**Thoughts and Analysis**

This blog post is intended to be more of a collection of my personal thoughts, opinions and questions on the research area I am currently working on rather than a microbiologically correct treatise on the subject.

As a theoretical physicist, my microbiological insight will be rather limited and sparing at best; hence the conjectural and interrogative tone of this account.

Please note that this account is chronological and is being continuously updated. (Posts at the end of the account will be most up-to-date with findings compared to earlier ones.)

As you will eventually gather by reading my posts, I am working on building an Individual-based Model to try and predict bacterial growth and inactivation. For his part, my project partner, Sergey Goryunov, will be concentrating on the top-down approach of Population-based Modelling. We are working under the guidance of Prof. Peter Török and Dr Carl Paterson, as our supervisor and assessor respectively, as well as Dr Baranyi, from the Institute of Food Research, Norwich.

*On Inactivation*

Another aspect that needs to be taken into account in our model is the fact that the time scales over which inactivation happens depends on how the cells are dying off.*[Would this be an issue?]*

If thermal death is occurring, then (according to ComBase data), bacterial populations die off quickly.

** Edit: **Apparently, the Weibull Distribution addresses this well (different mechanisms + different time scales)…

- The Weibullian distribution applied to microbial survival curves still uses parameters (e.g. b) to characterise the shape of the curve being fitted to data.

- This was confirmed by using MatLab’s own Weibullian fit to ComBase data; some points at extremities were excluded from fit…

Different bacteria, different conditions of experiment à weibullian parameters will have to be adjusted yet again…

No distinct rule as to how this adjustment should happen…

- Several models to explain inactivation…especially its ‘finer’ features involving an ‘inactivatory lag’ (shoulder), activation of dormant spores, improved dispersion etc.

- It is easy to get lost in the various acrobatics which current mathematical models employ to accommodate for the above finer details…

However, it is important to begin capturing all these intricacies for incorporation into our own ‘generic’ model (which ideally should be able to tackle the full growth and inactivation spectrum). We believe it is crucial to move away from the tendency to use crude mathematical parameters (that characterise curvature arbitrarily for instance) and try to encode some form of microbiological meaning behind any parameters we use in the terms for each phase in the growth/inactivation function.

- Baranyi’s 1995 Smooth Combined Model[i] first scales the specific (growth/death) rates, µ, using their so-called L-Transformation function which uses a signum function:

- They define 2 turning points (T
_{1}and T_{2}), where growth kind of slows down and death starts to occur (the growth-death boundary).

They ensure a smooth transition in that boundary by using a smoothing function (if conditions for continuity and differentiability are not met).

- Specifically, the smoothing function, (and to some extent, our own proposed model) incorporates some of the components of Ratkowsky’s 1982 New Empirical Non-Linear Regression model[ii] (modification of the Arrhenius Law).

- Models such as the Ratkowsky model (as named by Zwietering[iii]) describe the
**growth rate**of bacteria as a function of temperature, whereas our proposed model looks at the actual population models.

It would be therefore good to ensure that our proposed model reconciles or is compatible with what appears to work well as per Zwietering.

*By ‘rogue cells’ is meant cells which deviate from normal behaviour, as observed for the bulk of the population…*

Friday 21^{st} Supervisor Meeting Update

*Peter has met with Dr Baranyi, who suggests that using the traditional population-based approach to describe bacterial evolution might not be the best way to tackle the problem (because of rogue cells for example). He also used the term ‘happy growth’, where cell division occurs optimally and predictably…*

*Peter suggested introducing confidence levels with predictions…*

*Peter suggested splitting up the work Sergey and I are doing so that we can focus on 2 slightly different angles and hopefully, towards the end, be in possession of two distinct approaches to explain the same problem, which would be good for verification purposes…*

*He recommended we look at work done by Prof. Kim Christensen, on the ‘Tangled Nature Model of Biological Evolution [iv]’…*

*He suggested Sergey carry on developing his mathematical model while I get in touch with Dr Baranyi and tackle the issue of discrepancies with the population based model…*

**‘Tangled Nature’ Model – A quick summary**

- Consideration of an emerging structure within ecology of co-existing microorganisms
- Consideration of the fact that extinctions can be local but can also affect the whole population
- All this ‘microscopic chaos’ synergises to produce a dynamical macroscopic structure that stabilises itself under the conditions it is subjected to…
- Talks of a mutation rate, which in this case, factors in somehow to reveal the various dynamical modes of the system…
- Evolutionary stages involves several quasi-stable periods which are separated by a notable chaotic phase, where, presumably critical microscopic changes are engaging to transform the structure of the system and give rise to a new dynamical macroscopic system.

By the way, the **‘Model of Punctuated Equilibrium’** [Existence and Death being ‘jerky’ processes whereby extinction happen in brief hectic periods…) of Bak and Sneppen[v], which attempts a different take on characterising the theory of (Darwinian) evolution, does not really apply to bacterial evolution (because ‘unfit’ cells might be dying for example, the ‘fittest’ are still multiplying as well as adapting to the new conditions imposed…).

Unless…bacterial lag phases are viewed as one of these ‘punctuations’ in bacterial evolution?

From what the work of Christensen at al. suggests, it seems that incorporating considerations of complexity, dynamical systems and chaos at some point might be inevitable if we are to give any kind of sensible microbiological basis to our proposed model. This would most probably and in part, include elements of Individual Based Modelling to describe the microscopic skeleton at work and maybe superimposing elements of macroscopic stochastic modelling to give form and flesh to this skeleton. However, most importantly, it will be the interactions between the discrete elements of this dynamical system that will need to be properly defined or modelled.

Of importance, is to decide whether we want to look at the variation of growth rates with temperature; or look at actual population levels.

Bear in mind that Baranyi’s 1995 paper applies the smoothing technique specifically to growth and death rates (and not the population levels).

Baranyi notes in his 1995 paper that thermal death is usually sigmoid or at least, generally non-linear. Sigmoid cases tend to imply that there are two strains of the same bacteria involved, one of which is more heat-resistant (or else, indicates a change in heat resistance of the bacterial population).

Different heating patterns seem to yield survival curves of different shapes…But also, apparently, identical heating patterns yielded slightly different curve characteristics e.g. Thermal Inactivation at 45°c, pH 7 and 0.5% NaCl sometimes yielded a curve having both shoulder and tail while other times, only the tail was observed.

Other than the above characteristics, sudden sharp non-exponential dips (sudden death of some cells) followed by exponential decreases were observed, as were curves with no tail whatsoever.

If anything the above seems to corroborate Baranyi’s suspicion of there being ‘rogue’ bacterial cells at work.

The above survival curves were fitted using the mirror image of the 1994 Baranyi Model[vi] [remember that the smoothing technique was only applied to investigate the growth rate – By the way, Baranyi’s Smoothing function contains similar features to the Ratkowsky Model 3 (as described by Zwietering), built for the same purpose of modelling the growth rate.]

So what we really are concerning ourselves here is to construct a mathematical model with a single equation (?) that describes the growth and inactivation stages; without having to resort to complicated transformations (e.g. Baranyi’s Mirroring and L Transformations etc.)

This is really what will motivate the need to incorporate IBM parameters/external factors in our model that can automatically take care of the different evolutionary stages. But can this be realistically done?

**Thursday 27 ^{th} January**

*Dr Baranyi replied yesterday! He addressed us with the word ‘colleagues’ – what an honour! Plus, he likes the term ‘rogue cells’!*

What he essentially said was that when it comes to determining the overall trend of the bacterial evolution curve (at least in terms of the growth rate), it seems that it is the characteristics from the ‘dominant’ subpopulation which have the upper hand. This appears to be similar to what goes on during Darwinian Evolution – a ‘survival of the fittest’ scenario involving determinants such as lag, resistance, adaptation, mutation etc.

However, he warns that while the behaviour and properties of rogue cells (e.g. dormant cells) might not matter during initial stages of growth/division (when the overwhelming majority of cells are happily multiplying), these very rogue cells might have a more noticeable effect during other stages of the bacterial evolution, e.g. during the survival phase, when they would exhibit signs of greater resistance compared to say, cells which have already divided.

He also highlights the ever-so-important role of the physiological state of the cells – which can have a heavy influence on the evolution curve. The fact of the matter is that we still have not been able to define a parameter in current models that can satisfactorily (while making microbiological sense) deal with the physiological state…

Dr Baranyi said he will be in London on the 17^{th} and 22^{nd} of February, and that he would be happy to talk more about this, and also discuss computational solutions and options…

**Friday 28 ^{th} Supervisor Meeting**

We will be meeting Dr Baranyi on February 17^{th} – time to be confirmed.

Peter has reiterated the fact that it’s probably best we work on different aspects of the problem. It turns out Sergey’s model is tending towards being a population based one, so Peter recommended I look into the Individual-based Modelling approach – also, I should try to contact the guys involved with Prof. Christensen regarding this.

So I should really be focussing on the cell itself – how it divides and evolves in its environment Eventually and hopefully, I may be able to gain some insight into what drives the ‘roguish’ behaviour of some cells – it would then be interesting to tie this information into Sergey’s Model with confidence levels to explain the phenomenon of rogue cells.

A good start as always is the paper on IbM by Clara Prats[vii] as well as contact Dr La Ragione at the University of Surrey for his microbiological expertise.

It might be worth starting with ‘Game of Life’ and/or ‘Cellular Automaton’ theory and adapting this to bacterial cellular evolution (?).

Found a review by Tom Portegys on ‘Game of Life’[viii] that mentions state spaces (as mentioned by Carl)…

Rather, let’s consider Prats’ IbM theory and INDISIM [Individual Discrete Simulator] – they appear to have had success with using INDISIM to various applications…

**Wednesday 2 ^{nd} Feb**

Some of the key elements that need to be addressed in our new IBM model:

- In Prats’ INDISIM Model, a cell is defined to be inactivated (reaching its limit of viability) when its mass falls below a certain threshold.

In the context of this, a cell needs to ‘process’ its own biomass to produce maintenance energy with which it ensures its own survival. If this becomes too ‘expensive’, then the cell will not have enough biomass to spare for reproduction purposes.

- The above then raises several questions. How do we determine this minimum mass? Is it well-established for the different types of bacteria? Is it arbitrarily decided? Does it depend on the physiological state of the cells?

Also, this leads to the question of how and why does a cell inactivate? (In this context, I guess inactivation implies ‘not reproducing’ as opposed to ‘dying’).

How do we account for cells which do have the necessary biomass to reproduce, but yet do not do so? [The ultimate ‘rogue’ cells which for instance, ‘decide’ to remain dormant during initial stages of population growth but then jump-start into action later on…]

*Edit: are these called ‘non-vegetative’ cells? (As opposed to bacterial spores)*

The above obviously implies that the individual biomass of cells is a dynamic quantity. Yet, there does seem to be a general trend which can be established for this quantity if the population as a whole is considered.

- Enzyme synthesis is an important factor to determine how bacterial cells adapt to a particular environment as well as to changes to this environment.

By the way, what follows from here has been tested and observed primarily in E. Coli.

*[Now reading the thesis by Clara Plats on IbM in the study of the Lag Phase. Also, from now on any analysis conducted will be done bearing in mind that the algorithmic logic for the model needs to sorted out and code eventually written, most probably in C++]*

**Thursday and Friday 3 ^{rd}/4^{th} Feb**

One approach that can be adopted is to investigate the pre-inoculation physiological state of the inoculum and then create individual rules for each bacterial ‘element/object’ based on that.

There might be some kind of a probability parameter involved – say, if cells were previously taken from an optimal environment, then, out of 50 cells, there will be a 95% chance of all of them multiplying. 5% of the cells can then be randomly selected to ‘inactivate’.

If taken from a stationary stage or less suitable environment, then the probability of division is much lower (we would expect a greater likelihood of dips in the lag phase for instance).

We could also do with a ‘shock factor’ that measures how different the pre- and post-inoculation environments are relative to one another. The greater the differences (as measured by water activity or pH differences for example), the bigger the shock factor.

This would then be correlated to the biological resistance of the particular bacterial strain (known) and the set of rules for the bacterial cells would then be modified accordingly.

Example:

Very resistant strain à greater uniformity in rules despite large shock factor

Low-resistance strain à more adjustments to rules with more limitations on biomass for example

We would then need to model and then check with experiment – the crucial thing here will be to obtain as much information as possible about the initial conditions of the experiment as well as the physiological history of the inoculum.

*Edit: Very naively, this physiological state of the cells might very simply correspond to a set of initial conditions affecting the inoculum [Temperature, phase of evolution, pH, nutrients available to them…]. Inocula could then probably be numbered in some way and this could form the basis of a very simple quantification system for the physiological state of the cells.*

A bit like chaos – small details at the start of the experiment would give significantly (and measurably) different evolution trends…which ultimately might lead to more insight into why some experiments feature dips in the lag phase and/or shoulders/tails in the survival curves…

One possible course of action from here on is to try and create such a scenario with cells in a grid (in matlab or c++), apply the rules to each ‘object’ with each iteration and from this, obtain a time evolution. Trends should then emerge as more and more cells or objects are simulated and hopefully, the most prominent features from the trends will match features observed from experimental plots of the growth of bacterial cultures and population-based models.

What this (might) offer is a set of microbiologically meaningful set of rules which cause some form of ‘self-organised criticality’ to emerge within the system. [A 1987 paper by Bak, Tang and Wiesenfeld[ix] has linked a simple cellular automaton to the phenomenon of self-organised criticality]

*Edit: Dr Baranyi also mentioned the idea of cellular automatons…*

Wikipedia says that *self-organised criticality** is a property of dynamical systems which have a critical point as an attractor.* *Their macroscopic behaviour thus displays the spatial and/or temporal **scale-invariance** characteristic of the **critical point** of a **phase transition**, but without the need to tune control parameters to precise values.*

*An attractor is a *

*set*

*towards which a*

*dynamical system*

*evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed.*

It might be worth investigating the development of resistance through mutation and bacterial sporulation.

*On Sporulation***[x]***:*

*Bacteria of the genera Bacillus and Clostridium can be found in two distinct states. In the vegetative state, the bacterium is metabolically active and uses available nutrients to grow and divide by binary fission, a process that generates two identical daughter cells. By contrast, when nutrients are scarce, a developmental program of endospore formation (sporulation) is initiated, resulting in the production of a highly resistant spore. In the spore state, the bacterium is metabolically dormant, and its genetic material, protected in the core of the spore, can endure a variety of challenges, including radiation, heat and chemicals. Sporulation is a complex process, which requires the generation of two distinct cell types: a forespore and a larger mother cell.*

**Friday 4 ^{th} Feb Supervisor Meeting**

Sergey talked to Peter about State Phases and MatLab modelling of this technique.

I told Peter of my thoughts (as outlined above) and he told me to start with the basics – basically begin constructing and coding a ‘happy growth’ IbM (in whatever language suits me). So need to start doing that now. **Create a grid, introduce cells, apply simple rules of growth, obtain graphs. Simple really.**

An idea which crossed my mind: maybe the roguish behaviour is fuelled through bacterial cell signalling mechanisms? For instance, a rogue cell might release some chemicals within the medium which diffuse and begin to influence other ‘susceptible’ cells into inactivating/adopting the same rogue behaviour.

But obviously, not all cells become affected by this, so there must some cells which are more susceptible to this influence than others. So what then defines this susceptibility?

Very simply, we can encode this phenomenon into a ‘probability’ of cells ‘defaulting’…

Peter also mentioned his plans on designing some kind of experiment to use fluorescent tagging techniques to track bacterial cells…

It would be worthwhile to understand how this tagging works, as this would be useful when it comes to tracking rogue cells and understand their behaviour.

**Development of the Individual-based Model in C++**

We’ll need to define the basic parameters which characterise the population dynamics. Some the parameters below will be initially fixed, or declared to be mean, constant values. But obviously, when we decide to depart from the ideal, these parameters will be allowed to vary. Keeping it simple to begin with, we’ll consider:

- Biomass
- Mean Mass for cell division to occur
- Mean Intra-cellular Enzyme [Enzyme will limit the nutrient use]
- Uptake Constant
- Mean Synthesis Rate per unit of biomass
- Nutrient particles per unit of biomass required for maintenance

**Rules and Calculations**

- Essentially, we will have the bacterium using its intracellular enzyme to synthesize the extracellular nutrient particles.

- The products of this synthesis will be the maintenance energy required to sustain the bacterium itself and new biomass generated.

- The biomass of the bacterium thus increases and if it exceeds a set threshold, the bacterium will then divide.

- Prats used a 300×300 grid of spatial cells to model the environment. This might be computationally expensive with our current limited resources, but we can always start with smaller grid sizes, say 50×50 instead.

- We will initially consider a
*homogeneous (equal biomasses) inoculum*of 100 bacterial cells.

- Prats specified a maximum number of bacterial cells per spatial cell of 4.

- In our model, we will assume that metabolic efficiency is 1, that is, every metabolised nutrient particle results in a new synthesized biomass unit.

- At the moment, we will not be considering maintenance energy requirements, for simplicity.

**Questions/Issues to think about**

- How is intracellular enzyme generated in a bacterial cell? Does the enzyme increase with biomass?

- Do large amounts of intracellular enzyme necessarily imply larger synthesis rates?

- Surely, nutrient uptake by the bacterial cell from its environment will start to decrease over time? It seems to make no sense for the cell to keep taking up nutrients when it can only ‘process’ a limited amount per unit time.

The uptake rate should therefore be linked to the synthesis rate, which itself might be changing if it depends explicitly on increasing biomass….

(But even if the synthesis rate remains constant, the uptake rate (should?) decrease…exponentially?)

**Friday 11 ^{th} Feb**

Crude Model compiling, graph obtained – simplistically, an exponential rise in population levels emerges…

A lot of improvements are now needed and at this point, it is crucial to start factoring in the correct microbiological behaviours and translate them in the best computational way possible so that we can as close as the real thing as possible, right at the start…

Consider a 300 by 300 grid of spatial cells with say (for ‘symmetry’), 9 bacteria at the centre of the grid. The individual ‘bacterium’ objects will then have values corresponding to their ‘physical addresses within the grid.

It will then be important to determine whether cell division and population growth occurs isotropically or anisotropically.

For instance, for the central bacteria, would there be reduced probability for it to divide because it is surrounded on all sides by other cells? (This might be the case, given that it would be more difficult for nutrients to reach the cell, let alone the cellular entrance sites.)

A paper on *Self-Organisation in High-Density Bacterial Colonies: Efficient Crowd Control* by HoJung Cho[xi] states that:

“Recent studies suggest that to cope with local environmental challenges, bacterial cells can actively seek out small chambers or cavities and assemble there, engaging in quorum sensing behaviour. The directions of orientation of cells, their growth, and collective motion are mutually correlated and dictated by the chamber walls and locations of chamber exits.”

As a simplified model, we could for example, declare that for ‘peripheral’ bacteria, new cells form in the ‘unoccupied space’ right next to the cells (instead of on the inside).

(Such considerations will become important later on when space for available growth becomes an issue).

As for the bacterium in the centre, it will not divide initially but will see its biomass increase much slowly compared to the rest, due to a significantly lower nutrient uptake.

Such ‘interior cells’ might even be subject to inactivation and eventual apoptosis if their maintenance requirements are not fulfilled.

Considerations of these aspects should eventually lead to shaping the lag phase of the growth.

**Monday 14 ^{th} Feb**

**Synthesis Rate and Uptake**

The simplest case would be the following equation applying to each time step:

But at this point in time, since we have assumed the uptaken nutrients to be constant, the grid and bacteria vectors are not really working alongside one another.

Picture a grid spatial cell with say 4 bacterial cells. Prats assumed 145,000 initial nutrient particles per spatial cell.

Let’s assume we begin with 100,000 nutrient particles. Now this seems to be a huge number so that the initial decrease in nutrient due to uptake by the 4 bacterial cells is almost negligible and the uptake can be declared to be constant.

Actually, when we are talking about ‘uptake’, we mean the number of nutrient particles absorbed per unit time step; so really it’s the uptake *rate*.

But even during the initial stages of growth, this uptake value will probably not hold for bacteria located at the centre of the inoculum…

What about the synthesis rate then? We’ll come back to this later…

Now, we need to know how to model the motion and uptake of the nutrient particles. The usual picture would be that of Brownian Motion (which would be strictly speaking, difficult to code and implement in a rigorous form; but using random number generators, one can take a ‘shortcut route’ to *simulating *Brownian motion…

But I am thinking that if you have a multitude of cells ‘absorbing’ nutrients from all sides, then surely this would perturb and introduce some disruption to the Brownian motion; imposing a kind of directional current to the motion of the nutrients?

This could be ‘simulated’ using a biased number generator in the vicinity of bacterial cells…?

Am I even making sense?

**Monday 14 ^{th} Feb Supervisor Meeting**

**Lifetime of cells, Available space for growth – **more important than nutrient availability?

Exposure to nutrient ‘space’ is really 3D in nature! Even the ‘central’ bacterium is surrounded with nutrients (from the top and bottom at least).

Has this been modelled in 3D before? With INDISIM?

**Apoptosis** is also known as ‘programmed cell death’ and is the way cells die naturally in a ‘controlled’ and regulated fashion as opposed to **Necrosis**, where cells die in an uncontrolled manner where lysis (cell break-down) occurs due to external, inflammatory conditions.

Apoptosis is a procedure whereby cells actually play a part in their own death – ‘cell suicide’.

We also have **Cellular Senescence** which loosely means “aging of the cell after its maturity due to gene expression changes or the accumulative damage of biological processes for example. Normal diploid cells for instance, would lose the ability to divide, normally after about 50 cell divisions in vitro.” *Update: this applies to eukaryotic cells only!*

Typically, “a cell that has accumulated a large amount of DNA damage, or one that no longer effectively repairs damage incurred to its DNA, can enter either senescence – an irreversible state of dormancy – or apoptosis.”

Some links on this:

http://www.scq.ubc.ca/apoptosis/

http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/A/Apoptosis.html

http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/E/Eubacteria.html

http://www.cellsalive.com/gallery.htm

At some point, we also need to consider **toxic waste products** as a limiting factor to growth!

*Update [C++ Model]*

I have modified the code to use a single vector and class instead of separately defining a bacterium and grid class.

Strictly speaking, all we need to describe population dynamics is just a ‘container’ to store the bacteria objects. But I am trying to implement things in a grid-like format; so we are able to visualise the evolution (say in Excel) and derive some sense of direction when it comes to growth.

At the moment, graphs being produced do tend to indicate a stage of exponential growth, but since I had specified the limitation of a 30×30 grid (in effect, maximum bacteria = 900), this exponential phase rapidly disappears to give some form of a stationary phase. At least, this somewhat factors in the element of availability of physical space to grow into.

Also, there is no lag phase as such as unrestricted binary fission has been assumed.

But the advantage of the grid layout (2D for the time being) means that we can impose certain conditions in terms of the probability for a particular bacterium to divide and if so, in which ‘direction’.

**Wednesday 16 ^{th} Feb – Meeting with Dr Baranyi**

*[Quick notes]:*

- Stochastic generation times, division not deterministic, a bit like radioactive decay?

- Gamma dist. Critical protein

- Apoptosis not really all that crucial during initial stages of growth – especially for prokaryotic cells (like bacteria)

- More likely to see inactivated cells transitioning into active cells during a dynamic temperature profile rather than the other way round…

- Fuzzy probabilistic region beyond 44

- Also, different responses depending on how we get to 44…abruptly/gradually?

- Way of implementing ‘cognitive’ behaviour to individual cells…

- Analogy of a canoe being ‘dropped’ into a river stream

If it is dropped in the same direction the river is flowing, then there is no problem, the trajectory of the canoe ‘grows’ naturally and instantaneously from that point in the direction the river is flowing.

If it is dropped at an awkward angle to the direction of flow, then the person will have to adopt a possibly complex route to steer the canoe and make it face the right direction into the flow.

In the context of bacterial growth, this would correspond to homogeneous and heterogeneous inocula…

**Sunday 20 ^{th} Feb – Tuesday 22^{nd} Feb**

Continued work on the IbM, mainly adding new restrictions and expanding the size of the grid/environment.

Some of the new conditions added include:

- A fixed restriction on the number of times a cell can divide (3).
*Edit: We do not really need this!* - A cell in the ‘interior’ of the culture can now divide (with equal probability) by pushing surrounding cells (in a random direction determined by the random number generator).
- The maximum biomass of a cell has also set at 30 000.
*Edit: We do not really need this!*

*Initial Observations*

Increasing the grid size, and thus, the available space for the bacterial cells to play in, yields a dataset that starts to closely resemble the typical bacterial growth curve, with a ‘clear’ exponential trend during the initial stages of growth (remember that no lag phase as such exists yet because lag conditions have not yet been imposed), with the graph curving towards some kind of an ‘artificial stationary phase’ towards the end, as space is exhausted.

I call this an artificial stationary phase because the only reason the population slows down its growth is purely because of the space limitations imposed. The traditional stationary phase of course primarily refers to microbiological mechanisms such as cell apoptosis kicking in and inhibiting population growth.

Next steps would now involve:

- Imposing a gamma distribution for the generation/cell division times
- Factoring in cell apoptosis
- Factoring in the fact that some vegetative cells do not divide despite being within the set limits of maximum permissible division count.
- In terms of considering growth proteins, consider the role of FtsZ…
- Also, does a cell start apoptosis because it has exhausted its ‘lifetime’ or is it really a case of lysis occurring due to their evolution in hostile or less-than-ideal environments?

[In this case, growth and death should be regarded as two processes happening simultaneously from the very start, at inoculation…]

- We need to finally settle the issue of exactly how the synthesis rate and uptake are involved/used in calculating the new biomass of a cell at each time step.

**But then, **if as Dr Baranyi mentioned, the generation times are really not constant at all but follow a gamma distribution, surely this would also pertain to the synthesis rate?

*Apparently:*

*“*Bacteria are immortal, at least in principle. If a bacterial cell is placed in favourable conditions it will grow and divide indefinitely.”

Cell division (as for prokaryotic cells) occurs as follows:

DNA Replication à Chromosome Segregation à Cytokinesis…

This process is much simpler than processes occurring in eukaryotic cells where complex mechanisms such as mitosis and meiosis take place.

*Interesting statement from a team in the US [xii] researching the Ageing of Cells, led by Dr Visick.*

“Aging E.coli?? *Most people don’t really think about bacteria aging. One “old” bacterial cell can divide to form two “young” ones, and (for nearly all species) it’s impossible to tell which the “original” cell was. Both daughter cells function normally, and each continues to divide as long as nutrients are available–in this sense, the bacteria are essentially “immortal!”*^{1}* However, cellular senescence does occur in bacteria*^{2}*: in stationary phase, when nutrients are limited and cells are dividing slowly or not at all, individual cells can lose their ability to function over time.*

Stationary phase *begins as a fast-growing bacterial culture (e.g., in a rich laboratory culture medium) begins to exhaust the available nutrients. Cell division slows and eventually ceases, and the number of cells in the culture stabilizes (see the left panel below). The bacteria reduce their metabolic rate and activate a variety of genes to enable them to survive nutrient limitation and cope with stresses such as heat and oxidation that they might encounter before the return of nutrients*^{3}*. Soon, the cell number begins to decrease, marking what was once referred to as “death phase”. More recently, however, we have realized that the cell number will eventually stabilize again and remain stable indefinitely, a state termed “long-term stationary phase” (right panel below). Although the number of cells changes little, this is actually a dynamic state in which some cells die, others can divide and mutants better equipped for survival can be selected.”*

Thoughts on the above: first, they describe cellular senescence as a process that typically affects bacterial cells (instead of conventional eukaryotic cells). But they describe it as being triggered by external factors rather than by an ‘internal bio-clock’ – in this case, they mention the lack of nutrients effectively causing the metabolic rate to slow down (does this correspond exactly to the synthesis rate?).

But as Peter mentioned, if in a realistic scenario, we are considering a raw piece of meat for instance, the amount of available of nutrients can be assumed to be large enough to max out population levels (This still says nothing about the *access* to the nutrients in an ‘overpopulated regime’).

Therefore, any senescence or inadaptability on the part of the cells we are modelling would come from other more important factors such as eventual lack of space and ‘shock effects’ as the inoculum gets transferred across environments – which brings us back to the open question of determining the physiological state of the cells.

The states of the cells are essentially what will ‘map out’ the ‘fitness levels’ of the cells and the gamma distribution of the generation times…[*which implies that we could essentially do away with random number generators if we are able to correctly quantify the physiological state of the cells…]*

We also need to understand **why** these times follow a gamma distribution.

[Could it really be that identical cells with the same physiological state and history and the same biomass decide arbitrarily whether or not to divide? Double-Slit Experiment style??

Apparently, there are ways of generating random numbers that have a gamma distribution.

But I am wondering about the ‘ethics’ of using random number generators to simulate bacterial behaviour…

The use of random number generators seems to be like applying a ‘stochastic envelope’ to the inoculum, thereby predefining trajectories in bulk rather than following each cell individually…is this right?

Back in 1994, Baranyi and Roberts established one form of a quantification of the physiological state of the cells through q(t) – the proportion of the critical substance required for division to a threshold value.

**Friday 25 ^{th} Feb Supervisor Meeting**

Things that need to be addressed:

- As per Carl’s advice, we need to write down a clear course of action in the form of a short abstract (or ‘table of contents’) of where exactly we stand with the project, what still needs to be addressed and what we want to achieve specifically by the end of the project.

- The restriction of the number of times a cell can divide should probably be lifted (or at least fixed to a much larger number). [Assuming, of course an unlimited supply of and access to nutrients…]

- There should also be no limit as such to the biomass of a cell – typically, cells should be continuously dividing, so there should not be the problem of the size of a vector element reaching ‘disproportionate levels’.

If for some reason, vegetative cells stop dividing (exhibit roguish behaviour), then by virtue of maintenance energy requirements (for example) and limited nutrient and space availability, and other extraordinary phenomena (such as cellular signalling) kicking in, the synthesis rate within such cells might start grinding to a complete stop and apoptosis might start kicking in.

These processes would on their own, limit the biomass of a cell.

- Also need to consider homogeneity or heterogeneity of the inoculum…

**MSci Viva set to be on the afternoon of Friday 25 ^{th} March.**

**High probability of Dr Baranyi attending.**

**References**

**[i]** Baranyi J., Jones A., Walker C., Kaloti A., Robinson T. P., Mackey B. M., 1995. A Combined Model for Growth and Subsequent Thermal Inactivation of Brochothrix Thermosphacta. Applied and Environmental Microbiology, Vol. 62, No. 3.

**[ii]** Ratkowsky D. A., Olley J., McMeekin T. A., Ball A., 1981. Relationship between Temperature and Growth Rate of Bacterial Cultures. Journal of Bacteriology,

**[iii]** Zwietering M. H., 1991. Modeling of Bacterial Growth as a Function of Temperature. Applied and Environmental Microbiology, Vol. 57, No. 4.

**[iv]**http://www.cmth.ph.ic.ac.uk/people/k.christensen/research/evolution.html

**[v]** Bak P., Sneppen K., 1993. Punctuated Equilibrium and Criticality in a Simple Model of Evolution. Physical Review Letters, Vol. 71, No. 24.

**[vi]** Baranyi J., Roberts T.A., 1994. A dynamic approach to predicting bacterial growth in food. International Journal of Food Microbiology 23, 277– 294.

**[vii]** Prats C., 2006. Individual-based Modelling of Bacterial Cultures to study the Microscopic Causes of the Lag Phase. Journal of Theoretical Biology 241 (2006) 939-953.

**[viii]** http://www.necsi.edu/events/iccs/openconf/author/papers/f356.pdf

**[ix]** Bak, P., Tang, C. and Wiesenfeld, K. (1987). “Self-organized criticality: an explanation of 1 / f noise”. Physical Review Letters 59: 381–384.

**[x]** Bacillus: Cellular and Molecular Biology. May 2007. Peter Graumanm, University of Freiburg, Germany. Caister Academic Press.

**[xi]** Cho H, Jönsson H, Campbell K, Melke P, Williams JW, et al. (2007). Self-organization in high-density bacterial colonies: efficient crowd control. PLoS Biol 5(11): e302.