vector x x1; x2; ; xn. The driving forces here indicateconcentration difference between the biolm bulk and outside. Herein, x1;x2; ; xn may view as theindividual elements in biolms, i.e. the microbial cell. Similar to the classical statistical theory, here wecan consider that all possible micro-states compose a continuous scope in G space. dx dx1dx2 dxnis a volume unit in G space. The probability for the state of the system existing within the volume unit dx attime t is

rx; tdx. (1)rx; t is distribution function of ensemble, which satises the normalization condition:Z

rx; tdx 1. (2)

Assuming that the biomass ux is J when the state of the system exists within the volume unit dx at time t,the averaged biomass ux over all possible micro-states is

J Z

rx; tJrdx. (3)

By use of Lagrange multiplier methods, let us maximize the averaged biomass ux (note: for biomass isdirectly changed from substrates, maximization of the biomass ux implies the highest product) in Eq. (3)under the following constraints (i.e., prices given):

hxii b1, (4)

hxixji b2, (5)

hxixjxki b3, (6)

hxixjxkxli b4. (7)We obtained that [33,34]

r ezP

isixi

Pijsijxixj

Pijksijkxixjxk

Pijkl

xixjxkxl. (8)

In other words, the ensemble distribution function on biomass is elegantly yielded by maximization of thebiomass ux functional.Dening the exponential term of Eq. (8) as a potential function [33,34]:

Fr;x zXi

sixi Xij

sijxixj Xijk

sijkxixjxk Xijkl

sijklxixjxkxl . (9)

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807 797r in the left term represents a vector, and s in the right term represents a scalar. Both the parameters, z ands, are produced by Lagrange optimization. Accordingly, by transformation of

xi Xk

ckixk, (10)

Eq. (9) can be changed as [33,34]

Fl; x zXk

lkx2k . (11)

As a general analysis, x was normalized by concentration difference in any reference point chosen, which is akind of order parameter dened in Synergetics by Haken [34]. In applications, x is usually normalized by theconcentration difference between the outside environment and the biolm bulk. Through transformation (10),we can take x as the potential biolm pattern because x is a linear combination of all the kinds of cell micro-state x. Thus, we can think x as the concentration distribution of microbial cells.Stable modes lko0 represent that the microbial micro-colonies cannot survive to grow up and form, and

unstable modes lk40 stand for the formation and growth of cell micro-colonies. Considering the identiablecontributions of stable modes and unstable modes, the potential function can be decomposed as [33,34]

Fl; x z Fulu; xu Fslu; ls; xu; xs. (12)By a series of deductions, we can derive [33]

rxu expWu, (13)

rxs=xu expWsxs=xu. (14)Eqs. (13) and (14) can be regarded as the solutions of FokkerPlanck equations, which are equivalent to the

following Langevin equations:

_xu luxu Suxu; xs Fut, (15)_xs lsxs Tsxu Fst. (16)

For Wu and Ws are known, the functions Su and Ts can be decided. Langevin equations are dynamic. Thestochastic force F(t), which often appears in the Langevin equation, makes the system to jump from one stateto another [34]. Because x represents all the potential patterns of biolm, Eqs. (15) and (16) may be taken asbiolm pattern dynamic equations. Therefore, we surprisingly obtained the typical evolution dynamicequations for biolm systems, by which self-organization of biolm systems could be investigated. However,Eqs. (15) and (16) are extremely difcult to be solved due to strong nonlinear characteristics and too manyunknown parameters. Therefore, we need turn to analyses in next sections.

3. Self-organized processes through potential biolm pattern interactions

In order to show the competitive process, i.e., adiabatic elimination process rst appearing in statisticalmechanics and quantum mechanics, we now turned to the dynamics analyses. We considered multiple possiblemicrobial colonies patterns interactions in a general way. As shown in Eqs. (15) and (16), interactions amongmicrobial colonies patterns are reected through only one variable-normalized concentration difference in achosen region, which is directly related to the dynamic pattern parameters, such as microbial colonies patternvariable x. Equations with form similar to Eqs. (15) and (16), which describe the dynamic interactions of allpossible microbial colonies patterns, are written as [33]

_xi lixi f ix1; x2; . . . ; xni 1; 2; . . . ; n. (17)fi is a function of multiple parameters when considering the interactions among numerous possible

microbial colonies patterns. Coefcients li indicate damping effect of possible microbial colonies patterns.During the competitive processes controlled by this set of equations, the controlling and being controlled of

the possible microbial colonies patterns, compromise prevails if coefcient li differs from each other [34].

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807798More specically, the development of microbial colonies modes with small damping coefcients (mean smallresistances) will dominate the development of microbial colonies modes with relatively large dampingcoefcients (mean large resistance). In fact, in the classical quantum mechanics realm, according to well-known adiabatic elimination principle [34], for the interacting multi-elements system described by the kind ofequations with form like Eq. (17), the self-organized evolving process prevails, providing that the dampingcoefcients li are non-uniform. Let us do more detailed analyses. According to the magnitude of coefcientsli, equations (17) can be divided into some groups as was done in part 2: equations of the weakly dampedmodes with small li denoted as i u 1; 2; . . . m, and equations of the stable modes with relatively large lidenoted as i s m+1, m 2; . . . ; n. For possible microbial colonies patterns with stable modes (large li)whose growths will be controlled by possible microbial colonies patterns with weakly damped modes (smallli), in other words, these controlled microbial colonies patterns cannot grow and _x 0, and hence Eqs. (17)can be changed as

lsxs f sx1; x2; . . . ; xm; xm1xi; . . . ; xn. (18)This is a kind of self-organized process. From a physical perspective, it can be imagined that when the

biolm bulk concentrations reach certain values, the rst group of microbial colonies patterns(i u 1; 2; . . . ; m) with small damping coefcients will be activated and become active modes frompotential modes. Then, by interactions, the development of microbial colonies patterns (i u 1; 2; . . . ; m)will control and dominate the development of microbial colonies patterns (i s m 1, m 2; . . . ; n) withrelatively large damping coefcients. If the damping coefcients satisfy

l1bl2bl3b , (19)the terms related to x1 can be eliminated without affecting the other terms. The terms related to x2 are theneliminated in succession until only one variable remains. When one mode does not dominate the other modes,ashing will occur, which is a uniform state. Flashing is likely to occur when

l1 l2 l3 ln. (20)However, ashing only occurs on special occasions such as strong or fast mass transferring, or some other

special cases that can make parameters of the biolmliquid surface zone uniform as soon as possible, whichcan conrm the condition of Eq. (20) to be satised. In general, for the existence of all kinds of stochasticfactors in a biolm system, the substratum parameters are always non-uniform. The disturbance is induced bysubstratum parameters, which affect the damped modes. In most initial colonization processes in a biolmsystem, as described above, only one or a few microbial colonies patterns in a specic unit will becomeunstable while most other microbial colonies patterns will remain damped. The unstable model will expandand grow up to form the larger microbial colonies, while the stable mode will vanish.Then let us discuss how the biomass ux is distributed, which is more important for the understanding of

self-organization of biolm. According to the above dynamic analysis and considering the decompositions ofJxu; xs, Eq. (3) can be changed as [33]

J Zxu;xs

Ys

rsxs=xurxu Jxu Xs

Jsxs=xu" #

dx. (21)

Then we considered the non-equilibrium-phase transition. If the system is regulated by an external controlparameter a, Eq. (21) is dependent on the control parameter by a diverse manner.We may perform a series of transformations over Eq. (21) and obtain that [33]

J Ju Xs

Js, (22)

where the second part does not depend on a, at least in the present approximation. Therefore, the uxchange close to the instability point is governed by that of unstable microbial colonies patterns or modesalone [33]:

Ja Ja J a J a . (23)1 2 u 1 u 2

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807 799Eq. (23) provides a clear physical picture for the freedom degree compression [35]. By self-organizedprocess, only unstable microbial colonies patterns or modes with stronger ability can get the ux and develop.For an open complex biolm system that consists of many possible colonies patterns, the ux is concentratedon one or a few patterns or modes, and some of them can use the biomass ux better. In other words, biomassux is concentrated in one or a few patterns or modes, which may dominate the macroscopic symmetry-breaking behavior and heterogeneous structure of the whole biolm system. These kinds of dominationsguarantees that most freedom degree will be compressed. So all subsystems will behave like a whole system.The basic ideas of degree compression are intuitive incorporate in Eq. (23). Biolm formation is a self-organized dynamic process, which means that biolm system is a typical ordered dissipative structure. Theself-organized analysis provides a clear picture of the formation of this kind of ordered dissipative structure.Our analyses illustrate that the symmetry-breaking heterogeneous biolm structure can occur spontaneously(without any supposition on the inuence of the bacteria directly). Thus, we give an explicit answer for thequestion of van Loosdrecht et al. [36], on which the heterogeneous biolm structure can occur spontaneously.As shown in Fig. 2, when the biolm system grows up, the hierarchical structure will emerge and form with

an expanding cell cluster. When the cell clusters grow up and spread, the boundary layer, in which thesubstrate diffuse into the cell cluster will thicken. From Fig. 3, the increasing thickness of the boundary layerimplies that the transfer resistance of the generalized ux from the environment to the cell cluster or colonyincreases. In a real biolm formation, the substrate diffusion into the bulk and conversion into microbial cellsalmost only occur in the boundary layer, which is not the hydrodynamic boundary layer that was largely dealtwith in biolm engineering during the past decades, but a layer reecting the penetration of cells (changedfrom substrates) into clusters during the growth. Therefore, the ow behaviors of the generalized ux in theboundary layer inuence strongly the transfer of the generalized ux, which largely determine the distributionof ux in the system and hence control the macroscopic structure of the biolm system.12

.

n-1

n.

.

dJ

Fig. 2. The illustration of growth of biolm system.4. Formations of hierarchical biolm structures

The above analyses shed new light on the microbial colonies patterns interactions and the formation ofhierarchical or fractal structure qualitatively. To describe the formation of hierarchical or fractal structurequantitatively, we need to solve the problem from a new point of view. Let me study how the ne structure ofthe microbial biolm grows up. A micro-volume inside the biolm is isolated to study the growth process ofthe micro-colony. The micro-colony grows up from one small scale to another larger scale as shown in Fig. 2.We consider a biolm micro-colony of scale l, as shown in Fig. 3. The term dl represents innite-small scaleincrement from scale such as l1 to near scale l2. The substrate diffuses from the uid bulk to the boundary layernear the interface between the micro-colony and the uid. The processes including proliferation and growth ofthe microbial cell and conversion of substrate occur in the boundary layer, where the substrate is continuouslydegraded and converted to the microbial cells. The boundary layer will thicken as the micro-colony grows. Thevertical axis represents the direction of the micro-colony spreading; the curve represents the boundary layer

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807800y

l

dJ 1 d l 3

2 4 us s

Flow

(a)scope. The boundary layer of u and x often have different thickness. The term u represents the motion velocityof microbial cells and x represents the concentration of microbial cells.We take account of the ux balance between the two scales, represented by the surrounded element

1234, as showed in Fig. 3. The ux input per time unit from the microbial cell bulk to the element throughinterface 34 is

dld

dl

Z dt0

xSudy. (24)

The ux input per time unit through interface 12 is

adl qxqy

w

. (25)

The ux increase per time unit along scale increment through interface 13 and 24 is

dld

dl

Z dt0

xudy. (26)

Thus, the cascade ux balance gives the scale-coupled equation (SCE):

dld

dl

Z dt0

xSudy dld

dl

Z dt0

xudy adl qxqy

w

0. (27)

The SCE is also a renormalized equation [37]:

xSd

dl

Z dt0

udy a qxqy

w

ddl

Z dt0

xudy. (28)

l t (b)

Fig. 3. (a) An illustration of biolm structure and (b) enlarged part of selected rectangle in (a) and the illustration of boundary layer. The

networks stand for the expanding micro-colony along l direction.

For arbitrary u and x distributions, Eq. (28) usually yields biomass ux transfer coefcient as [37]

hnlp. (29)Depending on the actual distributions of u and x, p is a specic parameter ranging from 0 to 1.The analyses in Section 3 show that the biomass ux is concentrated in one or a few patterns or modes,

though many possible modes exist simultaneously. In this way, we have the expression [37]:

Jna1 Jna2 Juna1 Juna2hnlDxn. (30)Considering that driving forces are chosen as the concentration difference between biolm bulk and

boundary layer, we can derive the fractal structure describing the distribution of biolm structures in thefollowing way [37]:

OnDJOn1DJ

OnOn1

Juna1 Juna2Jun1a1 Jun1a2

hnl2nxn

hn1l2n1xn1

lnln1

2p(31)

Parameter 2-p is fractal dimension, whose value indicates the spatial distribution of biolm structures.Fractal dimension physically embodies dynamics features of evolutional complex biolm systems. Once we

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807 801have obtained the u and x distributions, we can determine the fractal dimension of biolm. Here, we roughlygive several typical results. If u and x have a trivial polynomial distribution, which may correspond to the caseof a laminar ow and mass transfer, we have p 1=2 and 2 p 3=2. If u and x have an exponentialdistribution, which may correspond to the case of turbulent ow and mass transfer, we have p 1=5 and2 p 9=5. If u and x have a trivial linear distribution, which may correspond to a sole conductive masstransfer, we have p 1 and 2 p 1 (it is no surprise that no fractal structures will form in this case). If u andx have almost uniform distribution (of course it is an ideal case), which may correspond to extremely turbulentmass transfer, we have p 0 and 2 p 2 (it is natural that solid 2D Euclidean structures will form in thiscase). All qualitative theoretical results for different cases are roughly shown in Fig. 4, in which ve typicalcurves divide the rst quadrant into four sections relating to different biolm structures and types. Obviously,we will have different p values and biolm structures for different u and x distributions. Logical conclusionscould be drawn that during high mass ux, ow induced by cells is very intense, more like violent turbulentow, and a high density (i.e., high fractal dimension) of cell clusters may be formed. While during low massux, ow induced by cells is less intense than violent turbulent, a low density (i.e., low fractal dimension) ofcell clusters may be formed.

c5, p=0

c3, p=1/2c4, p=1/5

u,

y

c1, p=1

No fractal structures

c2, p=1

Fig. 4. Inuence of the u and x distribution on p and biolm structure (c1 stands for no distributions of u and x; c2 stands for lineardistributions of u and x; c3 stands for polynomial distributions of u and x; c4 stands for exponential distributions of u and x and c5 stands

for independence on y of u and x distributions).

In fact, the generalized ux is often hybrid of the different types of uxes, such as the combination of thelaminar and turbulent ow. Therefore, the p value is not the value in the ideal or extreme case. The above fourvalues p 0, 0.2, 0.5, 1, which is derived in the ideal or extreme case, may divide the interval [0, 1] into threesubintervals [0, 0.2], [0.2, 0.5], [0.5, 1]. Correspondingly, the fractal dimension 2-p in the interval [1, 2] bedivided into three subintervals [1.8, 2], [1.5, 1.8], [1, 1.5], which exactly correspond to the three sections of Fig.4. Since the generalized ux is often a hybrid of the two types of ux, we may assume the ux comprises thefraction b of the ux that is related to the index p1, and the fraction 1b of the ux that is related to the indexp2, where p1 and p2 may be the two extreme points of the three subintervals. Based on the conservation of thegeneralized ux, we can derive

lp blp1 1 blp2 , (32)

where b is related to the intrinsic and extrinsic factors of the system, and reect the distribution of the elementsof the system in essence. b is actually a parameter that can reect all kinds of the ux pattern and thedistributions of u and x.By the above Eq. (32), we can derive the equation as follows:

p lnblp1 1 blp2

ln l. (33)

ARTICLE IN PRESS

1.841.861.881.901.921.941.961.98

2

2-p

105103

10

10-1

10-3

10-5

1.55

1.60

1.65

1.70

1.75

1.80

2-p

105

103

10

10-1

10-3

10-5

L.M. Chen, L.H. Chai / Physica A 370 (2006) 7938078020 0.2 0.4 0.6 0.8 11.801.82

0 0.2 0.4 0.6 0.8 11.50

0 0.2 0.4 0.6 0.8 11

1.051.101.151.201.25

1.301.351.401.451.50

2-p

105103

10

10-1

10-3

10-5

(a) (b)

(c)Fig. 5. The inuence of b on the fractal dimension 2-p, when p is between (a) 00.2; (b) 0.20.5 and (c) 0.51, separately.

Through Eq. (33), we can discuss the inuence of the fraction b and the scale l on the index p and the fractaldimension 2-p.As shown in Figs. 5 and 6, the fractal dimension 2-p varies with the fraction b and the scale l. These results

can be viewed as a physical explanation to the dynamic origin of fractal and multi-fractal dimension. In theabove framework, the multi-fractal of the structure of a given system depends on the variations of the uxpattern and the scale, which results from dynamic interactions of the elements and openness of the biolmsystem. Zahid and Ganczarzyk [38] and Moghabghab [39] had found the transition point in the observation ofbiolm from rotating biological contactors (RBCs). From Fig. 6, we can draw the conclusion that only whenb 0 orb 1, the fractal dimension is invariable and equal to p1 (b 1) or p2 (b 0). In the case of b 0(p1 0:2) or b 1 (p2 0:5), the system forms the fractal structure with uniform fractal dimension by self-organization. In other cases, the system forms the multi-fractal structure by self-organization. The aboveresults are derived from physical interactions among cells; in this connection, it is a logical conclusion that wemay reveal the theoretical secret on the origin of the fractal and the multi-fractal. According to Figs. 5 and 6,once we measure the fractal dimension 2-p of the system or structure, we can know the prole of b andcorrespondingly understand internal dynamics of the system. Conversely, once we know the dynamiccomponents interactions of the system under the input ux, we can mechanistically derive the fractaldimension 2-p.These results are qualitatively in agreement with the experimental observations. To conduct out analyses

more thoroughly, the following section will further compare our theoretical results with available experimentalresults.

ARTICLE IN PRESS

1.80

1.85

1.90

1.95

2.00

2.05

2.10

2-p

0.2

0.40.6

0.81

1.551.601.651.701.75

1.801.851.90

2-p

0.2

0.4

0.6

0.8

1

L.M. Chen, L.H. Chai / Physica A 370 (2006) 793807 803-5 -4 -3 -2 -1 0 1 2 3 4 51.70

1.75

logl

0

-5 -4 -3 -2 -1 0 1 2 3 4 51.401.451.50

logl

0

-5 -4 -3 -2 -1 0 1 2 3 4 50.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

logl

2-p

0.2

0.4

0.6

0.8

1

0

(a) (b)

(c)Fig. 6. The inuence of the b on the fractal dimension 2-p, when p is between (a) 00.2; (b) 0.20.5 and (c) 0.51, separately.

5. Comparisons with the available experimental results

It is necessary to compare our forgoing renewed theoretical analyses with other researches in literatures. Theavailable researches have partially illustrated the interactions between the microbial cells and their localenvironment through the various numerical simulations including CA and Monte Carlo, etc. [5]. The factorsinuencing the structure of the microbial biolm may be classied into intrinsic and extrinsic factors. Theintrinsic factors mainly include growth, division, deactivation and death of microbial cells, and microbial type,etc. The extrinsic factors include the environment that is the concentration distributions of substrate andinhibitory matter, hydrodynamic condition, and surface characteristic of substratum, etc. In our newframework, these parametric factors may often be included into u and x distributions. Distributions of u and x(corresponding structural index p) largely depend on these intrinsic and extrinsic factors. If the changes ofthese extrinsic and intrinsic factors are benecial for the conversion of substrate ux in the boundary layer, thep value will decrease and fractal dimension 2-p will increase. The different levels of substrate conversion in theboundary layer will produce different biolm structures. The better the substrate utilized in the boundarylayer, the smaller the p or the larger the fractal dimension. For example, the larger the substrate concentrationin the liquid bulk, the faster the substrate ux transfers to the boundary layer, and the better the substrate isutilized. Thus, the ux pattern in the boundary layer may shift from the polynomial distribution toexponential distribution, which results in the value of p to decrease and the fractal dimension increasescorrespondingly.

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807804Heterogeneities in biolm systems are characteristic of very different spatial and temporal scales, whichare reected by fractal dimensions. There were many experimental results on fractal dimensions in litera-tures, as shown in Table 1. In terms of our above analyses, based on fractal dimension and operationalconditions, we can conjecture the u and x distributions, and then analytically correlate the values offractal dimension, as shown in Table 1. For example, Jackson et al. [29] obtained the fractal dimensionabout 1.5 in 2D for 4 days biolm that was near the steady state. This illuminated that the microbialcells met polynomial distribution under the experimental condition of Jackson et al. [29]. Yang et al. [28]measured the fractal dimension 1.34 of 2D for the 18-days biooms that was nearly steady. This illuminatedthat the biolm cells was near polynomial distribution in this case. Nevertheless, in most cases, hybriddistributions of u and x based on linear superimpositions of different-typed distributions are often required, asshown in Table 1.The measured fractal dimension of biolms is correlated with many factors, such as, distance to the

substratum, ow direction, scale, growth history of biolm, biolm characteristic and measurement method,

Table 1

Fractal dimension of experimental and theoretical results of biolm structure

Descriptions of samples Fractal dimensions Sources and methods Theoretical distributions of u and

x (written as formsb*c1+(1b)*c2)

3-day biolm 1.3720 28, (ISA, Minkowski Sausage) 0.94*c3+0.06*c2

7-day biolm 1.3650 0.93*c3+0.07*c2

45-day PAO biolm 1.53 (laminar) 30, (ISA, Minkowski Sausage) 0.7*c3+0.3*c4

1.37(turbulent) 0.94*c3+0.06*c2

45-day JP1 biolm 1.56(laminar) 0.46*c3+0.54*c4

1.34(turbulent) 0.92*c3+0.08*c2

24 h biolm 2.12.7(large scale420 mm)* 26,(correlation dimension)(volume fractals or mass fractals)

vary between c2 and c5

2.42.9(small scale o5 mm)*48 h biolm 2.42.9(large scale 420 mm)*

2.92.95(small scale o5 mm)* vary between c4 and c5018-day biolm 1.021.34** (turbulent) 27, (Minkowski Sausage) vary between c2 and c3

14-day biolm 1.31.5** 29, (ISA, Minkowski Sausage) vary between c2 and c3Note: *The fractal dimensions vary with the distance from substratum and direction; ** The fractal dimension varies with growth time.

nonlinear characteristics and time dependence. In other words, by renormalization, nonlinear stochastic

articial sites and environmental condition, which mean that we can determine the distributions of u and x for

ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807 805the articial sites, we then can decide its distributions. The result may provide important instructions fordesigns and distributions of articial cavities on packing or carrier surface.Different types of biolm structures are needed for different engineering applications like wastewater

treatment [5]. To obtain the desired biolm structures and types, we can control and optimize the inuencingfactors, by analyzing of the inuence of all kinds of factors on the conversion of substrate in the boundarylayer qualitatively. The prediction of biolm behavior can be derived theoretically by the above theoreticalframework. Thus, we can control and predict the biolm process, and then can get the desired biolm types orstructures from the basic principle and microscopic dynamics in a given reactor system. Thus, we can not onlyoptimize the design of novel biolm reactors, but also promote optimization and reconstruction of the existingpartial differential equation is transformed to the ux pattern analyses in the boundary layer. Moreover, bythe time-independence SCE obtained, we can obtain the fractal structures for the given or known distributionsof u and x, and in turn conjecture the distributions of u and x in case of known fractal dimensions of biolmfractures.

6.2. Engineering instructions

The packing or carrier performance inuences the efciency of biolm reactors. Therefore, the design ofpacking or carrier plays a key role in the design of the effective biolm reactors. The articial cavitiesdistribution will inuence the packing or carrier performance. Enhancement of biolm process by means ofarticial cavities is a very important aspect of mass transfer research in biolm reactors. The distribution ofarticial cavities is a key factor for optimizing and controlling bioms. Placing articial cavities based on atheoretically optimized distribution of active points can greatly improve the initial colonization of microbialcells, which may shorten the formation time of biolms and speed start-up of biolm reactors. Accordingly,the efciency of biolm reactor may be improved. Our foregoing investigation results in a natural theoreticaldistribution of active sites under the optimization that maximizes mass ux. It is shown that the optimaldistribution of active sites is dependent on the behavior of the active site itself. After knowing the feature ofetc. as showed in Table 1. These factors may be attributed to different distributions of u and x. The fractaldimension is characteristic of the heterogeneous and anisotropic biolms. The different fractal dimensions indifferent scales show that multi-fractal occurs in biolms [27], and the different ux patterns (correspondingdifferent distributions of u and x ) in the boundary layer. The fractal dimensions decreases with the increasingdistance perpendicular to the substratum, i.e., the biolm gradually loosens. According to our forgoinganalyses, this implies that the u and x distributions change with the distance to the substratum. The differentfractal dimensions in the different ow directions [27] mean that the distributions of u and x must be different.It is no surprise that the adaptive fractal structures of biolms are favorable for the mass transportation ofsubstrate and oxygen to biolms and the inhibitory products out of biolms.

6. Important implications

6.1. Academic implications

Though the available mathematical models play an important role on understanding the biolm process, itis not sufcient for the understanding of the whole biolm dynamic process, especially the correlationsbetween the microscopic cell interactions and macroscopic biolm behaviors. Further investigations aredenitely required. Our new framework may provide an access to establishing a novel hierarchical multi-scalemodel, which will promote the understanding on the relation between the microscopic cell interactions andmacroscopic control parameters in detail. The SCE will provide potentially an approach to establish therelation between the microscopic activities and macroscopic properties.In fact, by the SCE, we could avoid solving Eqs. (15) and (16), which are difcult to be solved due to strongbiolm reactors for desired engineering biolm structure.

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[32] A. Bejan, Advanced Engineering Thermodynamics, Wiley, New York, 1997 (Chapter 13).

[33] L.H. Chai, M. Shoji, ASME J. Heat Transfer 124 (2002) 505.

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The formation and structure of biolm is a classical puzzle during the past decades. This paper proposed anovel theoretical analysis on the interactions among microbial cells and their environment, and clearlyillustrated that the heterogeneous biolm structure can occur spontaneously (without any supposition oninuence of the bacteria directly). The present studies may give more rational and physical descriptions onbiolm systems. The formation of the fractal structure of biolm is essentially a process of self-organizationand non-equilibrium phase change. The possible application is correspondingly discussed. The presentinvestigation, although preliminary, provides a renewed theoretical effort to understand the underlyingmechanisms of the biolm process and the emergence of a complex structure.

Acknowledgments

The Project is currently sponsored by the National Natural Science Foundation of China through theContract # 50406018 and the Scientic Research Foundation for the Returned Overseas Chinese Scholars.Suggestions from anonymous reviews are highly appreciated.

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ARTICLE IN PRESSL.M. Chen, L.H. Chai / Physica A 370 (2006) 793807 807

A theoretical analysis on self-organized formation of microbial biofilmsIntroductionNon-equilibrium statistical descriptions on biofilm formationSelf-organized processes through potential biofilm pattern interactionsFormations of hierarchical biofilm structuresComparisons with the available experimental resultsImportant implicationsAcademic implicationsEngineering instructions

ConclusionsAcknowledgmentsReferences