Interpolation

My understanding is that interpolation typically involves reconstruction of a function over an entire region based on its values at a discrete / countable (not necessarily finite ??) set of sampled points. Here, the independent variable and the dependent variable are typically continuous. In our case, usually both the independent variable (the amino acid or the nucleotide base) and the dependent variable are typically discrete (though sometimes frequency / odds ratio of disease are somtimes used in lieu of continuous variables). Typically, the problem is looked at in terms of classification , i.e. determining the class biundaries. But again, intuituve interpretation of the models is difficult (even with frequency / odds approach). In the case of the dependet variable, to get more "resolution" , one could try to order the nucleotide bases based on some kind of substitution matriix which allows us to embed them on a real interval. Also, one could look at many sets of bases together , possibly looking at the above as a polynomial

Excuse me, sir. I do not uderstand your question. It is possible you may use 📷

nih.gov

https://blast.ncbi.nlm.nih.gov

Converting discrete models in biology to continuous models can be achieved through various mathematical techniques, such as differential equations, agent-based modeling, or stochastic processes.

For example, instead of just looking at the presence or absence of a particular gene or allele, we could use differential equations to model the dynamic changes in the concentration of different biomolecules over time. This would allow us to establish continuous trackers or markers that correlate with the propensity to contract a disease.

Similarly, morphological parameters such as fractal or scaling dimensions of tumors can also be modeled using continuous mathematics. For example, fractal dimensions can be calculated using techniques such as box-counting or the Richardson-Lucy algorithm, and can provide a continuous measure of the complexity of a tumor's structure.

Pathway analysis can also be used to identify key biomolecules and their interactions, which can then be incorporated into a continuous model. This can help to identify potential biomarkers for a disease or trait, and can provide insights into the underlying molecular mechanisms.

Overall, converting discrete models to continuous models in biology can provide a more intuitive, easy to understand, and extendable framework for analyzing complex biological systems.

In my (a bit crazy) opinion category theory provides a powerful framework for abstracting and generalizing mathematical structures across different fields, including biology. However, using category theory to convert discrete models in biology to continuous models requires a deep understanding of both category theory and the specific biological problem being modeled.

One approach to using category theory to convert discrete models in biology to continuous models is to use the concept of sheaf theory. In this approach, discrete data, such as gene expression levels, can be treated as a sheaf over a topological space, with the continuous data being the sections of the sheaf. The sheaf can be constructed using various techniques, such as cohomology or étale cohomology, and can provide a way to represent discrete data as a continuous function over a space.

Another approach is to use the concept of functors, which are maps between categories that preserve structure. In this approach, a functor can be used to map a discrete model to a continuous model, such as mapping a Boolean network to a system of differential equations. The functor can then be used to analyze the properties of the continuous model, such as stability and attractors, and to compare it with the discrete model.

Finally, the concept of adjunctions can also be used to connect discrete and continuous models. An adjunction is a pair of functors that are inverses of each other, and can be used to relate different mathematical structures. In the context of biology, an adjunction could be used to relate a discrete model, such as a genetic network, to a continuous model, such as a system of differential equations.

• Dear Agnieszka Matylda Schlichtinger , why You do not like difference equations?

Daniil ( Даниил ) Александрович Fedorov ( Федоров )

Why do you think I don't like difference equations?

Dear Agnieszka Matylda Schlichtinger , "Overall, converting discrete models to continuous models in biology can provide a more intuitive, easy to understand, and extendable framework for analyzing complex biological systems." Why difference equations is not good? Excuse me for my English, please.

Daniil ( Даниил ) Александрович Fedorov ( Федоров )

Difference equations can be useful for modeling discrete events in biological systems, such as the discrete stages of a cell cycle or the discrete events of a neural network firing. However, they may not be ideal for modeling continuous processes, such as the dynamics of a chemical reaction or the movement of a fluid through a network of vessels. In these cases, continuous models, such as differential equations or agent-based models, may provide a more accurate representation of the underlying biology.

One limitation of difference equations is that they assume that the changes in a system occur in discrete steps, and do not account for small changes that occur continuously over time. Additionally, they may not be able to capture the effects of small perturbations in the system, which can have a significant impact on the overall behaviour of the system.

Dear Frederic,

continuous and discrete models are equal in the fact that each of them has their pros and cons. I have to say thus firstly that I strongly disagree that continuous models are better. They are certainly not intuitive at all, since the numbers we are counting in for the longest time are discrete, and also for other reasons.

Necessarily correcting the basis, let's move on to the merit, which is Your question. Since it is hard, we have to first make it simple using the following spell:

Divide et Impera!

Into the three questions:

1) "So let us say, we are trying to find a biomarker for a disease \ trait. Instead of just looking at absence / presence or frequencies, could one try to establish continuous trackers / markers that positively or negatively correlate with the propensity to contract the disease , such as concentration levels of a chosen set of biomolecules ??"

Computer Models are good for modelling dynamics systems and experimenting with input parameters. BUT your research question is asking for statistical methods, since you are comparing two static distributions of gathered data and are looking for explaining their difference.

2) "Possibly, this may involve some kind of preliminary pathway analysis."

In statistical methods, there are good preliminary techniques such as ANOVA decomposition, where you can see statistical significance of the influence of individual factors (size, color, whatever) on the resulting distributions (in tumor vs in healthy kidney, for example)

3) "Could one also look at morphological parameters (fractal / scaling dimension of tumors etc ??) ??"

Yes, you could, but statistics works with numbers, so you need to quantify parameters.

Again divide your question into:

1) what is the nature of your data - for tumor histopathology or other image analysis you can use machine learning, or you can make a metric from the image, for example NUMBER of abnormal cells per one high power field.

2) Then you ask the question: is the data

categorial,

e. g. (apple, orange, or pear),

ordinal,

e. g. (first-smallest, second,..., last and largest, but by how much, nobody knows, since it is a ranking only),

interval,

(I know how big interval there is between, for example, degrees Celsius, but I cannot make ratios, cca twice as hot, since -20 is maybe two times hotter than -40, but 40 is not -1 times hotter than 40)

or rational,

(in which their ratio makes sense, ie. volume of 2 litres is 2 times larger than volume of 1 litre)?

For that, and all other statistical methods, consult a good statistician or his, or her, work.

And please remember, especially in the dawn of new machine-learning based false idols, that only there is wrong and good, good and evil, nothing between, unless proven otherwise. Until then:

"All models are wrong, but some are useful" - George Box, statistician

Your sincerely,

Faustus Tuus, Jan Mičan, Medicinae Universeae Doctor

Laboratoriorum Johanni Josephi Loschmidti

Facultatis Scientium et RECETOXi

Universitatis Masarykianae Brunensis

Philosophiae Doctorandus, sic, Studentus Humilis