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To begin this question, I will quote Molecular Biology of the Cell (page 38):

... Biological systems are, ..., full of feedback loops, and the behavior of even the simplest of systems with feedback is remarkably difficult to predict by intuition alone; small changes in parameters can cause radical changes in outcome. To go from a circuit diagram to a prediction of the behavior of the system, we need detailed quantitative information, and to draw deductions from that information we need mathematics and computers.

... You might think that, knowing how each protein influences each other protein, and how the expression of each gene is regulated by the products of others, we should soon be able to calculate how the cell as a whole will behave, just as an astronomer can calculate the orbits of the planets, or a chemical engineer can calculate the flows through a chemical plant. But any attempt to perform this feat for anything close to an entire living cell rapidly reveals the limits of our present knowledge. The information we have, plentiful as it is, is full of gaps and uncertainties. Moreover, it is largely qualitative rather than quantitative.

(Johnson, A. D., Roberts, K., Lewis, J., Morgan, D., Raff, M. C., Walter, P., Alberts, B. (2015). Molecular Biology of the Cell. United States: Garland Science, Taylor and Francis Group.)

Thus comes a fundamental question: what does it mean when cell biologists want to quantitatively describe a cell?

In my understanding, a cell is a complex system, and a quantitative description of it invokes treatment of the cell as a mechanistic, mathematically determinate system. For example, a projectile in the context of Newtonian physics is a product of such treatment. Physicists use a series of equations to represent motion of the projectile. They also plug in initial values such as velocities and positions to supplement the equations.

https://dr282zn36sxxg.cloudfront.net/datastreams/f-d%3A032a554642a41232285e6e13aba720cece7b3c87d0e78f2a90ddb95b%2BIMAGE_TINY%2BIMAGE_TINY.1

As an analogy, quantifying a cell might mean finding a series of mathematical expressions for every chemical reaction or every biochemical inside a cell, and using a set of initial conditions to calculate the overall state of the cell at its very beginning. In this way, cell biologists can theoretically calculate figures of interest for whatever chemical reaction at any point during the cell's life. Of course, such an effort would be astronomically large and complex, and not feasible at present.

However, if this is what quantification of a cell means, why do Bruce Alberts and other authors cite feedback loops as a hurdle that stands in the way of quantifying a cell? How do feedback loops affect quantification of a cell?

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  • $\begingroup$ I don’t think cell biologists have or ever had any interest in “quantifying“ cells, whatever that may mean — and it will certainly mean little to people describing themselves as cell biologists. By definition and the origin of the term, these are people whose interest lies not at the molecular or chemical level, but in the behaviour of those components visible through a microscope. The book section quoted is at best an apologia for the non-molecular or numerical nature of the discipline. It worst it is pretentious nonsense and in any case it is best ignored. $\endgroup$
    – David
    Jul 28 at 10:22
  • $\begingroup$ The 2017 article "The quantified cell" ( molbiolcell.org/doi/full/10.1091/mbc.e14-09-1347 ) seems appropriate to cite here $\endgroup$
    – timeskull
    Jul 28 at 18:38
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Quantitative in the context of biology is similar to chemistry, and means "how much of something there is" - for example, how much of a particular protein is produced under what conditions.

Now, you might think this is a simple problem, just measure the protein/RNA/DNA and find out. However, it isn't quite as simple as that. Even if the best methods we have were truly quantitative (e.g. see Western blot quantitation for measurement of protein changes), which a lot of them aren't, the changes you see in one cell type might be different in another cell type (or even between cell lines of the same type of cell) and certainly between species.

You then have to consider - how well does a lab strain of something (say a bacterium like Escherichia coli or a cell line or cultured virus) differ from those found in the wild, and how does that vary geographically.

Life consists of huge numbers of feedback loops - stimulus results in action, which removes stimulus, which returns to baseline, allowing stimulus to function again. You experience this all day, every day of your life. It is how things like nerves function for example. In almost all situations it is not a single input results in a single action, rather that multiple inputs result in multiple outcomes, meaning an interconnected web of feedback loops. Have a look at this figure for a well known pathway and well studied, Caspase 3 signalling, from this article (I have no affiliation to the study):

enter image description here

This is one of the simple ones, which results in just one form of cell death - notice how many inputs there are and all the players. In general we can model 2-3 part systems fairly easily, so to work this out, each individual part of it needed to be studied and worked out empirically, determining which parts interact with which and why they do that under what stimuli.

Go and have a play at string.db with something like the well known and heavily studied tumour suppressor p53 (hit search, type in p53, choose human (Homo sapiens), then hit continue). Keep clicking the "more" button on the results page, and it will give you some idea of just how complex this is. Similarly, have a look at a metabolic pathways chart - this happens in every single cell.

We cannot currently reduce life to equations.

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  • $\begingroup$ Even more fun: many of the feedback loops share the same intermediate signals so that when one loop is running, it implicitly modifies the behavior of the other loop. The biggest difference between biological and engineered systems is that biological systems have absolute no design constraint for comprehensibility. Evolution can and will build ridiculously complex systems that stretch belief as to how they can possibly work, yet somehow they do or they wouldn't have been selected for. $\endgroup$
    – Dan Bryant
    Jul 28 at 13:41
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Quantitative vs. qualitative
Quantitative description usually implies numbers, rather than just saying which response follows which stimulus. For example, describing a chain of biochemical transformations is qualitative description - we know what processes happen, in which order, etc. Quantifying how many of the molecules would actually undergo this transformation up to a certain step (or to the very end) would be an example of a quantitative description.

Types of quantitative description
Quantitative description usually implies doing some math to relate the input and the output parameters, although the origin of the equations can be different. One typically distinguishes:

  • first principles description - when the model is based on the fundamental laws of chemistry and physics;
  • phenomenological description - when one uses the laws attested experimentally.

Examples of quantitative description of a cell
One should not really be looking for one fits all sizes mathematical model. Rather, the mathematical/quantitative description is usually tailored to a problem in hand. I give below a few examples, but this list is by no means exhaustive:

  • elastic properties - one can use elasticity theory to describe the deformations of a cellular membrane in response to pressure, attempts to pierce it, etc. In the former case one may be able to provide the mathematical expressions for the shape of the cell in response to specific level of pressure, in the latter one aims at determining the breakdown point
  • transport of chemicals/molecules - most of molecules absorbed by a cell or produced in it, travel to their destinbation via diffusion or with convection flows. Such flows can be modeled using diffusion equation and hydrodynamics equations respectively. Alternatively, if we are dealing with more directed transport chains, various random walk models become of great use. This allows to extimate the efficiency and the speed with which chemicals are transported to their destination.
  • transcription/translation are routinely described quantitatively, e.g., using rate equations models for the transcript/protein molecular growth.
  • Protein and RNA folding are a domain where thermodynamically motivated energy minimization has been of use for many decades.
  • Cellular dynamics is often modeled using non-linear dynamics, e.g., an HIV infection goes through a few phases: early on the auxhiliary proteins for reaction catalysis and interfering with the cellular machinery are produced, which is followed by synthesis of the essential proteins and then the new viral genomes. The transition between different stages is triggered by concentration of earlier synthesized proteins, and is nicely modeled by non-linear equations similar to Lottka-Volterra type. The feedback loops mentioned in the OP are often studied using non-linear dynamics.

One could go on further, to propose quantitative models for describing optical, thermal, chemical, replication, growth, and other processes - whatever is of interest. I suggest consulting the books I cite below for information on more specific topics.


References

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Let’s start by considering a very simple feedback loop, namely a system with just two genes A and B which repress each other. Such a system can be in two possible states (it is bistable):

  • A is expressed and thus suppresses B, which is not or hardly expressed.
  • B is expressed and thus suppresses A, which is not or hardly expressed.

We may perfectly understand this system on the biochemical and genetic level, which allows us to quantify the two above states. However, without further information about the system’s history (initial state), we cannot say in which state the system will be at a given time. Moreover depending on the strength of the expression it may be that the system occasionally switches between the two states just due to noise.

(Note that for the above bistability, a second ingredient is needed, namely non-linearity, but we get that from the thermodynamics of chemical reactions and genetic expression anyway.)

Now, in a cell you have a myriad of such feedback loops intertwined, which grants you a bazillion potential states to consider through combinatorial explosion. The challenge is now to figure out which of these states are generally and historically plausible. This is horribly difficult even when you perfectly understand all the individual molecular processes. Note that through the complexity you have more than just binary on and off states like in my introductory example, but a fine spectrum of potential states which gets fully continuous through molecular noise – thus affecting quantification.

A helpful analogy might be a modern computer running several interacting programs compiled with a highly efficient compiler. You can perfectly understand single operations on the level of a transistor and similar fundamental components, but this doesn’t get you very far with understanding the programs that are running. Also, to even remotely predict the computer’s behaviour, you need its history, i.e., what is currently stored in memory.

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