Biological Calls Interact Qualitative . . . Analyticity Attraction-Repulsion Forces Tidal Forces Scale Invariance Between Biological Cells: Definitions Main Results A Theoretical Explanation Proofs of Empirical Formulas Home Page Title Page Olga Kosheleva, Martine Ceberio, and ◭◭ ◮◮ Vladik Kreinovich ◭ ◮ University of Texas at El Paso Page 1 of 14 500 W. University El Paso, Texas 79968, USA Go Back olgak@utep.edu, mceberio@utep.edu, vladik@utep.edu Full Screen Close Quit
Biological Calls Interact 1. Biological Calls Interact Qualitative . . . Analyticity • Biological cells attract and repulse each other. Tidal Forces • For each type of cell, there is a certain distance R 1 at Scale Invariance which there is no interaction. Definitions Main Results • When r < R 1 , the cells repulse with the force Proofs � 1 � r − 1 = r def Home Page f = − k 1 · · e , where e r. R 1 Title Page • When r > R 1 , the cells attract each other with the ◭◭ ◮◮ force f = k 2 · ( r − R 1 ) · e . ◭ ◮ • As a result of these two forces, the cells stay at the same Page 2 of 14 – biologically optimal – distance from each other. Go Back • In this paper, we provide a theoretical explanation for Full Screen the above empirical formulas. Close Quit
Biological Calls Interact 2. Qualitative Requirements: Monotonicity Qualitative . . . Analyticity • We want to find the dependence f ( r ) of the interactive Tidal Forces force f on the distance r between the two cells. Scale Invariance • To find such a dependence, let us consider natural re- Definitions quirements on f ( r ). Main Results Proofs • The larger the difference between the actual distance Home Page r and R 1 , the larger should be the force. Title Page • So, the repulsion force should increase when the dis- tance r decreases. ◭◭ ◮◮ ◭ ◮ • The attraction force should increase as the distance r increases. Page 3 of 14 • It should be mentioned that the empirical formulas sat- Go Back isfy this property. Full Screen Close Quit
Biological Calls Interact 3. Analyticity Qualitative . . . Analyticity • All dependencies in fundamental physics are analytical Tidal Forces functions, i.e., can be expanded in Laurent series: Scale Invariance f ( r ) = a 0 + a 1 · r + a 2 · r 2 + . . . + a − 1 · r − 1 + a − 2 · r − 2 + . . . Definitions Main Results • In fundamental physics, this phenomenon is usually Proofs explained by the need to consider quantum effects: Home Page – quantum analysis means extension to complex Title Page numbers, and ◭◭ ◮◮ – analytical functions are, in effect, differential func- ◭ ◮ tions of complex variables. Page 4 of 14 • It is worth mentioning that both empirical formulas are analytical. Go Back Full Screen Close Quit
Biological Calls Interact 4. Tidal Forces Qualitative . . . Analyticity • The main objective of the forces between the two cells Tidal Forces are to keep the cells at a certain distance. Scale Invariance • There is also an undesired side effect, caused by the Definitions fact that cells are not points. Main Results Proofs • Different parts of the cell have slightly difference force Home Page acting on them. Title Page • We have tidal forces that make the parts of the cell move with respects to each other. ◭◭ ◮◮ ◭ ◮ • So, cells compress or stretch. Page 5 of 14 • In general, the tidal forces are proportional to the gra- = d f def Go Back dient of the force field F ( r ) dr. Full Screen • From the biological viewpoint, tidal forces are undesir- able, so they should be as small as possible. Close Quit
Biological Calls Interact 5. Scale Invariance Qualitative . . . Analyticity • Physical laws are formulated in terms of the numerical Tidal Forces values of physical quantities. Scale Invariance • However, these numerical values depend on what mea- Definitions suring unit we select to describe this quantity. Main Results Proofs • If we first measure distances in m, and then start using Home Page cm, then all the numerical values multiply by 100. Title Page • In particular, 2 m becomes 200 cm. ◭◭ ◮◮ • In most fundamental physical laws, there is no physi- ◭ ◮ cally preferred unit. Page 6 of 14 • It thus make sense to require that the physical law not depend on the choice of the unit. Go Back • If we change the unit of one of the quantities, then we Full Screen have to change the units of related quantities. Close Quit
Biological Calls Interact 6. Scale Invariance (cont-d) Qualitative . . . Analyticity • After an appropriate re-scaling of all the units, all the Tidal Forces formulas should remain the same. Scale Invariance • Scale-invariance of the dependence b = B ( a ) means Definitions that for every λ , there exists a µ ( λ ) such that: Main Results – if we change a to a ′ = λ · a and b to b ′ = µ ( λ ) · b , Proofs Home Page – the dependence remain the same: if b = B ( a ), then we should have b ′ = B ( a ′ ), i.e., µ ( λ ) · b = B ( λ · a ). Title Page ◭◭ ◮◮ • For the dependence f ( r ), there is no scale-invariance: there is a special distance R 1 (when the force is 0). ◭ ◮ • However, for the tidal force F ( r ), both F ( r ) ∼ r − 2 for Page 7 of 14 small r and F ( r ) = const for large r are scale invariant. Go Back Full Screen Close Quit
Biological Calls Interact 7. Definitions Qualitative . . . Analyticity • By a force function , we mean a function f ( r ) from pos- Tidal Forces itive numbers to real numbers. Scale Invariance • We say that a force function unction f ( r ) is analytical Definitions if it can be expanded in Laurent series Main Results f ( r ) = a 0 + a 1 · r + a 2 · r 2 + . . . + a − 1 · r − 1 + a − 2 · r − 2 + . . . Proofs Home Page • We say that a force function is monotonic-at-0 if for Title Page sufficiently small r , | f ( r ) | increases as r decreases. ◭◭ ◮◮ • We say that a force function is monotonic-at- ∞ if for sufficiently large r , | f ( r ) | increases as r increases. ◭ ◮ • By a tidal force function corresponding to the force Page 8 of 14 function f ( r ), we mean its derivative F ( r ) = d f dr. Go Back Full Screen • We say that a tidal force function is scale-invariant if Close ∀ λ > 0 ∃ µ ( λ ) ∀ r ∀ a ( a = F ( r ) ⇒ µ ( λ ) · a = F ( λ · r )) . Quit
Biological Calls Interact 8. Definitions (cont-s) Qualitative . . . Analyticity • Let f ( r ) be an analytical monotonic-at-0 force function Tidal Forces f ( r ), let F ( r ) be its tidal force function. Scale Invariance • We say that F ( r ) grows fast if there exists another Definitions analytical monotonic-at-0 force function g ( r ): Main Results – with scale-invariant tidal force function G ( r ), Proofs Home Page – for which F ( r ) G ( r ) → ∞ as r → 0. Title Page • Let f ( r ) be an analytical monotonic-at- ∞ force func- ◭◭ ◮◮ tion f ( r ), let F ( r ) be its tidal force function. ◭ ◮ • We say that F ( r ) grows fast if there exists another Page 9 of 14 analytical monotonic-at- ∞ force function g ( r ), Go Back – with scale-invariant tidal force function G ( r ), Full Screen – for which F ( r ) G ( r ) → ∞ as r → 0. Close Quit
Biological Calls Interact 9. Main Results Qualitative . . . Analyticity • Proposition 1. Every analytical monotonic-at-0 force Tidal Forces function f ( r ) Scale Invariance – for which the tidal force function F ( r ) is scale- Definitions invariant and does not grow fast, Main Results – has the form f ( r ) = c 0 r + c 1 for some c 0 and c 1 . Proofs Home Page • Proposition 2. Every analytical monotonic-at- ∞ Title Page force function f ( r ) ◭◭ ◮◮ – for which the tidal force function F ( r ) is scale- invariant and does not grow fast, ◭ ◮ – has the form f ( r ) = c 0 · r + c 1 for some c 0 and c 1 . Page 10 of 14 • These are exactly the empirical formulas that we Go Back wanted to explain. Full Screen • Thus, we have a theoretical explanation for these for- Close mulas. Quit
Biological Calls Interact 10. Proofs Qualitative . . . Analyticity • The tidal force function F ( r ) is scale-invariant: Tidal Forces F ( λ · r ) = µ ( λ ) · F ( r ) . (1) Scale Invariance Definitions • The function F ( r ) is analytical, thus smooth. Main Results • Thus, µ ( λ ) = F ( λ · r ) Proofs is smooth as the ratio of two F ( r ) Home Page smooth functions. Title Page • Differentiating both sides of (1) by λ and taking λ = 1, ◭◭ ◮◮ we get r · dF = dµ def dr = α · F , where α dλ | λ =1 . ◭ ◮ • Moving all terms with r to one side and all terms with Page 11 of 14 F to another, we get dF F = α · dr r . Go Back Full Screen • Integrating both sides, we get ln( F ( r )) = α · ln( r ) + C , for some integration constant C . Close Quit
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