1. Totally Positive Differential Systems: A Tutorial and New Results
Michael Margaliot, Tel Aviv University
Joint work with:
Eduardo D. Sontag (Northeastern University)

2. Outline
Motivation: the ribosome flow model (RFM), and stability analysis of nonlinear, tridiagonal, cooperative systems (Smillie 1984, Smith 1991,…).
Totally nonnegative (TN) and totally positive (TP) matrices (Schoenberg, Gantmacher, Krein, Karlin,…)
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
Linear totally positive differential systems (TPDSs) (Schwarz 1970)
Putting it all together

3. Motivation: Ribosome Flow Model (RFM)
site 1 is completely free; site 1 is completely full.
controls the transition rate from
site i to site i+1.

4. Motivation: Ribosome Flow Model (RFM)
site 1 is completely free; site 1 is completely full

5. Ribosome Flow Model (RFM)
Write this as The Jacobian is:
𝑱 𝒙 ≔ 𝝏 𝒇 𝟏 𝒙 𝝏 𝒙 𝟏 𝝏 𝒇 𝟏 𝒙 𝝏 𝒙 𝟐 𝝏 𝒇 𝟏 𝒙 𝝏 𝒙 𝟑 𝝏 𝒇 𝟐 𝒙 𝝏 𝒙 𝟏 𝝏 𝒇 𝟐 𝒙 𝝏 𝒙 𝟐 𝝏 𝒇 𝟐 𝒙 𝝏 𝒙 𝟑 𝝏 𝒇 𝟑 𝒙 𝝏 𝒙 𝟏 𝝏 𝒇 𝟑 𝒙 𝝏 𝒙 𝟐 𝝏 𝒇 𝟑 𝒙 𝝏 𝒙 𝟑 .

6. Ribosome Flow Model (RFM)
Write this as The Jacobian is:
𝑱 𝒙 = − 𝝀 𝟎 − 𝝀 𝟏 (𝟏− 𝒙 𝟐 ) 𝝀 𝟏 𝒙 𝟏 0 𝝀 𝟏 (𝟏− 𝒙 𝟐 ) − 𝝀 𝟏 𝒙 𝟏 − 𝝀 𝟐 (𝟏− 𝒙 𝟑 ) 𝝀 𝟐 𝒙 𝟐 0 𝝀 𝟐 (𝟏− 𝒙 𝟑 ) − 𝝀 𝟑 −𝝀 𝟐 𝒙 𝟐 .
A tridiagonal matrix with nonnegative entries on the super- and sub-diagonals.

7. Motivation
Smillie (1984) studied nonlinear systems in the form: 𝒙 =𝒇 𝒙 , 𝒙(𝒕)∈ 𝑹 𝒏 , where the Jacobian: 𝑱 𝒙 ≔ 𝝏 𝒇 𝟏 (𝒙) 𝝏 𝒙 𝟏 ⋯ 𝝏 𝒇 𝟏 (𝒙) 𝝏 𝒙 𝒏 ⋮ ⋱ ⋮ 𝝏 𝒇 𝒏 (𝒙) 𝝏 𝒙 𝟏 ⋯ 𝝏 𝒇 𝒏 (𝒙) 𝝏 𝒙 𝒏 is: (1) tridiagonal; and (2) strongly cooperative.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

8. Motivation
𝒙 =𝒇(𝒙) with the Jacobian 𝑱 𝒙 ≔ 𝝏 𝒇 𝟏 (𝒙) 𝝏 𝒙 𝟏 ⋯ 𝝏 𝒇 𝟏 (𝒙) 𝝏 𝒙 𝒏 ⋮ ⋱ ⋮ 𝝏 𝒇 𝒏 (𝒙) 𝝏 𝒙 𝟏 ⋯ 𝝏 𝒇 𝒏 (𝒙) 𝝏 𝒙 𝒏 tridiagonal: 𝒇 𝒊 𝒙 depends only on 𝒙 𝒊−𝟏, 𝒙 𝒊, 𝒙 𝒊+𝟏 strongly cooperative: 𝝏 𝒇 𝒊 (𝒙) 𝝏 𝒙 𝒊−𝟏 , 𝝏 𝒇 𝒊 (𝒙) 𝝏 𝒙 𝒊+𝟏 >𝟎
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
𝒙 𝒊 =

9. Motivation
𝒙 =𝒇(𝒙) with a tridiagonal and strongly cooperative Jacobian. Smillie’s Theorem (1984): every trajectory either leaves every compact set or converges to an equilibrium. Many applications in models from systems biology, dynamical neural networks, neuroscience, …
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

10. Motivation
There are also several generalizations. Theorem (Smith, 1991) Consider 𝒙 =𝒇(𝒕,𝒙) that is tridiagonal, strongly cooperative, and T-periodic (𝒇(𝒕,𝒙)=𝒇(𝒕+𝑻,𝒙)). Every trajectory either leaves every compact set or converges to a T-periodic solution.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

11. Motivation
Theorem (Smith, 1991) 𝒙 =𝒇(𝒕,𝒙) T-periodic, tridiagonal, and strongly cooperative. Every trajectory either leaves every compact set or converges to a T-periodic solution.
T-periodic control
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
𝒙 =𝒇(𝒖,𝒙) 𝒖
The system entrains to the periodic excitation.

12. Entrainment in the RFM
Here n=3,

13. Smillie’s Proof Let 𝝈 𝒗 denote the number of sign variations in 𝒗 e.g.
𝝈 𝟏 −𝟏 𝟐 =𝟐.
Consider 𝒙 =𝒇 𝒙 . Let 𝒛:= 𝒙 =𝒇(𝒙). Then 𝒛 = 𝝏 𝝏𝒙 𝒇 𝒙 𝒙 =𝑱 𝒙 𝒛. Thus, 𝒛(𝒕) satisfies a “linear time-varying differential equation”.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
Smillie used 𝝈 𝒛(𝒕) as an integer-valued Lyapunov function.

14. Smillie’s Proof
Let 𝒛:= 𝒙 =𝒇(𝒙). Then 𝒛 =𝑱 𝒙 𝒛. Proposition 𝝈 𝒛(𝒕) is non-increasing. Proof. Assume 𝒏=𝟑. Suppose that:
𝒕 𝟎 +
𝒕 𝟎
𝒕 𝟎 − +
𝒛 𝟏 (𝒕) -
𝒛 𝟐 (𝒕)
𝒛 𝟑 (𝒕)
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
𝝈 𝒛 𝒕 =𝟎
𝝈 𝒛 𝒕 =𝟐

15. Smillie’s Proof
𝒕 𝟎 +
𝒕 𝟎
𝒕 𝟎 − +
𝒛 𝟏 (𝒕) −
𝒛 𝟐 (𝒕)
𝒛 𝟑 (𝒕)
But 𝒛 =𝑱 𝒙 𝒛 gives 𝒛 𝒕 𝟎 = ∗ ∗ ∗ + 𝟎 + = ∗ + ∗ , so the case described in the table above cannot take place.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

16. Smillie’s Proof Smillie then showed that the behavior
of 𝝈 𝒛(𝒕) implies that for any 𝒂 the omega limit set of 𝒙 =𝒇 𝒙 , 𝒙 𝟎 =𝒂, cannot contain two different points.
Smillie’s proof is based on direct analysis of 𝒛 =𝑱 𝒙 𝒛. This requires iterated differentiations, so he assumed 𝒇 𝒙 ∈ 𝑪 𝒏−𝟏 . Our goal: link this to a classical result from linear algebra: the sign variation diminishing property of totally nonnegative matrices.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

17. Outline
Motivation: Stability analysis of nonlinear, tridiagonal, cooperative systems (Smillie 1984, Smith 1991,…).
Totally nonnegative (TN) and totally positive (TP) matrices (Schoenberg, Gantmacher, Krein, Karlin,…)
Linear totally positive differential systems (TPDSs) (Schwarz 1970)
Putting it all together
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

18. Totally Nonnegative (TN) and Totally Positive (TP) Matrices
A minor is the det( submatrix). has minors. A is called TN [TP] if all its minors are nonnegative [positive].
𝑘×𝑘
det =−6.
𝑘=1 𝑛 𝑛 𝑘 2
𝐴∈ 𝑅 𝑛×𝑛
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

19. Totally Nonnegative (TN) and Totally Positive (TP) Matrices
A is called TN [TP] if all its minors are nonnegative [positive]. In particular, all entries (=1x1 minors) are nonnegative [positive], so we are in the realm of Perron-Frobenius theory. But TN [TP] matrices have a much richer structure.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

20. Totally Nonnegative (TN) and Totally Positive (TP) Matrices
Example: All 1x1 minors are positive; All 2x2 minors are positive, e.g. The 3x3 minor is Thus, 𝐴 is TP. A=
det =0.101;
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
det =1.

21. Totally Nonnegative (TN) and Totally Positive (TP) Matrices
The sum of two TN [TP] matrices is not necessarily TN [TP]. The product of two TN [TP] matrices is TN [TP]. This follows from the Cauchy-Binet formula: every minor(AB) is a sum of products: minor(A)*minor(B). For example, det(AB)=det(A)*det(B).
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

22. Sign Variation Diminishing Properties of TN and TP Matrices
A very important fact: Multiplying a vector by a TP matrix cannot increase the number of sign variations in the vector.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

23. Number of Sign Variations in a Vector
Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. For a vector 𝒙∈𝑹 𝒏 with no zero entries: 𝝈 𝒙 ≔ 𝒊: 𝒙 𝒊 𝒙 𝒊+𝟏 <𝟎 . For example, 𝝈 𝟏 −𝟏 𝟐 =𝟐.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

24. Number of Sign Variations
The domain of definition of 𝝈 ∙ is: 𝐕≔{𝒙∈ 𝑹 𝒏 | 𝒙 𝟏 ≠𝟎, 𝒙 𝒏 ≠𝟎, and if 𝒙 𝒊 =𝟎 then 𝒙 𝒊−𝟏 𝒙 𝒊+𝟏 <𝟎}. For example, 𝝈 −𝟏 𝝐 𝟐 =𝟏 for any 𝝐∈𝑹.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

25. Two More Definitions for the Number of Sign Variations
For a vector 𝒙∈𝑹 𝒏 : 𝒔 − 𝒙 ≔𝝈 𝒙 after deleting all zero entries in 𝒙; 𝒔 + 𝒙 ≔𝐦𝐚𝐱{𝝈 𝒚 }, where 𝒚 is obtained by replacing every zero entry in 𝒙 by either -1 or +1.
Example:
𝒔 − 𝟏 𝟎 𝟐 =𝟎, 𝒔 + 𝟏 𝟎 𝟐 =𝟐.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
Note that 𝒔 − 𝒙 ≤ 𝒔 + 𝒙 , and
that 𝒔 − 𝒙 , 𝒔 + 𝒙 ∈{𝟎,𝟏,…,𝒏−𝟏}.

26. Relation Between Definitions of the Number of Sign Variations
For a vector 𝒙∈𝑹 𝒏 : 𝒔 − 𝒙 ≔𝝈 𝒙 after deleting all zero entries in 𝒙; 𝒔 + 𝒙 ≔𝐦𝐚𝐱{𝝈 𝒚 }, 𝒚 obtained from 𝒙 by replacing every 0 by -1 or +1.
Domain of definition of 𝝈 ∙ :
𝐕≔{𝒙∈ 𝑹 𝒏 | 𝒙 𝟏 ≠𝟎, 𝒙 𝒏 ≠𝟎,
if 𝒙 𝒊 =𝟎 then 𝒙 𝒊−𝟏 𝒙 𝒊+𝟏 <𝟎} = 𝒙∈ 𝑹 𝒏 | 𝒔 − 𝒙 = 𝒔 + 𝒙 .
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

27. Relation Between Definitions of the Number of Sign Variations
The domain of definition of 𝝈 ∙ is: 𝐕≔{𝒙∈ 𝑹 𝒏 | 𝒙 𝟏 ≠𝟎, 𝒙 𝒏 ≠𝟎, and if 𝒙 𝒊 =𝟎 then 𝒙 𝒊−𝟏 𝒙 𝒊+𝟏 <𝟎}. For example, −𝟏 𝟎 𝟐 ∈𝑽, and 𝒔 + −𝟏 𝟎 𝟐 = 𝒔 − −𝟏 𝟎 𝟐 .
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
O𝐧 𝐭𝐡𝐞 𝐨𝐭𝐡𝐞𝐫 𝐡𝐚𝐧𝐝, 𝟏 𝟎 𝟐 ∉𝑽, and
𝒔 − 𝟏 𝟎 𝟐 < 𝒔 + 𝟏 𝟎 𝟐 .

28. Variation Diminishing Properties of TN and TP Matrices
Theorem: If 𝑨∈𝑹 𝒏×𝒏 is TP then for any 𝒙∈ 𝑹 𝒏 ∖ 𝟎 : 𝒔 + 𝑨𝒙 ≤ 𝒔 − 𝒙 .
Example: 𝑨= 𝟐 𝟏 𝟒 𝟑 is TP. Take 𝒙= −𝟏 𝟏 . Then 𝒔 − 𝒙 =𝟏 and
𝒔 + 𝑨𝒙 = 𝒔 + 𝟐 𝟏 𝟒 𝟑 −𝟏 𝟏 = 𝒔 + −𝟏 −𝟏 =𝟎.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

29. Variation Diminishing Properties of TN and TP Matrices
Theorem: If 𝑨∈𝑹 𝒏×𝒏 is TP then for
any 𝒙∈ 𝑹 𝒏 ∖ 𝟎 : 𝒔 + 𝑨𝒙 ≤ 𝒔 − 𝒙 .
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
Question: How is all this related to Smillie’s Theorem?

30. Outline
Motivation: Stability analysis of nonlinear, tridiagonal, cooperative systems (Smillie 1984, Smith 1991,…).
Totally nonnegative (TN) and totally positive (TP) matrices (Schoenberg, Gantmacher, Krein, Karlin,…)
Linear totally positive differential systems (TPDSs) (Schwarz*, 1970)
Putting it all together
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

31. Totally Positive Differential Systems (Schwarz, 1970)
Consider 𝒙 =𝑨𝒙. A𝐬𝐬𝐮𝐦𝐞 𝐞𝐱𝐩 𝑨𝝐 is TP for all 𝝐>𝟎 sufficiently small. Then
𝒙 𝒕+𝝐 =𝐞𝐱𝐩 𝑨𝝐 𝒙 𝒕
yields 𝒔 + 𝒙 𝒕+𝝐 ≤ 𝒔 − 𝒙 𝒕 .
Q1: What does this mean for 𝒙(𝒕)?
Q2: When is 𝐞𝐱𝐩(𝑨𝝐) TP?
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

32. Totally Positive Differential Systems (Schwarz, 1970)
Theorem 1. Consider 𝒙 =𝑨𝒙, 𝒙 𝟎 ≠𝟎, with 𝐞𝐱𝐩 𝑨𝝐 TP for all 𝝐>𝟎 sufficiently small. Then
(𝟏) 𝒔 + 𝒙 𝒕 , 𝒔 − 𝒙 𝒕 are nonincreasing;
(2) 𝒙 𝒕 ∈𝑽 for all 𝒕 except perhaps for up to 𝒏−𝟏 points.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
𝝈(𝒙(𝒕)) 𝑡

33. Totally Positive Differential Systems (Schwarz, 1970)
𝒙 =𝑨𝒙, 𝒙 𝟎 ≠𝟎, with 𝐞𝐱𝐩 𝑨𝝐 TP for all 𝝐>𝟎 sufficiently small.
I𝐝𝐞𝐚 𝐨𝐟 𝐏𝐫𝐨𝐨𝐟. 𝐏𝐢𝐜𝐤 𝒕 𝟎 <𝒕. Then 𝒔 + 𝒙 𝒕 ≤ 𝒔 − 𝒙 𝒕 𝟎
if 𝒔 − 𝒙 𝒕 𝟎 < 𝒔 + 𝒙 𝒕 𝟎
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
if 𝒔 − 𝒙 𝒕 𝟎 = 𝒔 + 𝒙 𝒕 𝟎
𝒔 + 𝒙 𝒕 < 𝒔 + 𝒙 𝒕 𝟎
𝒔 + 𝒙 𝒕 ≤ 𝒔 + 𝒙 𝒕 𝟎

34. Totally Positive Differential Systems (Schwarz, 1970)
𝒙 =𝑨𝒙, 𝒙 𝟎 ≠𝟎, with 𝐞𝐱𝐩 𝑨𝝐 TP for all 𝝐>𝟎 sufficiently small.
I𝐝𝐞𝐚 𝐨𝐟 𝐏𝐫𝐨𝐨𝐟.
𝒔 + 𝒙 𝒕 < 𝒔 + 𝒙 𝒕 𝟎
𝒔 + 𝒙 𝒕 ≤ 𝒔 + 𝒙 𝒕 𝟎
We conclude that 𝒔 + 𝒙 𝒕 is nonincreasing and strictly decreases as 𝒙 𝒕 goes through a point not in 𝑽. But 𝒔 + 𝒙 𝒕 ∈ 𝟎,𝟏,…,𝒏−𝟏 .
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

35. Totally Positive Differential Systems (Schwarz, 1970)
𝒙 =𝑨𝒙, 𝒙 𝟎 ≠𝟎, with 𝐞𝐱𝐩 𝑨𝝐 TP for all 𝝐>𝟎 sufficiently small.
𝝈(𝒙(𝒕)) 𝑡
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
No more than 𝒏−𝟏 “jump” points.

36. Totally Positive Differential Systems (Schwarz, 1970)
Question 2: When is the transition matrix of 𝒙 𝒕 =𝑨 𝒕 𝒙 𝒕 TP for all 𝝐>𝟎 sufficiently small? Schwarz answered this for the case 𝒕→𝑨 𝒕 continuous.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
The paper by Schwarz was cited 22 times since its publication in Probably because he considered only linear systems.

37. Totally Positive Differential Systems
Theorem 2. Consider the linear system:
(*) 𝒙 𝒕 =𝑨 𝒕 𝒙 𝒕 ,
with 𝑨 𝒕 measurable and locally (essentially) bounded. Assume 𝑨 𝒕 is tridiagonal, 𝒂 𝒊𝒋 𝒕 ≥𝜹>𝟎 for all 𝒊−𝒋 =𝟏. Then the transition matrix 𝚯 𝒕, 𝒕 𝟎 of (*)
is TP for all 𝒕> 𝒕 𝟎 . (Here 𝒙 𝒕 =𝚯 𝒕, 𝒕 𝟎 𝒙 𝒕 𝟎 .)
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

38. Totally Positive Differential Systems
Theorem 2. 𝑨 𝒕 tridiagonal, 𝒂 𝒊𝒋 𝒕 ≥𝜹>𝟎 for all 𝒊−𝒋 =𝟏. Then the transition matrix 𝚯 𝒕, 𝒕 𝟎 of 𝒙 𝒕 =𝑨 𝒕 𝒙 𝒕 is TP for all 𝒕> 𝒕 𝟎 . A=
Example: Then and this is TP.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
exp(0.05A)=

39. Outline
Motivation: Stability analysis of nonlinear, tridiagonal, cooperative systems (Smillie 1984, Smith 1991,…).
Totally nonnegative (TN) and totally positive (TP) matrices (Schoenberg, Gantmacher, Krein, Karlin,…)
Linear totally positive differential systems (TPDSs) (Schwarz 1970)
Putting it all together
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

40. Putting it all Together
Smillie analyzed 𝒙 =𝒇(𝒙) with a tridiagonal and strongly cooperative Jacobian. As we saw, this means that 𝒛 =𝑱 𝒙 𝒛 is a TPDS. Thus,
𝝈(𝒛(𝒕)) 𝑡
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
No more than 𝒏−𝟏 “jump” points.

41. Putting it all Together
Advantages of the TPDS framework:
(1) Smillie showed: structure of 𝑱 implies TPDS yields the converse result.
𝒔 + 𝒛 𝒕 ≤ 𝒔 − 𝒛 𝒕 𝟎 𝐟𝐨𝐫 𝐚𝐥𝐥 𝒕> 𝒕 𝟎 .
(2) Simpler proofs.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).
𝑨(𝒕)
(3) It is possible to generalize from continuous to measurable.
Very useful when 𝒙 =𝒇 𝒙,𝒖 .
𝑨(𝒕)

42. Conclusions
Powerful results on tridiagonal strongly cooperative dynamical systems are based on using the number of sign variations as a Lyapunov function. We showed that these results can be generalized, and their proofs simplified, using the beautiful (yet forgotten) theory of totally positive differential systems.
Translation consumes most of the cell energy.
Need computational models to understand all the biological findings (experiments).
Ribosome profiling – a snapshot of the elongation step at nucleotide resolution. Measure ribosome density over codons.
Analysis can help improve the fidelity and predictively of the model.
Simple vs. complex models. Generic vs. specific phenomena. Micro vs. Macro models. Deterministic vs. stochastic (unknown are describes via probabilities). Lagrange representation (particle-based) vs. Eulerian (site-based).

43. More Details Pinkus. Totally Positive Matrices, 2010.
Fallat & Johnson. Totally Nonnegative Matrices, 2011.
Schwarz. Totally positive differential systems, Pacific J. Math., 1970.
Stated first by Francis Crick, is the explanation of the unidirectional flow of genetic information within a biological system.
Cells express different subsets of the genes in different tissues and under different conditions and times (response to stimuli). This may also explain the sometimes apparently contradiction theories/findings presented in the literature.
DNA replication occurs during the Synthesis phase of the cell cycle.
In cells that are not actively dividing, the DNA is contained within chromosomes (23 pairs in the human body).
Transcription – safe and serves as an amplifier.
Protein – sequence of amino-acids (small molecules).
DNA double helix – about 10 nucleotide pairs per helical turn.
mRNA is single stranded (and contains U instead of T).
Four types of RNA: (messenger) mRNA, (transfer) tRNA, (ribosomal) rRNA and (micro) miRNA.
In viruses, the virus’s RNA is inverse-transcribed to DNA which is then (gene expression) transcribe and translated to proteins (thus, virus replication).

44. More Details
Smillie. Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal., 1984.
Smith. Periodic tridiagonal competitive and cooperative systems of differential equations, SIAM J. Math. Anal., 1991.
Margaliot and Sontag. Revisiting totally positive differential systems: a tutorial and new results. Available online:
Stated first by Francis Crick, is the explanation of the unidirectional flow of genetic information within a biological system.
Cells express different subsets of the genes in different tissues and under different conditions and times (response to stimuli). This may also explain the sometimes apparently contradiction theories/findings presented in the literature.
DNA replication occurs during the Synthesis phase of the cell cycle.
In cells that are not actively dividing, the DNA is contained within chromosomes (23 pairs in the human body).
Transcription – safe and serves as an amplifier.
Protein – sequence of amino-acids (small molecules).
DNA double helix – about 10 nucleotide pairs per helical turn.
mRNA is single stranded (and contains U instead of T).
Four types of RNA: (messenger) mRNA, (transfer) tRNA, (ribosomal) rRNA and (micro) miRNA.
In viruses, the virus’s RNA is inverse-transcribed to DNA which is then (gene expression) transcribe and translated to proteins (thus, virus replication).

45. Acknowledgments Tamir Tuller Yoram Zarai Tsuff Ben-Avraham Guy Sharon
Stated first by Francis Crick, is the explanation of the unidirectional flow of genetic information within a biological system.
Cells express different subsets of the genes in different tissues and under different conditions and times (response to stimuli). This may also explain the sometimes apparently contradiction theories/findings presented in the literature.
DNA replication occurs during the Synthesis phase of the cell cycle.
In cells that are not actively dividing, the DNA is contained within chromosomes (23 pairs in the human body).
Transcription – safe and serves as an amplifier.
Protein – sequence of amino-acids (small molecules).
DNA double helix – about 10 nucleotide pairs per helical turn.
mRNA is single stranded (and contains U instead of T).
Four types of RNA: (messenger) mRNA, (transfer) tRNA, (ribosomal) rRNA and (micro) miRNA.
In viruses, the virus’s RNA is inverse-transcribed to DNA which is then (gene expression) transcribe and translated to proteins (thus, virus replication).