mathematics

Kinetic logic

Kinetic logic is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Kinetic logic rather than just read about it. In short: Kinetic logic, developed by René Thomas, is a qualitative modeling approach feasible to model impact, feedback, and the temporal evolution of the variables. It uses symbolic descriptions and avoids continuous descriptions such as differential equations.

Key takeaways

Kinetic logic belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kinetic logic to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kinetic logic from memory before moving on to harder problems.

Reference excerpt

Kinetic logic, developed by René Thomas, is a qualitative modeling approach feasible to model impact, feedback, and the temporal evolution of the variables. It uses symbolic descriptions and avoids continuous descriptions such as differential equations. The derivation of the dynamics from the interaction graphs of systems is not easy. Many parameters must be inferred, for differential description, even if the type of each interaction is known in the graph. Even small modifications in parameters can lead to a strong change in the dynamics. Kinetic logic is used to build discrete models, in which such details of the systems are not required. The information required can be derived directly from the graph of interactions or from a sufficiently explicit verbal description. It only considers the thresholds of the elements and uses logical equations to construct state tables. Through this procedure, it is a straightforward matter to determine the behavior of the system.

Formalism

Following is René Thomas's formalism for Kinetic Logic : In a directed graph G = (V, A), we note G− (v) and G+ (v) the set of predecessors and successors of a node v ∈ V respectively. Definition 1: A biological regulatory network (BRN) is a tuple G = (V, A, l, s, t, K) where (V, A) is a directed graph denoted by G, l is a function from V to N, s is a function from A to {+, −}, t is a function from A to N such that, for all u ∈ V, if G+(u) is not empty then {t(u, v) | v ∈ G+(u)} = { 1, . . . , l(u)}. K = {Kv | v ∈ V} is a set of maps: for each v ∈ V, Kv is a function from 2G− (v) to {0, . . . , l(v)} such that Kv(ω) ≤ Kv(ω_) for all ω ⊆ ω_ ⊆ G−(v). The map l describes the domain of each variable v: if l (v) = k, the abstract concentration on v holds its value in {0, 1, . . . , k}. Similarly, the map s represents the sign of the regulation (+ for an activation, − for an inhibition). t (u, v) is the threshold of the regulation from u to v: this regulation takes place iff the abstract concentration of u is above t(u, v), in such a case the regulation is said active. The condition on these thresholds states that each variation of the level of u induces a modification of the set of active regulations starting from u. For all x ∈ [0, . . ., l(u) − 1], the set of active regulations of u, when the discrete expression level of u is x, differs from the set when the discrete expression level is x + 1. Finally, the map Kv allows us to define what is the effect of a set of regulators on the specific target v. If this set is ω ⊆ G− (v), then, the target v is subject to a set of regulations which makes it to evolve towards a particular level Kv(ω). Definition 2 (States): A state μ of a BRN G = (V, A, l, s, t, K) is a function from V to N such that μ (v) ∈ {0 .., l (v)} for all variables v ∈ V. We denote EG the set of states of G. When μ (u) ≥ t (u, v) and s (u, v) = +, we say that u is a resource of v since the activation takes place. Similarly when μ (u) < t (u, v) and s (u, v) = −, u is also a resource of v since the inhibition does not take place (the absence of the inhibition is treated as an activation). Definition 3 (Resource function): Let G = (V, A, l, s, t, K) be a BRN. For each v ∈ V we define the resource function ωv: EG → 2G− (v) by: ωv (μ) = {u ∈ G−(v) | (μ(u) ≥ t(u, v) and s(u, v) = +) or (μ (u) < t (u, v) and s (u, v) = −)}. As said before, at state μ, Kv (ωv(μ)) gives the level towards which the variable v tends to evolve. We consider three cases,

if μ(v) < Kv(ωv(μ)) then v can increase by one unit if μ(v) > Kv(ωv(μ)) then v can decrease by one unit if μ(v) = Kv (ωv (μ)) then v cannot evolve. Definition 4 (Signs of derivatives): Let G = (V, A, l, s, t, K) be a BRN and v ∈ V. We define αv: EG → {+1, 0, −1} by αv(μ) = +1 if Kv (ωv(μ)) > μ(u) 0 if Kv (ωv(μ)) = μ(u) −1 if Kv (ωv(μ)) < μ(u) The signs of derivatives show the tendency of the solution trajectories. The state graph of BRN represents the set of the states that a BRN can adopt with transitions among them deduced from the previous rules: Definition 5 (State graph): Let G = (V, A, b, s, t,K) be a BRN. The state graph of G is a directed graph G = (EG, T) with (μ, μ_) ∈ T if there exists v ∈ V such that: αv (μ) ≠ 0 and μ’ (v) = μ (v) + αv (μ) and μ (u) = μ’ (u), ∀u ∈ V \ {v}.

Critical Assumptions The critical assumptions of Kinetic Logic are:

The elements of system have slight effect on each other until they reach a threshold. At high levels the effect on each other tends to reach a plateau. So an element is present when greater than the threshold level and absent when it is below the threshold level.

Steps of application Following are the steps of Application of Kinetic Logic (Also shown in figure A).

Biological Regulatory Network (BRN) Keeping the research problem in mind, the behavior of elements in the system and their interactions are studied. Elements of a system can interact positively or negatively, that is, the level of an element may activate or reduce the rate of production of other elements or of itself. These interactions are represented as positive (activation) or negative (inhibition). When elements are connected in a topologically circular way, they exert an influence on their own rate of synthesis and they form a feedback loop. A feedback loop is positive or negative according to whether it contains an even or odd number of negative interactions. In a positive loop, each element of the system exerts a positive effect on its own rate of synthesis, whereas in a simple negative loop, each element has a negative effect on its own rate of synthesis. A simple positive feedback loop results in epigenetic regulation and have multiple steady states and a simple negative feedback loop results in homeostatic regulation. Abstraction: A chain of positive interactions is equivalent to a direct positive interaction between the two extreme elements, and any two negative interactions cancel out each other's effect. In this way, any simple feedback loop can be abridged to a one-element loop, positive or negative according to the number of negative interactions (even or odd) in the original loop. Accordingly, through extensive literature survey and the application of the above-mentioned rules, a BRN is abstracted.

… excerpt ends here. Continue reading the full article.

Illustrations

Kinetic logic: B. Sigmoid Curve — B. Sigmoid Curve

Kinetic logic: D. Naive Kinetic Logic — D. Naive Kinetic Logic

Worked examples

Example 1 — a first encounter with Kinetic logic

Start with the simplest possible case. Write down what Kinetic logic claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kinetic logic before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kinetic logic ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kinetic logic

In research: Kinetic logic appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kinetic logic in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kinetic logic is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical and theoretical biology, so understanding it makes those chapters shorter.
In everyday life: Look for Kinetic logic outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Kinetic logic in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kinetic logic means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kinetic logic out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kinetic logic in simple terms?

Why does Kinetic logic matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kinetic logic?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kinetic logic.