[PPT] - Integrating Theoretical Algorithmic Ideas in Empirical Biological PowerPoint Presentation

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Integrating Theoretical Algorithmic Ideas in Empirical Biological Study

Amos Korman In collaboration with Ofer Feinerman (Weizmann Institute)

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Scientific frameworks

Outline

1

Scientific frameworks

2

How can an algorithmic perspective contribute?

3

A novel scientific framework

4

Searching for a nearby treasure

5

Information lower bounds for probabilistic search (DISC 2012)

6

Conclusions

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Scientific frameworks

Classical scientific frameworks in biology Experimental framework:

1

Preprocessing stage: observe and analyze

2

“Guess” a mathematical model

3

Data analysis: tune the parameters

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Scientific frameworks

Example: the Albatross (Nature 1996, 2007)

Pr(l =d) ¡≈ ¡1/dα ¡

The Albatross is performing a Lévy flight

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Scientific frameworks

Example: the Albatross (Nature 1996, 2007)

Pr(l =d) ¡≈ ¡1/dα ¡

The Albatross is performing a Lévy flight What is α? do statistics on experiments and obtain e.g., α = 2

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Scientific frameworks

Theoretical framework:

1

Guess an abstract mathematical model

2

Analyze the model

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Scientific frameworks

Theoretical framework:

1

Guess an abstract mathematical model

2

Analyze the model

Find parameters maximizing a utility function

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Scientific frameworks

Theoretical framework:

1

Guess an abstract mathematical model

2

Analyze the model

Find parameters maximizing a utility function Example: if you perform a Lévy flight search under some certain food distribution then α = 2 is optimal [Viswanathan et al. Nature 1999]

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Scientific frameworks

Theoretical framework:

1

Guess an abstract mathematical model

2

Analyze the model

Find parameters maximizing a utility function Example: if you perform a Lévy flight search under some certain food distribution then α = 2 is optimal [Viswanathan et al. Nature 1999] “Explain” a known phenomena Example: Kleinberg’s analysis of the greedy routing algorithm in small world networks “explains” Milgram’s experiment [Nature 2000]

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How can an algorithmic perspective contribute?

Outline

1

Scientific frameworks

2

How can an algorithmic perspective contribute?

3

A novel scientific framework

4

Searching for a nearby treasure

5

Information lower bounds for probabilistic search (DISC 2012)

6

Conclusions

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How can an algorithmic perspective contribute?

An algorithmic perspective Recently, CS theoreticians have tried to contribute from an algorithmic perspective [Alon, Chazelle, Kleinberg, Papadimitriou, Valiant, etc.].

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How can an algorithmic perspective contribute?

An algorithmic perspective Recently, CS theoreticians have tried to contribute from an algorithmic perspective [Alon, Chazelle, Kleinberg, Papadimitriou, Valiant, etc.]. Guiding principle Algorithms’ people are good at:

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How can an algorithmic perspective contribute?

An algorithmic perspective Recently, CS theoreticians have tried to contribute from an algorithmic perspective [Alon, Chazelle, Kleinberg, Papadimitriou, Valiant, etc.]. Guiding principle Algorithms’ people are good at:

1

Formulating sophisticated guesses (algorithms)

2

Analyzing the algorithms

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How can an algorithmic perspective contribute?

Algorithmic perspective in classical frameworks Experimental framework:

1

Preprocessing stage: observe and analyze

2

Guess a mathematical model [Afek et al., Science’11]

3

Data analysis: tune the parameters

Theoretical framework:

1

Guess a mathematical model

2

Analyze the model Maximize a utility function [Papadimitriou et al., PNAS 2008] Explain a known phenomena [Kleinberg, Nature, 2000]

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How can an algorithmic perspective contribute?

Algorithmic perspective in classical frameworks Experimental framework:

1

Preprocessing stage: observe and analyze

2

Guess a mathematical model [Afek et al., Science’11]

3

Data analysis: tune the parameters

Theoretical framework:

1

Guess a mathematical model

2

Analyze the model Maximize a utility function [Papadimitriou et al., PNAS 2008] Explain a known phenomena [Kleinberg, Nature, 2000]

Can an algorithmic perspective contribute

therwise?

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How can an algorithmic perspective contribute?

Systems biology Biology is lacking tools for dealing with large, complex and interactive systems.

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How can an algorithmic perspective contribute?

Systems biology Biology is lacking tools for dealing with large, complex and interactive systems. Early 90’s – Systems Biology (a holistic approach)

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How can an algorithmic perspective contribute?

Systems biology Biology is lacking tools for dealing with large, complex and interactive systems. Early 90’s – Systems Biology (a holistic approach) Mathematics tools: differential equations.

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How can an algorithmic perspective contribute?

Systems biology Biology is lacking tools for dealing with large, complex and interactive systems. Early 90’s – Systems Biology (a holistic approach) Mathematics tools: differential equations. Distributed computing: closer to mainstream CS than to physics.

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How can an algorithmic perspective contribute?

Systems biology Biology is lacking tools for dealing with large, complex and interactive systems. Early 90’s – Systems Biology (a holistic approach) Mathematics tools: differential equations. Distributed computing: closer to mainstream CS than to physics.

How can a distributed algorithmic perspective contribute to biology?

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How can an algorithmic perspective contribute?

The big challenge: reduce the parameter space

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How can an algorithmic perspective contribute?

The big challenge: reduce the parameter space In physics: rules of nature Obtain equation (or connection) between parameters. E.g., E = MC2, ∆U = Q + W, σx · σp ≥ , etc.

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How can an algorithmic perspective contribute?

The big challenge: reduce the parameter space In physics: rules of nature Obtain equation (or connection) between parameters. E.g., E = MC2, ∆U = Q + W, σx · σp ≥ , etc. What about biology? 1st solution: borrow connections from physics. 2nd solution: ignore seemingly negligable parameters.

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How can an algorithmic perspective contribute?

The big challenge: reduce the parameter space In physics: rules of nature Obtain equation (or connection) between parameters. E.g., E = MC2, ∆U = Q + W, σx · σp ≥ , etc. What about biology? 1st solution: borrow connections from physics. 2nd solution: ignore seemingly negligable parameters. We propose: obtain connections between parameters using an algorithmic approach. Tradeoffs: use lower bounds from CS to show that, e.g., any algorithm that runs in time T must use x amount of resources (x > f(T)).

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A novel scientific framework

Outline

1

Scientific frameworks

2

How can an algorithmic perspective contribute?

3

A novel scientific framework

4

Searching for a nearby treasure

5

Information lower bounds for probabilistic search (DISC 2012)

6

Conclusions

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A novel scientific framework

Connecting parameters using an algorithmic perspective

Algorithmic ¡tradeoff ¡ ¡

Time ¡vs. ¡Informa4on ¡capacity ¡

Measurements ¡ ¡

Time ¡

Lower ¡bound ¡on ¡

Informa4on ¡capacity ¡ Ants ¡ Food ¡

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A novel scientific framework

Remarks: simplified experimental verifications

Algorithmic ¡tradeoff ¡ ¡ Measurements ¡ ¡ Biological ¡bound ¡

Se7ng ¡

¡Simple ¡
¡Realis+c ¡

100% ¡correct ¡ ¡ Requires ¡ verifica+on ¡

Tradeoffs are invariant of the algorithm =

⇒ Instead of verifying setting+algorithm, only need to verify the setting!

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A novel scientific framework

A proof of concept This talk Introduce the model (semi-realistic) Discuss the theoretical tradeoffs Experimental part: on-going

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A novel scientific framework

A proof of concept This talk Introduce the model (semi-realistic) Discuss the theoretical tradeoffs Experimental part: on-going Remark The work is not complete. This presentation is a proof of concept

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Searching for a nearby treasure

Outline

1

Scientific frameworks

2

How can an algorithmic perspective contribute?

3

A novel scientific framework

4

Searching for a nearby treasure

5

Information lower bounds for probabilistic search (DISC 2012)

6

Conclusions

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Searching for a nearby treasure

Inspiration: the Cataglyphis niger and Honey bee The Cataglyphis niger:

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Searching for a nearby treasure

Inspiration: the Cataglyphis niger and Honey bee The Cataglyphis niger: Desert ant– does not leave traces, more individual

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Searching for a nearby treasure

Inspiration: the Cataglyphis niger and Honey bee The Cataglyphis niger: Desert ant– does not leave traces, more individual Relatively smart– big brain, good navigation abilities

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Searching for a nearby treasure

Good distance and location estimations [Wehner et al.]

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Searching for a nearby treasure

Goal: find nearby treasures fast Reasons for proximity Increasing the rate of food collection in case a large quantity of food is found [Orians and Pearson, 1979], Decreasing predation risk [Krebs, 1980], The ease of navigating back after collecting the food using familiar landmarks [Collett et al., 1992], etc.

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Searching for a nearby treasure

Central place foraging

Goal: find nearby treasures fast (biologically motivated) No communication once out of the nest Grid network: the visual radius determines the grid resolution Fact: The expected running time is Ω(D + D2/k)

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Searching for a nearby treasure

Searching with one ant (k = 1) An optimal algorithm Perform a spiral search from the nest (takes O(D2) time).

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Searching for a nearby treasure

Searching with one ant (k = 1) An optimal algorithm Perform a spiral search from the nest (takes O(D2) time). Random walk is not efficient Expected time to visit any given node is ∞. Even in a bounded region– scales badly with # agents.

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Searching for a nearby treasure

Optimal algorithm (PODC 2012) [Feinerman, Korman, Lotker, Sereni] Lemma There exists an algorithm running in time O(D + D2/k)

2i ti

= 22i+2/k

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Searching for a nearby treasure

Optimal algorithm (PODC 2012) [Feinerman, Korman, Lotker, Sereni] Lemma There exists an algorithm running in time O(D + D2/k)

2i ti

= 22i+2/k

Observe: algorithm assumes that agents know k. Is it really necessary to know k? How much initial information is necessary?

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Information lower bounds for probabilistic search (DISC 2012)

Outline

1

Scientific frameworks

2

How can an algorithmic perspective contribute?

3

A novel scientific framework

4

Searching for a nearby treasure

5

Information lower bounds for probabilistic search (DISC 2012)

6

Conclusions

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Information lower bounds for probabilistic search (DISC 2012)

What is the amount of information that agents need initially? The oracle (modelling the pre-processing stage inside the nest)

Being extremely liberal: oracle is a probabilistic centralized algorithm. Input: k agents Output: An advice Ai for each agent a.

4" 3" 4" 7"

?

3" 6" 23 / 31

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Information lower bounds for probabilistic search (DISC 2012)

Information theoretic approach Advice complexity Given k agents, the advice complexity f(k) is the maximum #bits used for representing the advice of an agent

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Information lower bounds for probabilistic search (DISC 2012)

Information theoretic approach Advice complexity Given k agents, the advice complexity f(k) is the maximum #bits used for representing the advice of an agent State complexity Note, a lower bound f on the advice complexity implies a lower bound of 2f on the # of possible advices (states) when coming out of the nest

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Information lower bounds for probabilistic search (DISC 2012)

Main theorem [Feinerman and Korman, DISC 2012] Theorem For every 0 < ǫ ≤ 1, if the search time is O(log1−ǫ k · (D + D2/k)) then the advice complexity is ǫ log log k − O(1)

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Information lower bounds for probabilistic search (DISC 2012)

Main theorem [Feinerman and Korman, DISC 2012] Theorem For every 0 < ǫ ≤ 1, if the search time is O(log1−ǫ k · (D + D2/k)) then the advice complexity is ǫ log log k − O(1) Corollary If time is T = O(log1−ǫ k · (D + D2/k)) then number of states when coming out of the nest is S = Ω(logǫ k)

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Information lower bounds for probabilistic search (DISC 2012)

Main theorem [Feinerman and Korman, DISC 2012] Theorem For every 0 < ǫ ≤ 1, if the search time is O(log1−ǫ k · (D + D2/k)) then the advice complexity is ǫ log log k − O(1) Corollary If time is T = O(log1−ǫ k · (D + D2/k)) then number of states when coming out of the nest is S = Ω(logǫ k) Remarks Results are asymptotically tight Hidden constants are small

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (for a weaker version of the lower bound (appeared in PODC 2012) Lemma If running time is o(log k · (D + D2/k)) then advice > 0.

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (for a weaker version of the lower bound (appeared in PODC 2012) Lemma If running time is o(log k · (D + D2/k)) then advice > 0. Proof

Assume all agents start with the same advice regardless of k

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (for a weaker version of the lower bound (appeared in PODC 2012) Lemma If running time is o(log k · (D + D2/k)) then advice > 0. Proof

Assume all agents start with the same advice regardless of k Assume running in time is (D + D2

k ) · φ(k) (and φ(·) is non-decreasing)

I.e., the expected time to visit u is Tu ≤ (d(u, s) + d(u,s)2

k

) · φ(k)

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (for a weaker version of the lower bound (appeared in PODC 2012) Lemma If running time is o(log k · (D + D2/k)) then advice > 0. Proof

Assume all agents start with the same advice regardless of k Assume running in time is (D + D2

k ) · φ(k) (and φ(·) is non-decreasing)

I.e., the expected time to visit u is Tu ≤ (d(u, s) + d(u,s)2

k

) · φ(k) Fix W (upper bound on # agents)

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (for a weaker version of the lower bound (appeared in PODC 2012) Lemma If running time is o(log k · (D + D2/k)) then advice > 0. Proof

Assume all agents start with the same advice regardless of k Assume running in time is (D + D2

k ) · φ(k) (and φ(·) is non-decreasing)

I.e., the expected time to visit u is Tu ≤ (d(u, s) + d(u,s)2

k

) · φ(k) Fix W (upper bound on # agents) Structure of proof: we show that by time T = 2W · φ(W), an agent is expected to visit many nodes: ≈ W · log(W). Since she can visit at most 1 node in 1 time unit, we cannot have φ(W) = o(log W).

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.)

√W

S1

Si

ki =22i |Si|=W∙ki

√W∙2i

Fix i = 1, 2, · · · , log W

2

− 1, and consider Si := {u | √ W · 2i−1 < d(u, s) ≤ √ W · 2i}. Note, |Si| ≈ W · 22i

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.)

√W

S1

Si

ki =22i |Si|=W∙ki

√W∙2i

Fix i = 1, 2, · · · , log W

2

− 1, and consider Si := {u | √ W · 2i−1 < d(u, s) ≤ √ W · 2i}. Note, |Si| ≈ W · 22i Assume now that ki = 22i. So, |Si| ≈ W · ki. Note that ki < W.

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.)

√W

S1

Si

ki =22i |Si|=W∙ki

√W∙2i

Fix i = 1, 2, · · · , log W

2

− 1, and consider Si := {u | √ W · 2i−1 < d(u, s) ≤ √ W · 2i}. Note, |Si| ≈ W · 22i Assume now that ki = 22i. So, |Si| ≈ W · ki. Note that ki < W. Moreover, ki = 2i+1 · 2i−1 ≤ √ W · 2i−1. I.e., ki ≤ d(u, s), ∀u ∈ Si.

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.)

√W

S1

Si

ki =22i |Si|=W∙ki

√W∙2i

Fix i = 1, 2, · · · , log W

2

− 1, and consider Si := {u | √ W · 2i−1 < d(u, s) ≤ √ W · 2i}. Note, |Si| ≈ W · 22i Assume now that ki = 22i. So, |Si| ≈ W · ki. Note that ki < W. Moreover, ki = 2i+1 · 2i−1 ≤ √ W · 2i−1. I.e., ki ≤ d(u, s), ∀u ∈ Si. Therefore, Tu ≤ (d(u, s) + d(u,s)2

ki

) · φ(ki) ≤ 2 · d(u,s)2

ki

· φ(ki) < 2W · φ(W) = T.

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.) So, probability of visiting u ∈ Si by time 2T is at least 1/2.

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.) So, probability of visiting u ∈ Si by time 2T is at least 1/2. Thus, the expected number of nodes in Si that all agents visit by time 2T is roughly |Si| ≈ W · ki. Hence, the expected number of nodes in Si that one agent visits by time 2T is |Si|/ki ≈ W.

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.) So, probability of visiting u ∈ Si by time 2T is at least 1/2. Thus, the expected number of nodes in Si that all agents visit by time 2T is roughly |Si| ≈ W · ki. Hence, the expected number of nodes in Si that one agent visits by time 2T is |Si|/ki ≈ W. Observe, this holds ∀i ∈ [1, log W

2

).

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Information lower bounds for probabilistic search (DISC 2012)

Simplified proof (cont.) So, probability of visiting u ∈ Si by time 2T is at least 1/2. Thus, the expected number of nodes in Si that all agents visit by time 2T is roughly |Si| ≈ W · ki. Hence, the expected number of nodes in Si that one agent visits by time 2T is |Si|/ki ≈ W. Observe, this holds ∀i ∈ [1, log W

2

). Hence, the expected number of nodes that a single agent visits by time 2T is ≈ W · log W. As T ≈ W · φ(W), this implies that we cannot have φ(W) = o(log W).

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Information lower bounds for probabilistic search (DISC 2012)

A novel scientific framework? Combine the theoretical lower bound with an experiment on living ants

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Information lower bounds for probabilistic search (DISC 2012)

A novel scientific framework? Combine the theoretical lower bound with an experiment on living ants

1

Measure the search time - approximate T as a function of k and D (relatively easy)

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Information lower bounds for probabilistic search (DISC 2012)

A novel scientific framework? Combine the theoretical lower bound with an experiment on living ants

1

Measure the search time - approximate T as a function of k and D (relatively easy)

2

If the search time T < log1−ǫ k · (D + D2/k) then the number of states of ants when coming out of the nest is Ω(logǫ k)

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Conclusions

Outline

1

Scientific frameworks

2

How can an algorithmic perspective contribute?

3

A novel scientific framework

4

Searching for a nearby treasure

5

Information lower bounds for probabilistic search (DISC 2012)

6

Conclusions

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Conclusions

Conclusions and future work This work is a proof of concept for a novel scientific framework

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Conclusions

Conclusions and future work This work is a proof of concept for a novel scientific framework To fully illustrate it there is a need for experimental work. This will undoubtedly require some tuning in model and theoretical results

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Conclusions

Conclusions and future work This work is a proof of concept for a novel scientific framework To fully illustrate it there is a need for experimental work. This will undoubtedly require some tuning in model and theoretical results The framework can be applied to other biological contexts. What about bacteria? tradeoffs between efficiency and communication?

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Conclusions

Conclusions and future work This work is a proof of concept for a novel scientific framework To fully illustrate it there is a need for experimental work. This will undoubtedly require some tuning in model and theoretical results The framework can be applied to other biological contexts. What about bacteria? tradeoffs between efficiency and communication?

Thanks!

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