# The General Growth Logistics of Cell Populations

## Abstract

An increment model based on thermodynamics lays bare that the cell size distributions of archaea, prokaryotes and eukaryotes are optimized and belong to the same universal class. Yet, when a cell absorbs mass or signals are processed, these conditions are disturbed. Relaxation re-installs ideal growth conditions via an exponential process with a rate that slows down with the cell size. In a growing ensemble, a distribution of relaxation modes comes in existence, exactly defined by the universal cell size distribution. The discovery of nano-mechanic acoustic activities in cells led us to assume that in a growing ensemble acoustic signals may contribute significantly to the transmission of essential information about growth-induced disturbances to all cells, initiating that way coordinated relaxation. The frequency increases with the cell number shortening the period between successive signals. The completion of rearrangements occurring at a constant rate is thus progressively impaired, until cellular growth stops, totally. Due to this phenomenon, the so-called “relaxation-frequency-dispersion” cell colonies should exhibit a maximum cell number. In populations with large cell numbers, subsystems, behaving similar-like colonies, should form network-like patterns. Based on these ideas, we formulate equations that describe the growth curves of all cell types, verifying that way the general nature of the growth logistics.

## Keywords

Cell multiplication Colonies Populations Structure and dynamics Relaxation Growth logistics## Introduction

The growth of multi-cellular living systems is linked via cell division to the growth of individual cells [1]. A phenomenological description of these processes should provide information about cooperation within cell ensembles. If synchronized sequences of cell growth and division occur in an unlimited manner, an exponential relation describes the number of cells and in a semi-log plot a straight-line results, not showing the sigmoidal bending of experimental patterns. Various treatments [2, 3, 4, 5] present different approaches to a deeper understanding of the observations by modelling universality and scaling. Yet, limits of the characterization are also made out like the question whether and how the cell ensemble structure exercises an influence on the growth process.

Earlier treatments [6, 7, 8] identify the cell size distributions deduced by Kilian et al. [8] and Schlote [9] as optimized patterns. Moreover, expressed in dimensionless variables all of them coincide with one another. In order to understand this striking symmetry, a quantitative description of growth is required which accounts for the essential features of the structural development. Of course, at the present state of knowledge, this treatment needs sensible simplifications and approximations.

One central assumption is that throughout all stages of their lifecycle, cells tend to keep optimal process conditions, which are disturbed, however, each time when mass is taken up or signals are processed. To re-establish the optimal state, intra-cellular relaxation has to occur. Since the relaxation time increases with the cell size, a relaxation-time-distribution results, determined by the size distribution of the cell ensemble. In addition, whenever a cell grows, the momentary size distribution deviates from its ideal version [6, 7, 8, 10]. All cells have to improve the situation, by collective relaxation.

During these events, communication should coordinate the numerous bio-molecular processes. Intercellular-signalling may occur via cell-to-cell coupling that supports synchrony across the whole population [11]. “Quorum-sensing” is the ability of cells to detect and respond to their population density. In any case, growth control needs collective signals that exhibit at least these features: (a) Signals should inform all cells of a growing ensemble about the actual situation. (b) They should thus be transmitted fast enough in respect to the mean rate of molecular reaction and reorganization. (c) Via absorption and emission, cells should be able to take individual action. (d) A frequency spectrum should allow to generate precise messages.

In order to develop under these conditions a phenomenological model, we refer to the observations of nano-mechanical motions of the cell wall of *Saccharomyces* *cerevisiae* [12]. Systematic studies lead to the conclusion that this “acoustic breathing” is a metabolically driven process. One might then assume that cellular absorption of biomass causes acoustic signals. Yet, the concerted activities of all the communicating cells in a colony, including complex effects of intercellular contacts [13], are not known to the last details. We must content ourselves with approximating the succession of signals as an oscillating stress-field with frequencies in the acoustic regime.

During growth, these frequencies increase steadily, thus shortening the period between successive signals. Molecular rearrangements that show a constant rate become more and more incomplete. Finally, any further growth is blocked. We call this whole phenomenon “relaxation-frequency-dispersion”. In colonies, the number of cells should in principle be restricted to a maximum value. Any population comprised of a number of cells beyond this limit should form a transient network of subsystems with properties resembling those of a single colony.

Based on the above ideas, equations are formulated that allow us to describe the growth of populations for different cell types and under any culture condition, that way revealing notable symmetries.

## Stationary Properties

### The Saturation Conditions

Ensembles of cultured cells from all biological kingdoms exhibit cell size distributions with conspicuously similar properties [9, 14]. By introducing an “increment model”, it can be discussed how they may come about [6, 7, 8]. We suppose that in a growing cell energetically equivalent molecule clusters are absorbed and relatively fast adjusted to optimal conditions. These “equilibrated clusters” are regarded as particles, dubbed increments of biomass (ICs). Thermodynamics legitimates the assumption that the nutrient solution should contain such increments as preformed particles, that way formulating a virtual state of reference. This characterization obviates the necessity of dealing with the formation of ICs via a not fully known intracellular biochemistry and processes of self-organization.

*u*

_{c}of an IC as difference between the energy

*u*

_{c}in the cell and the energy

*u*

_{n}in the nutrient is

There are many reasons to suspect that keeping optimal growth conditions needs unceasing re-adjustments. They seem to happen analogous to the adaptation of equilibrium in a streaming gas. Within every volume element, relaxation installs thermal equilibrium, independent of the stationary collective movement of all particles. In cells, disturbance of the optimal internal structure occurs when biomass is absorbed. Virtually independent of the quasi-stationary collective growth of the total ensemble, relaxation should always arrange optimal process conditions.

*T*and pressure

*p*, the differential of the Gibbs-free-energy (

*d*g

_{ y })

_{ T,p }within a growing cell comprised of

*y*

_{ a }ICs with

*y*=

*y*

_{ a }− 1 contacts, can according to [15] be defined by

*ξ*

_{ y }(

*t*) is the reaction coordinate. The affinity

*A*

_{ y }

*> 0 measures the distance from stationary conditions at*

*A*

_{ y }

*= 0. If, on average, each increment experiences the same growth-induced disturbance, the affinity can be written as*

*A*

_{ y }

*=*

*yA*

_{0}. The affinity per increment

*A*

_{0 }is then equal to

*μ*

_{n}, and in the cell

*μ*

_{c}.

*ν*

_{ j }are the stoichiometric coefficients: the negative term (

*ν*

_{n}

*= −1) indicates that an IC disappears from the reservoir when it is linked in a cell (*

*ν*

_{c}= +1). With the definition

*μ*

_{ j }

*= u*

_{ j }−

*Ts*

_{ j }(

*j*= c, n) the relations per IC emerge

*u*

_{c}und the affinity

*A*

_{0}are supposed to be invariant, the reaction energy Δ

*u*

_{ y }of a cell with

*y*contacts is equal to Δ

*u*

_{ y }=

*y*Δ

*u*

_{0}. The reaction entropy Δ

*s*

_{ y }=

*s*

_{ y }−

*s*

_{ yn }is the difference of the entropy

*s*

_{ y }of

*y*

_{ a }ICs in the cell with

*y*contacts and the entropy

*s*

_{ yn }of these ICs in the reservoir. Postponing the explicit formulation of Δ

*s*

_{ y }, the “saturation conditions” emerge

The smallest cells have *y* _{min} contacts; *y* _{max} counts the contacts in cells of maximum size. Under optimal process conditions, the entropy term *T*Δ*s* _{ y } and the reaction energy Δ*u* _{ y } should become equal for each cell size (*y* _{min} ≤ *y* ≤ *y* _{max}). Via ever-present structural fluctuations, the entropy should rapidly adapt to the value demanded according to Eq. 4.

### Cell Size Distributions

*s*

_{ y }must be given in greater detail. Assumed to be comprised of the intracellular contribution Δ

*s*

_{ yi }and the mixing entropy Δ

*s*

_{ ymix}

*s*

_{ yi }. The number of microstates Ω

_{ yi }in the direction

*i*should then be related to the number of increments

*y*

_{ a }=

*y*+ 1 according to

*α*

_{ i }accounts for the way how a stationary number of microstates in the direction

*i*is produced. The total number of microstates Ω

_{ y }in a cell is then

*y*= 1, the intracellular reaction entropy Δ

*s*

_{ yi }is defined by

*k*

_{B}is Boltzmann’s constant. Since cell size distributions are experienced to belong without any exception to the (

*p*= 3)-class, the intracellular entropy Δ

*s*

_{ yi }may be represented as if exactly

*y*configurations are independently activated in each direction

*i*so that Ω

_{ y }=

*y*

^{3}(

*p*= 3) [6, 8].

*s*

_{mix}arises now since according to thermodynamics, a growing cell colony is a multi-component system with cells in different size as its components [16]. When a cell divides, two smaller daughter cells replace the bigger mother cell. Within more or less tightly packed patterns, growth of a cell enforces cooperative rearrangements of the positions of neighbouring cells that are growing too. Not dominated by diffusion, these not strictly synchronized processes accomplish de facto an ideal mixing of cells of different size. The mixing entropy Δ

*s*

_{mix}[16] is then given by

*x*

_{ y }is defined by relating the number of cells with

*y*contacts

*n*

_{ y }to the total cell number in the ensemble Σ

*n*

_{ y }. Since at the state of reference (

*∂s*

_{mix}

*/*∂

*n*

_{ y })

_{ r }= 0, the mixing entropy contribution of a cell with

*y*contacts Δ

*s*

_{ ymix}is equal to

*y*

_{min}contacts are in existence, the parameter

*y*should be replaced by Δ

*y*=

*y*−

*y*

_{min}. Equations 6 and 8 inserted into Eq. 4, the fraction

*x*

_{ y }of cells with Δ

*y*contacts

*x*

_{ y }is obtained to be equal to [7]

*p =*3, Δ

*u*

_{0},

*A*

_{0}and

*y*

_{min}, the set of fractions

*x*

_{ y }describes a cell-size distribution with optimal properties. When Eq. 9 is re-formulated in terms of the dimensionless variable

*η*given by relating the reaction energy per cell (

*y*−

*y*

_{min}) Δ

*u*

_{0}to the thermal energy

*k*

_{B}

*T*

*η*) emerges (7)

*p*= 3)-version of Φ(

*η*), i.e.

*p*should not depend on kinetic or genetic factors. Fluctuation of the intracellular structure according to the (

*p*= 3)-version of Eq. 12 indicates an unexpected symmetry of the intra-cellular dynamics.

## The Growth Process

### The Relaxation Mode Factor

*τ*

_{ y }of a cell with

*y*contacts is written as product of the dimensionless “relaxation mode factor”

*τ*

_{0y }and the kinetic factor

*τ*

_{kin}[8, 18]

The mode factor *τ* _{0y } should be proportional to the number of microstates Ω_{0y } within a cell with *y*-contacts that increases with the size in qualitative accordance with Eq. 6. In contrast, motions of molecular segments should only affect next neighbours, so that *τ* _{kin} should not depend on the ensemble structure. *a* is Onsager’s coefficient. The definition of the parameter *μ* is given in the Appendix.

### The Relaxation-mode-spectra

*s*

_{ y }to a state of reference with Ω

_{ n }= 1

*p*, the identities emerge

Φ(*η*) = Φ(Δ*s* _{ y }/*k* _{B}) confirms the interrelation between reaction energy and entropy as demanded according to Eq. 4. To fulfil the condition Φ(*η*) = Φ(*τ* _{0y }), the course of the mode spectrum Φ*(τ* _{0y }) has to coincide with the universal (*p* = 3)-cell size distribution Φ(*η*), i.e. all cell types should exhibit the same universal (*p* = 3)-relaxation-mode spectrum Φ(*τ* _{0y }).

### The Connection Between *ω* _{c} and *t*

*ω*

_{0, }the frequency

*ω*

_{c}of all signals in a colony comprised of \(n^{\prime}_{{\text{c}}}\)(

*t*) cells

*t*related to

*t*

_{c}, the period of a cell cycle, set equal to

*ρ*

*ρ*is always observed to be approximately equal to ln(2) so that we are led to

*t*) and \(n^{\prime\prime}_{{\text{c}}}\)(

*t*) are therefore obtained by plotting \(n^{\prime}_{{\text{c}}}\)(

*ω*

_{c}) and \(n^{\prime\prime}_{{\text{c}}}\)(

*ω*

_{c}) against

*t*given by

### A Principal Limitation

When a cell absorbs mass, its energy increases, including strain energy that characterizes the initial disturbance via cell deformation; the stress increases spontaneously, evoking acoustic signals which should convey information on the actual local situation. Since in a colony with many cells, nutrient is absorbed in an almost regular sequence [19], these events may be described by a periodic stress-field with the frequency *ω* _{c}. If the number of communicating cells grows, *ω* _{c} increases, and the time interval between two successive signals, i.e. the time for molecular reorganization shortens. Since the molecular relaxation processes occur at a constant rate, during growth optimization of the whole ensemble is increasingly constrained. When a network structure develops, reorganization-modes “freeze” selectively starting with the slowest highly cooperative events. The growing reaction energy of the ICs drives then *A* _{0} to increasing positive values (Eq. 3). According to Fig. 13, this effect is experimentally confirmed. Finally, at the highest frequency *ω* _{cmax}, cell growth and division are totally blocked. Usually, a tissue like pattern with an optimal ensemble structure emerges.

### An Instructive Analogy

*γ*the total deformation energy

*w**per chain is as a function of the elastic strain energy

*w*′ and the dissipated part

*w*″ defined by

*G**per chain is then comprised of the real part

*G*′ and the imaginary part

*G*″. Since network chains are energy-equivalent subsystems, the strain energy

*W**of the whole network grows proportional to the number of chains

If periodic deformation is applied, the reorganization of the conformation of chains at increasing frequencies is constrained steadily, and finally these processes freeze totally: the network is in the glassy state. Apparently, the phenomenon of relaxation-frequency-dispersion typifies the glass transition of a macromolecular network [18, 21].

A growing cell ensemble shows now topologically analogous features since cells operate as equivalent widely autonomous visco-elastic subsystems. They are locally deformed whenever a cell absorbs biomass. Strain-energy is stored, and then in parts dissipated during relaxation. As approximation growing cell ensembles are supposed to be automatically submitted to a collective quasi-periodic deformation, represented by an oscillating stress-field with the increasing frequency *ω* _{c}. Optimal process conditions should be adjusted via relaxation, exhibiting analogous to a freezing macromolecular network, the phenomenon of relaxation-frequency-dispersion. We may thus adopt ideas of the theory of rubber elasticity and of the phenomenological classification of relaxation [22].

*σ*(

*t*) at the amplitude

*σ*

_{0}

*ω*

_{c}(Eq. 17), the accompanying strain

*γ*(

*t*) is

*δ*originates from the dissipation of strain-energy. In a growing cell with

*y*contacts, the real part

*G*

_{ y }′ (

*ω*

_{ c }) and the imaginary component

*G*

_{ y }″ (

*ω*

_{c}) of the complex modulus

*G*

_{ y }*(

*ω*

_{c}) should be defined by [22]

*G*′ =

*G*′

_{max}−

*G*′

_{min}is the “relaxation-strength”, as the difference between the maximum value of the modulus at the largest frequency and its minimum under static conditions. In order to describe the growth of a cell ensemble, it is appropriate to introduce the complex function \(X^{\ast}_{{{y}}}\)(

*ω*

_{c}

*τ*

_{y}) per cell

*ω*

_{c}

*τ*

_{ y }) characterizes the phenomenon of relaxation-frequency dispersion of a cell with

*y*contacts and the relaxation time

*τ*

_{ y }linked in a colony, while a stress-field with the frequency

*ω*

_{c}is active. According to Fig. 2a, the real part

*X*

_{ y }′(

*ω*

_{c}

*τ*

_{ y }), giving the relative strain-energy exhibits a sigmoidal pattern while the relative loss

*X*

_{ y }″(

*ω*

_{c}

*τ*

_{ y }) goes through a maximum at

*ω*

_{c}

*τ*

_{ y }= 1 (grey bar).

### The Growth of a Cell Colony

*p*= 3)-relaxation-mode distribution

*n*

_{c}′(

*ω*

_{c}) is obtained by adding all the fractions

*h*

_{ y }(

*τ*

_{0y }) of differently large cells, each one multiplied by the conjugated factor in Eq. 21. Calling the maximum number of newly formed cells in the grownup colony Δ\(n^{\prime}_{{\text {c}}}\) and accounting for the existence of the mother cell, the total cell number \(n^{\prime}_{{\text {c}}}\)(

*ω*

_{c}) and the corresponding loss \(n^{\prime\prime}_{{\text {c}}}\)(

*ω*

_{c}) are equal to

At the lower limit \(n^{\prime}_{{\text {c}}}\)(*ω* _{c}) is equal to 1, i.e. the colony develops from a single mother cell. At high frequencies, the maximum cell number comes out to be \(n^{\prime}_{{\text {cmax}}}\) (*ω* _{c}) = 1 + Δ\(n^{\prime}_{{\text {c}}}\). Plotting *n* _{col}′(*ω* _{c}) = \(n^{\prime}_{{\text {c}}}\)/\(n^{\prime}_{{\text {cmax}}}\) and \(n^{\prime\prime}_{{\text {col}}}\)(*ω* _{c}) = \(n^{\prime\prime}_{{\text {c}}}\)/\(n^{\prime}_{{\text {cmax}}}\) against *t*/*t* _{c}, Fig. 2b illustrates that the width of the dispersion regime embraces the principal contribution according to Eq. 21 (Fig. 2a) plus the additional broadening caused by the existence of the spectrum *h* _{ y }(*τ* _{0y }) (Eq. 22).

The growth curve \(n^{\prime}_{{t}}\)(*t*) is obtained by presenting \(n^{\prime}_{{\text {c}}}\)(*ω* _{c}) in dependence of the time *t* that is related to *ω* _{c} as formulated in Eq. 19; the width of the dispersion regime depends strongly on *t* _{c. }

### The Growth of Cell Populations

Perfect coordination of all processes is only possible in colonies with a number of cells below a maximum value. A cell population with optimal properties should thus be comprised of *n* _{0} equivalent subsystems that show qualities similar to a cell colony.

*n*

_{0}cells, each one operating as the mother cell of a subsystem. If coordinated growth of the subsystems leads to an affine transformation of the ensemble structure,

*n*

_{0}stays constant. In a growing population, the cell number \(n^{\prime}_{{\text {c}}}\)(

*ω*

_{c}) and \(n^{\prime\prime}_{{\text {c}}}\)(

*ω*

_{c}) are then equal to

*n*

_{0}. If the number of subsystems is large (

*n*

_{0}»1), the maximum cell number \(n^{\prime}_{{\text {cmax}}}\) =

*n*

_{0}(1 + Δ\(n^{\prime}_{{\text {c}}}\)) may exceed by far the maximum size of a single subsystem (1 + Δ\(n^{\prime}_{{\text {c}}}\)). Since the maximal number of newly formed cells Δ\(n^{\prime}_{{\text {c}}}\) is usually very large, the approximation

*= n*

_{0}Δ\(n^{\prime}_{{\text {c}}}\) but different combinations of

*n*

_{0}and Δ\(n^{\prime}_{{\text {c}}}\) are identical, i.e. we cannot determine the absolute value of the structure-parameter

*n*

_{0}by describing a growth curve with the use of Eq. 24.

### The Ψ′- and Ψ″-functions

^{ * }-function by relating \(n^{\prime}_{{\text {c}}}\)(

*ω*

_{c}) and \(n^{\prime\prime}_{{\text {c}}}\)(

*ω*

_{c}) in Eq. 24 to the maximum cell number \(n^{\prime}_{{\text {cmax}}}\) =

*n*

_{0}(1 + Δ\(n^{\prime}_{{\text {c}}}\)) and by plotting this ratio against the dimensionless variable

*t/t*

_{c}= ln(

*ω*

_{c}

*/ω*

_{0}).

*n*

_{0}drops out: according to our model, cell populations or colonies should exhibit the same growth characteristics. Since Δ\(n^{\prime}_{{\text {c}}}\) is large 1/(1 + Δ\(n^{\prime}_{{\text {c}}}\)) ≈ 0 and Δ\(n^{\prime}_{{\text {c}}}\)/(1 + Δ\(n^{\prime}_{{\text {c}}}\)) ≈ 1, so that the relations emerge

Ψ′(*ω* _{c} *τ* _{ y }), giving the relative strain-energy, exhibits a sigmoidal pattern while the loss Ψ″(*ω* _{c} *τ* _{ y }) goes through a maximum at *ω* _{c} *τ* _{ y } = 1 (Fig. 2b). Since the (*p *= 3)-relaxation-mode-spectrum *h* _{ y }(*τ* _{0y }) is invariant, the growth logistics should be the same for cells of every kind.

The general applicability of the model is easily tested experimentally by checking whether at the same parameter *τ* _{kin}, the ratio \(n^{\prime}_{{\text {c}}}\)(*ω* _{c})/\(n^{\prime}_{{\text {cmax}}}\) plotted against of *t*/*t* _{c} coincides with the universal (*p *= 3)-Ψ′**-**function.

## Results

### Eukaryotic Cells

#### Adherent Mammalian Cells

*t*) and the individual DNA content of Chinese Hamster Ovary (CHO)-cells were determined by flow cytometry [23]. Each growth curve in Fig. 3a, b starts at the cell number

*n*

_{initial}listed in Table 3. Since the

*n*

_{initial}cells adjust rapidly optimal conditions, each experimental data set in Fig. 3 can completely be fitted with Eqs. 19 and 24, using apart from

*τ*

_{kin}the same set of parameters (Table 3). The growth curves have then the same shape but different positions on the time axis. The position of each pattern is fitted by assigning the kinetic factor to values as shown in Table 3. According to Fig. 3e, the kinetic factor increases nearly linear with

*n*

_{initial}. Apparently, at

*t*= 0 the virtual age of the patterns increases with the initial concentration of the nutrient

**.**

Since the growth curves show altogether the same shape, a master curve is obtained by “simply shifting” each original data set along the time-axis until the pattern in Fig. 3c, d results that corresponds to a theoretical curve calculated at *τ* _{kin} = 4 × 10^{−6} h.

Each dotted line in Fig. 3b–d describes the energy loss during growth. Unfortunately, the dissipation heat has not been measured until now. Yet, the calculation reveals that in the regime of exponential growth (in Fig. 3d between 30 and 60 h) ln(\(n^{\prime}_{{\text {c}}}\)(*t*)) and ln(\(n^{\prime\prime}_{{\text {c}}}\)(*t*)) converge more or less, suggesting that a network-like structure is developing as in a gel [24].

### The Growth of a Cell Colony

Because of being unable to determine the absolute number *n* _{0} of subsystems by fitting a growth curve (Eq. 24), we studied growing colonies of adherent A549 human lung cancer cells. Cells from an exponentially growing culture were counted with a flow cytometer. They were seeded in 25 cm^{2} culture vessels at approximately ten cells in 10 ml nutrient and incubated for up to 3 weeks. After 7 and 14 days, cultures were fed with fresh medium or were stained to determine cell numbers in the colonies (see the legend of Fig. 4a).

Micrographs from different times are depicted in Fig. 4. Calculations with the use of Eqs. 19 and 24 reproduce the data satisfyingly (Fig. 5a, b). When replacing the nutrient repeatedly, marginal extra-fluctuations of data points are observed occasionally, may be caused by the technical interventions. The protracted convergence to a final maximum cell number may be due to loosely packed cells on the boundary (arrow in Fig. 4c) where occasional cell divisions may occur. Still, the estimated cell number in a 3-week-old colony (≈650,000) significantly lags behind the 2–5 Mio cells obtained in the same culture vessel within one week after seeding 10–50,000 cells.

### Cell Populations

Consistent with the model’s predictions, we succeed in describing the growth pattern of a population by using apart from *n* _{0} and *τ* _{kin} the same parameters as in the preceding section. Within the limits of accuracy, the plots in Fig. 5c, d demonstrate the satisfying fit achieved with Eqs. 19 and 24. Setting Δ\(n^{\prime}_{{\text {colmax}}}\) equal to 6.5 × 10^{5} cells, the population embracing \(n^{\prime}_{{\text {cmax}}}\) = 1.17 × 10^{7} cells should be formed by *n* _{0} = \(n^{\prime}_{{\text {cmax}}}\)/\(n^{\prime}_{{\text {colmax}}}\) = 18 subsystems linked in a network.

Showing the growth curve of a colony and of a population at the same position on the time axis (Fig. 5e), the remarkable similarity demonstrates that the growth-logistics in populations and colonies is identical. The subsystems in cell populations seem to exhibit about the same properties as colonies. According to Fig. 5f represented in terms of the dimensionless variables (\(n^{\prime}_{{\text {c}}}\)/\(n^{\prime}_{{\text {cmax}}}\); *t*/*t* _{c}), all the data coincide with the (*p *= 3)-Ψ′-function, proving the validity of our concept. If in correspondence with Pelling et al. [12] the frequency *ω* _{0} of signals emitted by a single cell (Eq. 17) is set equal to *ω* _{0} ≈ 1 kHz the values of *ω* _{c} indeed match the range of acoustic waves in liquids at room temperature [25].

Most likely, relaxation-frequency-dispersion limits the size of a colony to a maximum cell number, usually smaller than the maximal cell number of a grown-up population. As an optimal structure, any cell population should form an assembly of equivalent subsystems into a transient network.

### The Influence of the Nutrient

We regard the growth process in CHO-cultures, started with 10^{5} cells at 100% nutrient concentration in the first culture (C1) and 30% in the second (C2). With Eqs. 19 and 24 and the same value of *t* _{c} = 0.55*d* both experimental patterns in Fig. 6 are fitted fairly well. Hence, the period of cell cycles does not depend on the amount of nutrient that is available. For different culture media, remarkable modifications are observed [26].

The maximum cell number in the (C1)-pattern \(n^{\prime}_{{\text {cmax}}}\) = *n* _{01}(1 + Δ\(n^{\prime}_{{\text {c}}}\)) is equal to 1.5 × 10^{7} cells. Assigning Δ\(n^{\prime}_{{\text {c1}}}\) = Δ\(n^{\prime}_{{\text {c2}}}\) of the subsystems to 6.5 × 10^{5} cells (as observed in the previous experiments), the (C1)-population should contain *n* _{01} = \(n^{\prime}_{{\text {cmax}}}\)/Δ\(n^{\prime}_{{\text {c1}}}\) = 16.1 subsystems. If the ratio *n* _{02}/*n* _{01} agrees with the ratio of the nutrient concentrations 30/100 the number of subsystems in the (C2)-population drops to *n* _{02} = 6.9 (\(n^{\prime}_{{\text {cmax}}}\) = 6.9 (1 + Δ\(n^{\prime}_{{\text {c}}}\)) = 4.5 × 10^{6} = 0.30 × 1.5 × 10^{7} cells). When a culture experiment is started, the initial concentration of nutrient seems to exercise a crucial influence on the optimal ensemble structure that is adjusted, rapidly.

In order to obtain in the calculation at *t *= 0 exactly *n* _{initial} = 10^{5} cells, the kinetic factor, must be assigned to the different values *τ* _{kinC1} = 3.5 × 10^{−5} *d* and *τ* _{kinC2} to 1.15 × 10^{−4} *d*. At *t *= 0 the virtual age of (C2)-subsystems is about thrice the one of (C1)-subsystems. Since at *t *= 0, the size of (C2)-subsystems is then about 3.34 times the size of (C1)-patterns, in both cultures the cell number is equal to \(n^{\prime}_{{{\text{c}}}}\) = *n* _{initial} = 10^{5} cells.

In terms of our model, the number of mother cells *n* _{0} seems to increase with the concentration of the nutrient. If the subsystem’s ensemble structure undergoes then at *n* _{0} = const a growth induced affine transformation, the underlying processes run off on optimal conditions. Of course, as long as the product of *n* _{0}Δ\(n^{\prime}_{{\text {c}}}\) remains constant, very intricate growth processes are theoretically equivalent.

### The Ψ′-function and the Cell Cycle

*f*

_{G1}(

*t*),

*f*

_{S}(

*t*) and

*f*

_{G2M}(

*t*), defined in this figure, are deduced from flow cytometric measurements of the DNA content [23]. According to Fig. 8, in the grown-up population about 95% of the cells are in the G1-phase, virtually no cells in the S-phase, and about 5% of the cells in the G2M-phase. The fraction

*f*

_{G1}(

*t*) and the total cell number (dotted line) run parallel, i.e.

*f*

_{G1}grows analogous to the (

*p*= 3)-Ψ′(

*ω*

_{c})-function (Eq. 26). During growth, the cell cycle distribution should then show the characteristics

When a culture experiment is started, and optimal process conditions are adjusted, the distribution is characterized by the parameters *ε* _{11}, *ε* _{12} and *ε* _{S} (*ε* _{G2M} = 1 − *ε* _{G12} − *ε* _{S}). In the grown-up colony *ε* _{G2M0} gives the fraction of cells in the G2M-phase. The solid lines in Fig. 8 are calculated with the use of Eqs. 19 and 27 and the parameters in Tables 3a,b and 8 (only *τ* _{kin} is slightly reduced: *τ* _{kin} = 2.5 × 10^{−6} h ⇒ *τ* _{kin} = 7 × 10^{−7} h).

During growth, the fraction of G1-cells, and possibly the transitions into the S-phase, are directly related to the phenomenon of relaxation-frequency-dispersion. On the biochemical level, cell cycle checkpoints with related regulatory proteins [27] are the means to translate the processes represented by the stress field into a frequency-dependent cellular response.

### Human Melanocytes

*τ*

_{kin}) always the same parameters (Table 9). In analogy to the precedent example of CHO-cell patterns,

*τ*

_{kin}is set to those individual values that are indicated with each curve. The conspicuous resemblance of the growth curves is a precondition to construct the master curve in Fig. 9b, c by shifting each data set along the time axis. In agreement with our model, rheological simplicity is again proven to typify the growth process of cell cultures.

### Murine Leukaemia Cells and *S. cerevisiae*

## Prokaryotic Cells

### *Escherichia coli*

*E. coli*population measured by Schlote [9]. The growth curve can be fitted with the use of Eqs. 19 and 24.

The inserts in Fig. 12 present the cell size distributions determined at the time points 3–6 along the growth curve [9]. The width of the distributions decreases in the course of time. Equation 9 allows us to reproduce these patterns, and to identify them as members of the universal Φ(*η*, *p* = 3)-class (Fig. 1).

*ß*Δ

*u*

_{0}(

*t*) fall below 0.023; compared with the mean thermal energy

*k*

_{B}

*T*the contact energy of ICs is relatively small. Hence, ever-present structural fluctuations support even complicated molecular rearrangements during relaxation.

*ß*Δ

*u*

_{0}(

*t*) usually drops within Δ

*t*≈

*t*

_{c}to optimized patterns with a minimal reaction energy, increasing then in correspondence with the growth process. Δ

*u*

_{0}(

*t*) is thus described by

The exponential term accounts for the rapid initial adaptation immediately after the experiment has been started. Afterwards, the density of constraints grows according to *A* _{0}(*t*), i.e. controlled by the relaxation-frequency-dispersion.

*y*> should decrease during growth. From Eq. 9 <

*y>*is deduced to be (7)

*y*>

_{min}/<

*y*>

_{max}to be equal to Δ

*u*

_{0max}/Δ

*u*

_{0min}. With the data of Fig. 13 we arrive at

The mean size of *E.* *coli* cells in a grown-up population is only about 37% of the initial value; these relatively small cells have to stay mainly in a G1-like state (see above). It seems that the initiation of DNA-replication is directly related to the conditions set up by the relaxation-frequency-dispersion.

## Concluding Remarks

Optimal growth of cell ensembles requires the balance of energy and entropy (according to Eq. 4). This implies a structure–dynamics interrelation as formulated in the increment model. The compulsory dynamics is perpetuated by thermally driven molecular rearrangements of the liquid-like fluctuating cytoplasm. As part of the controlled process of growth and cell division these rearrangements are tantamount to a relaxation process which enables optimization of the individual cell as well as of the whole cell ensemble. The universal validity of the (*p* = 3)-cell size distribution confirms this principal characteristics.

Our description of the growth of cell ensembles shows that complete co-ordination is accomplished under the following premises: with an individual cell size-dependent relaxation time, each cell acts as a largely independent visco-elastic element linked in an ensemble characterized by a universal relaxation mode distribution. All cells participate in an all-embracing signal–response system, always related to a thermodynamically defined state of reference, an appropriate situation specificity and an ample adaptability to complex global conditions of the ensemble.

Besides the various known biochemical processes which are governed by diffusion or active molecular translocation, intercellular communication might rely on the much faster acoustic signals as observed by Pelling et al. [12], emitted and perceivable by every growing cell in a population. The overlaid signals seem to be represented by a quasi-periodic field that allows for a collective response via relaxation. This relaxation occurs at the restricted rate of the molecular reconfigurations, while the signal sequence accelerates steadily with the growing cell number. The resulting phenomenon of relaxation-frequency-dispersion then becomes the central controlling and limiting mechanism of the growth process. It is important to note that it also enables the organization of cells in statistically equivalent optimized tissue-like structures. The process is identical in cells of all kinds as it always follows the same rules. The highly co-operative events are the first to be blocked. Therefore, they typify the topological features of the structure of every grown-up cell ensemble. However, less complex rearrangements take place under optimal conditions, still.

*p*= 3)-version of the Ψ′-function typifes the topologically uniform growth behaviour of cell ensembles of widely different species. The same laws should likely control the growth and development of higher, i.e. multicellular organisms, as well.

Hence, throughout growth of cell colonies or populations, the outlined relations define an invariable logistic framework that guarantees reproducible structural properties, installed under nearly optimal process conditions, independent of exogenous or endogenous factors. Different genetic patterns can thus develop on an individual as well as on an evolutionary scale. These results may allow us to extend the Taylor and Weibel principle [31]: “Biological structures are formed, *controlled by a general growth logistics* to meet but not to exceed maximum requirements”.

## Notes

### Acknowledgements

We have had stimulating discussions with H. Ringsdorf (Mainz) and H. Gruler (Ulm). We are grateful for getting valuable information by J.P. Spatz (MPI Metal Research, Stuttgart).

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