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Introduction Mass-action Kinetics Negative Numbers Conclusion Chemical Analog Computing with Negative Numbers Hendrik Happe hendrik.happe@uni-jena.de Nineteenth International Conference on Membrane Computing September 11, 2018 Chemical

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Analog Computing with Negative Numbers

Hendrik Happe hendrik.happe@uni-jena.de

Nineteenth International Conference on Membrane Computing

September 11, 2018

Chemical Analog Computing with Negative Numbers Hendrik Happe 1/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Introduction Mass-action Kinetics Representation of Negative Numbers Conclusion

Chemical Analog Computing with Negative Numbers Hendrik Happe 2/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Advantages of chemical computing ✌ ✌ ✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30

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SLIDE 4

Introduction Mass-action Kinetics Negative Numbers Conclusion

Advantages of chemical computing ✌ High storage density ✌ ✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30

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SLIDE 5

Introduction Mass-action Kinetics Negative Numbers Conclusion

Advantages of chemical computing ✌ High storage density ✌ High energy efficient ✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30

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SLIDE 6

Introduction Mass-action Kinetics Negative Numbers Conclusion

Advantages of chemical computing ✌ High storage density ✌ High energy efficient ✌ High parallelism ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Advantages of chemical computing ✌ High storage density ✌ High energy efficient ✌ High parallelism ✌ Recyclable

Chemical Analog Computing with Negative Numbers Hendrik Happe 3/ 30

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SLIDE 8

Introduction Mass-action Kinetics Negative Numbers Conclusion

Mass-action Kinetics

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867)

  • Ý

Ñ ✒ r s ✒ r s ñ ✒ r s ☎ r s ♣r sq ✏ ☎ r s ☎ r s ♣r sq ✏ ✏ ♣r sq ✁ ♣r sq ✏ ☎ r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30

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SLIDE 9

Introduction Mass-action Kinetics Negative Numbers Conclusion

Mass-action Kinetics

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A B

ˆ k

Ý Ñ C . . . ZC ✒ rAs and ZC ✒ rBs ñ ZC ✒ rAs ☎ rBs ♣r sq ✏ ☎ r s ☎ r s ♣r sq ✏ ✏ ♣r sq ✁ ♣r sq ✏ ☎ r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30

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SLIDE 10

Introduction Mass-action Kinetics Negative Numbers Conclusion

Mass-action Kinetics

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A B

ˆ k

Ý Ñ C . . . ZC ✒ rAs and ZC ✒ rBs ñ ZC ✒ rAs ☎ rBs Production rate generating C: Vprod♣rCsq ✏ ˆ k ☎ rAs ☎ rBs ♣r sq ✏ ✏ ♣r sq ✁ ♣r sq ✏ ☎ r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30

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SLIDE 11

Introduction Mass-action Kinetics Negative Numbers Conclusion

Mass-action Kinetics

Assumption: number of effective reactant collisions Z proportional to reactant concentrations (Guldberg 1867) A B

ˆ k

Ý Ñ C . . . ZC ✒ rAs and ZC ✒ rBs ñ ZC ✒ rAs ☎ rBs Production rate generating C: Vprod♣rCsq ✏ ˆ k ☎ rAs ☎ rBs Consumption rate of C Vcons♣rCsq ✏ 0 d C d t ✏ Vprod♣rCsq ✁ Vcons♣rCsq ✏ ˆ k ☎ rAs ☎ rBs

Chemical Analog Computing with Negative Numbers Hendrik Happe 4/ 30

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SLIDE 12

Introduction Mass-action Kinetics Negative Numbers Conclusion

Mass-action Kinetics: General ODE Model

Considering scheme of r reaction and p involved species

  • Ý

Ñ

  • Ý

Ñ

  • Ý

Ñ

  • ✾

r s ✏ r s ✏ ➳

✏

✄ ☎ ♣ ✁ q ☎ ➵

✏

r s ☛ ✏

Chemical Analog Computing with Negative Numbers Hendrik Happe 5/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Mass-action Kinetics: General ODE Model

Considering scheme of r reaction and p involved species a1,1S1 a2,1S2 . . . ap,1Sp

k1

Ý Ñb1,1S1 b2,1S2 . . . bp,1Sp a1,2S1 a2,2S2 . . . ap,2Sp

k2

Ý Ñb1,2S1 b2,2S2 . . . bp,2Sp . . . a1,rS1 a2,rS2 . . . ap,rSp

kr

Ý Ñb1,rS1 b2,rS2 . . . bp,rSp Results in system of ordinary differential equations (ODEs): ✾ r s ✏ r s ✏ ➳

✏

✄ ☎ ♣ ✁ q ☎ ➵

✏

r s ☛ ✏

Chemical Analog Computing with Negative Numbers Hendrik Happe 5/ 30

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SLIDE 14

Introduction Mass-action Kinetics Negative Numbers Conclusion

Mass-action Kinetics: General ODE Model

Considering scheme of r reaction and p involved species a1,1S1 a2,1S2 . . . ap,1Sp

k1

Ý Ñb1,1S1 b2,1S2 . . . bp,1Sp a1,2S1 a2,2S2 . . . ap,2Sp

k2

Ý Ñb1,2S1 b2,2S2 . . . bp,2Sp . . . a1,rS1 a2,rS2 . . . ap,rSp

kr

Ý Ñb1,rS1 b2,rS2 . . . bp,rSp Results in system of ordinary differential equations (ODEs): ✾ rSis ✏ drSis d t ✏

r

➳

h✏1

✄ kh ☎ ♣bi,h ✁ ai,hq ☎

p

➵

j✏1

rSjsaj,h ☛ with i ✏ 1, . . . , p

Chemical Analog Computing with Negative Numbers Hendrik Happe 5/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Addition

✾ r s ✏ ✾ r s ✏ ✾ r s ✏ r s r s ✁ r s

X1 X2 Y ❍ k3 k1 k2

✏ ✏ r s♣✽q ✏

Ñ✽♣ ✁ ✁

q ☎ ♣r s♣ q r s♣ qq ✏ r s♣ q r s♣ q ñ r s ✏r s r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Addition

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k1rX1s k2rX2s ✁ k3rY s

X1 X2 Y ❍ k3 k1 k2

✏ ✏ r s♣✽q ✏

Ñ✽♣ ✁ ✁

q ☎ ♣r s♣ q r s♣ qq ✏ r s♣ q r s♣ q ñ r s ✏r s r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30

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SLIDE 17

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Addition

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k1rX1s k2rX2s ✁ k3rY s

X1 X2 Y ❍ k3 k1 k2

asymptotic solution in a stationary state with k1 ✏ k2 ✏ k3 rY s♣✽q ✏ lim

tÑ✽♣1 ✁ e✁k1tq ☎ ♣rX1s♣tq rX2s♣tqq ✏ rX1s♣0q rX2s♣0q

ñ r s ✏r s r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Addition

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k1rX1s k2rX2s ✁ k3rY s

X1 X2 Y ❍ k3 k1 k2

asymptotic solution in a stationary state with k1 ✏ k2 ✏ k3 rY s♣✽q ✏ lim

tÑ✽♣1 ✁ e✁k1tq ☎ ♣rX1s♣tq rX2s♣tqq ✏ rX1s♣0q rX2s♣0q

ñ rY s ✏rX1s rX2s

Chemical Analog Computing with Negative Numbers Hendrik Happe 6/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Non-negative Subtraction

✾ r s ✏ ✾ r s ✏ ✾ r s ✏ ✁ r sr s ✁ r s r s ✾ r s ✏ r s ✁ r sr s

X2 X1 Z Y ❍ ❍ k2 k1 k1 k1

✏ r s♣✽q ✏ ★ r s♣ q ✁ r s♣ q r s♣ q → r s♣ q ñ r s ✏r s ✾ ✁r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Non-negative Subtraction

✾ rX1s ✏ ✾ rX2s ✏ ✾ rY s ✏ ✁k2rY srZs ✁ k1rY s k1rX1s ✾ rZs ✏ k1rX2s ✁ k2rY srZs

X2 X1 Z Y ❍ ❍ k2 k1 k1 k1

✏ r s♣✽q ✏ ★ r s♣ q ✁ r s♣ q r s♣ q → r s♣ q ñ r s ✏r s ✾ ✁r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Non-negative Subtraction

✾ rX1s ✏ ✾ rX2s ✏ ✾ rY s ✏ ✁k2rY srZs ✁ k1rY s k1rX1s ✾ rZs ✏ k1rX2s ✁ k2rY srZs

X2 X1 Z Y ❍ ❍ k2 k1 k1 k1

asymptotic solution in a stationary state with k1 ✏ k2 rY s♣✽q ✏ ★ rX1s♣0q ✁ rX2s♣0q if rX1s♣0q → rX2s♣0q else ñ r s ✏r s ✾ ✁r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Non-negative Subtraction

✾ rX1s ✏ ✾ rX2s ✏ ✾ rY s ✏ ✁k2rY srZs ✁ k1rY s k1rX1s ✾ rZs ✏ k1rX2s ✁ k2rY srZs

X2 X1 Z Y ❍ ❍ k2 k1 k1 k1

asymptotic solution in a stationary state with k1 ✏ k2 rY s♣✽q ✏ ★ rX1s♣0q ✁ rX2s♣0q if rX1s♣0q → rX2s♣0q else ñ rY s ✏rX1s ✾ ✁rX2s

Chemical Analog Computing with Negative Numbers Hendrik Happe 7/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Multiplication

✾ r s ✏ ✾ r s ✏ ✾ r s ✏ r sr s ✁ r s

X1 X2 Y ❍ k2 k1

✏ r s♣✽q ✏

Ñ✽♣ ✁ ✁

q ☎ ♣r s♣ q ☎ r s♣ qq ✏ r s♣ q ☎ r s♣ q ñ r s ✏r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 8/ 30

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SLIDE 24

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Multiplication

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k1rX1srX2s ✁ k2rY s

X1 X2 Y ❍ k2 k1

✏ r s♣✽q ✏

Ñ✽♣ ✁ ✁

q ☎ ♣r s♣ q ☎ r s♣ qq ✏ r s♣ q ☎ r s♣ q ñ r s ✏r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 8/ 30

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SLIDE 25

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Multiplication

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k1rX1srX2s ✁ k2rY s

X1 X2 Y ❍ k2 k1

asymptotic solution in a stationary state with k1 ✏ k2 rY s♣✽q ✏ lim

tÑ✽♣1 ✁ e✁k1tq ☎ ♣rX1s♣tq ☎ rX2s♣tqq ✏ rX1s♣0q ☎ rX2s♣0q

ñ r s ✏r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 8/ 30

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SLIDE 26

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Multiplication

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k1rX1srX2s ✁ k2rY s

X1 X2 Y ❍ k2 k1

asymptotic solution in a stationary state with k1 ✏ k2 rY s♣✽q ✏ lim

tÑ✽♣1 ✁ e✁k1tq ☎ ♣rX1s♣tq ☎ rX2s♣tqq ✏ rX1s♣0q ☎ rX2s♣0q

ñ rY s ✏rX1s ☎ rX2s

Chemical Analog Computing with Negative Numbers Hendrik Happe 8/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Division

✾ r s ✏ ✾ r s ✏ ✾ r s ✏ r s ✁ r sr s

X2 X1 Y ❍ k1 k2

✏ r s♣✽q ✏ ✩ ✫ ✪

Ñ✽

✁ ♣ ✁

✁

q ☎ r

s♣ q r s♣ q

✠ r s♣ q →

Ñ✽♣

➩ r s♣ q q r s ✏r s④r s r s →

Chemical Analog Computing with Negative Numbers Hendrik Happe 9/ 30

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SLIDE 28

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Division

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k2rX2s ✁ k1rX1srY s

X2 X1 Y ❍ k1 k2

✏ r s♣✽q ✏ ✩ ✫ ✪

Ñ✽

✁ ♣ ✁

✁

q ☎ r

s♣ q r s♣ q

✠ r s♣ q →

Ñ✽♣

➩ r s♣ q q r s ✏r s④r s r s →

Chemical Analog Computing with Negative Numbers Hendrik Happe 9/ 30

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SLIDE 29

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Division

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k2rX2s ✁ k1rX1srY s

X2 X1 Y ❍ k1 k2

asymptotic solution in a stationary state with k1 ✏ k2 rY s♣✽q ✏ ✩ ✫ ✪ lim

tÑ✽

✁ ♣1 ✁ e✁k1tq ☎ rX1s♣tq

rX2s♣tq

✠ if rX2s♣tq → 0 lim

tÑ✽♣

➩ k2rX2s♣tq d tq else r s ✏r s④r s r s →

Chemical Analog Computing with Negative Numbers Hendrik Happe 9/ 30

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SLIDE 30

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: Division

✾ rX1s ✏0 ✾ rX2s ✏0 ✾ rY s ✏k2rX2s ✁ k1rX1srY s

X2 X1 Y ❍ k1 k2

asymptotic solution in a stationary state with k1 ✏ k2 rY s♣✽q ✏ ✩ ✫ ✪ lim

tÑ✽

✁ ♣1 ✁ e✁k1tq ☎ rX1s♣tq

rX2s♣tq

✠ if rX2s♣tq → 0 lim

tÑ✽♣

➩ k2rX2s♣tq d tq else rY s ✏rX1s④rX2s ifrX2s → 0

Chemical Analog Computing with Negative Numbers Hendrik Happe 9/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: IF-ELSE

✌ rTs, rFs P t♣1, 0q, ♣0, 1q✉ ✌ r s ✏ r s ✏ r s ☎ r s r s ✏ r s r s r s ✏ r s ☎ r s r s ✏ r s r s ✏ r s ☎ r s ☎ r s r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 10/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: IF-ELSE

✌ rTs, rFs P t♣1, 0q, ♣0, 1q✉ ✌ i.e. if rTs ✏ 1 : rCs ✏ rAs ☎ rBs, else : rCs ✏ rAs if rTs then rCs ✏ rAs ☎ rBs else rCs ✏ rAs end r s ✏ r s ☎ r s ☎ r s r s ☎ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 10/ 30

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SLIDE 33

Introduction Mass-action Kinetics Negative Numbers Conclusion

Chemical Calculations: IF-ELSE

✌ rTs, rFs P t♣1, 0q, ♣0, 1q✉ ✌ i.e. if rTs ✏ 1 : rCs ✏ rAs ☎ rBs, else : rCs ✏ rAs if rTs then rCs ✏ rAs ☎ rBs else rCs ✏ rAs end rCs ✏ rAs ☎ rBs ☎ rTs rFs ☎ rAs

Chemical Analog Computing with Negative Numbers Hendrik Happe 10/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Copasi: Software for Chemical Reaction’s Simulation

✌ Copasi: Complex Pathway Simulator ✌ Freely available at www.copasi.org ✌ Very stable and reliable tool, convenient user interface ✌ Particle numbers (multiset of molecular counts) as output

Chemical Analog Computing with Negative Numbers Hendrik Happe 11/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Estimation of Time Course using Copasi

✌ Numerical ODE solver based on adaptive Runge-Kutta method ✌ Variable time discretisation stepsize according to volatility ✌ High internal numerical precision ✌ Output rounded to obtain integer numbers for molecular counts

Chemical Analog Computing with Negative Numbers Hendrik Happe 12/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Problem With This Model

✌ Representation of negative numbers ✌ ✁ ✌

✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 13/ 30

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SLIDE 37

Introduction Mass-action Kinetics Negative Numbers Conclusion

Problem With This Model

✌ Representation of negative numbers ✌ What happens if you want to calculate 2 ✁ 3? ✌

✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 13/ 30

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SLIDE 38

Introduction Mass-action Kinetics Negative Numbers Conclusion

Problem With This Model

✌ Representation of negative numbers ✌ What happens if you want to calculate 2 ✁ 3? ✌ I developed two possible representation forms

✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 13/ 30

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SLIDE 39

Introduction Mass-action Kinetics Negative Numbers Conclusion

Problem With This Model

✌ Representation of negative numbers ✌ What happens if you want to calculate 2 ✁ 3? ✌ I developed two possible representation forms

✌ Signed Number Representation ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 13/ 30

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SLIDE 40

Introduction Mass-action Kinetics Negative Numbers Conclusion

Problem With This Model

✌ Representation of negative numbers ✌ What happens if you want to calculate 2 ✁ 3? ✌ I developed two possible representation forms

✌ Signed Number Representation ✌ Bi-Concentration Representation

Chemical Analog Computing with Negative Numbers Hendrik Happe 13/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation

Idea ✌

✌ r s ✏ ★ r s → r s ✌ r s ✏ ★ r s → r s

✌

✌ r s ✏ ✩ ✬ ✫ ✬ ✪ r s ✁ r s r s → r s r s ✁ r s r s → r s

ñ ✁ ♣

q Ñ ✂ t♣

q ♣ q✉

Chemical Analog Computing with Negative Numbers Hendrik Happe 14/ 30

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SLIDE 42

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation

Idea ✌ Use two substances (P/N) for the sign

✌ r s ✏ ★ r s → r s ✌ r s ✏ ★ r s → r s

✌

✌ r s ✏ ✩ ✬ ✫ ✬ ✪ r s ✁ r s r s → r s r s ✁ r s r s → r s

ñ ✁ ♣

q Ñ ✂ t♣

q ♣ q✉

Chemical Analog Computing with Negative Numbers Hendrik Happe 14/ 30

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SLIDE 43

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation

Idea ✌ Use two substances (P/N) for the sign

✌ rPs ✏ ★ 1 if rX1s → rX2s else ✌ r s ✏ ★ r s → r s

✌

✌ r s ✏ ✩ ✬ ✫ ✬ ✪ r s ✁ r s r s → r s r s ✁ r s r s → r s

ñ ✁ ♣

q Ñ ✂ t♣

q ♣ q✉

Chemical Analog Computing with Negative Numbers Hendrik Happe 14/ 30

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SLIDE 44

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation

Idea ✌ Use two substances (P/N) for the sign

✌ rPs ✏ ★ 1 if rX1s → rX2s else ✌ rNs ✏ ★ 1 if rX2s → rX1s else

✌

✌ r s ✏ ✩ ✬ ✫ ✬ ✪ r s ✁ r s r s → r s r s ✁ r s r s → r s

ñ ✁ ♣

q Ñ ✂ t♣

q ♣ q✉

Chemical Analog Computing with Negative Numbers Hendrik Happe 14/ 30

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SLIDE 45

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation

Idea ✌ Use two substances (P/N) for the sign

✌ rPs ✏ ★ 1 if rX1s → rX2s else ✌ rNs ✏ ★ 1 if rX2s → rX1s else

✌ Use another substance (Y ) for the number

✌ r s ✏ ✩ ✬ ✫ ✬ ✪ r s ✁ r s r s → r s r s ✁ r s r s → r s

ñ ✁ ♣

q Ñ ✂ t♣

q ♣ q✉

Chemical Analog Computing with Negative Numbers Hendrik Happe 14/ 30

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SLIDE 46

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation

Idea ✌ Use two substances (P/N) for the sign

✌ rPs ✏ ★ 1 if rX1s → rX2s else ✌ rNs ✏ ★ 1 if rX2s → rX1s else

✌ Use another substance (Y ) for the number

✌ rY s ✏ ✩ ✬ ✫ ✬ ✪ rX1s ✁ rX2s if rX1s → rX2s rX2s ✁ rX1s if rX2s → rX1s else

ñ ✁ ♣

q Ñ ✂ t♣

q ♣ q✉

Chemical Analog Computing with Negative Numbers Hendrik Happe 14/ 30

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SLIDE 47

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation

Idea ✌ Use two substances (P/N) for the sign

✌ rPs ✏ ★ 1 if rX1s → rX2s else ✌ rNs ✏ ★ 1 if rX2s → rX1s else

✌ Use another substance (Y ) for the number

✌ rY s ✏ ✩ ✬ ✫ ✬ ✪ rX1s ✁ rX2s if rX1s → rX2s rX2s ✁ rX1s if rX2s → rX1s else

ñ ✁s : ♣Rq2 Ñ R ✂ t♣1, 0q, ♣0, 1q✉

Chemical Analog Computing with Negative Numbers Hendrik Happe 14/ 30

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SLIDE 48

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Reactive Network

✁s:

X1 A P N X2 rAs♣0q ✏ 1

Chemical Analog Computing with Negative Numbers Hendrik Happe 15/ 30

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SLIDE 49

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Reactive Network

✁s:

X1 A P N X2 Y V Z ❍ ❍ ❍ r s♣ q ✏

Chemical Analog Computing with Negative Numbers Hendrik Happe 15/ 30

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SLIDE 50

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Example

rX1s ✏ 5 rX2s ✏ 3

Chemical Analog Computing with Negative Numbers Hendrik Happe 16/ 30

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SLIDE 51

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Example

rX1s ✏ 5 rX2s ✏ 3

Chemical Analog Computing with Negative Numbers Hendrik Happe 16/ 30

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SLIDE 52

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Example

rX1s ✏ 3 rX2s ✏ 5

Chemical Analog Computing with Negative Numbers Hendrik Happe 16/ 30

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SLIDE 53

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Example

rX1s ✏ 3 rX2s ✏ 5

Chemical Analog Computing with Negative Numbers Hendrik Happe 16/ 30

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SLIDE 54

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Addition

Input : X1 ✏ ♣rX1s, rX1Ps, rX1Nsq, X2 ✏ ♣rX2s, rX2Ps, rX2Nsq Output: Y ✏ ♣rY s, rYPs, rYNsq ✏ X1 X2 if rX1Ps AND rX2Ps then rY s ✏ rX1s rX2s, rYPs ✏ 1, rYNs ✏ 0 end if rX1Ns AND rX2Ns then rY s ✏ rX1s rX2s, rYPs ✏ 0, rYNs ✏ 1 end if rX1Ps AND rX2Ns then rY s, rYPs, rYNs ✏ rX1s ✁s rX2s end if rX1Ns AND rX2Ps then rY s, rYNs, rYPs ✏ rX2s ✁s rX1s end return rY s, rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 17/ 30

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SLIDE 55

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Subtraction

Input : X1 ✏ ♣rX1s, rX1Ps, rX1Nsq, X2 ✏ ♣rX2s, rX2Ps, rX2Nsq Output: Y ✏ ♣rY s, rYPs, rYNsq ✏ X1 ✁ X2 if rX1Ps AND rX2Ps then rY s, rYPs, rYNs ✏ rX1s ✁s rX2s end if rX1Ns AND rX2Ns then rY s, rYNs, rYPs ✏ rX2s ✁s rX1s end if rX1Ps AND rX2Ns then rY s ✏ rX1s rX2s, rYPs ✏ 1, rYNs ✏ 0 end if rX1Ns AND rX2Ps then rY s ✏ rX1s rX2s, , rYPs ✏ 0, rYNs ✏ 1 end return rY s, rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 18/ 30

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SLIDE 56

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Multiplication

Input : X1 ✏ ♣rX1s, rX1Ps, rX1Nsq, X2 ✏ ♣rX2s, rX2Ps, rX2Nsq Output: Y ✏ ♣rY s, rYPs, rYNsq ✏ X1 ☎ X2 rY s ✏ rX1s ☎ rX2s rYPs ✏ rX1Ps ☎ rX2Ps rX1Ns ☎ rX2Ns rYNs ✏ rX1Ps ☎ rX2Ns rX1Ns ☎ rX2Ps return rY s, rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 19/ 30

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SLIDE 57

Introduction Mass-action Kinetics Negative Numbers Conclusion

Signed Number Representation: Division

Input : X1 ✏ ♣rX1s, rX1Ps, rX1Nsq, X2 ✏ ♣rX2s, rX2Ps, rX2Nsq Output: Y ✏ ♣rY s, rYPs, rYNsq ✏ X1④X2 rY s ✏ rX1s④rX2s rYPs ✏ rX1Ps ☎ rX2Ps rX1Ns ☎ rX2Ns rYNs ✏ rX1Ps ☎ rX2Ns rX1Ns ☎ rX2Ps return rY s, rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 20/ 30

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SLIDE 58

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation

Idea ✌

✌ r s ✏ r s ✾ ✁r s

✌

✌ r s ✏ r s ✾ ✁r s

ñ ✁ ♣

q Ñ ♣ q

Chemical Analog Computing with Negative Numbers Hendrik Happe 21/ 30

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SLIDE 59

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation

Idea ✌ One substance (YP) for the positive part

✌ r s ✏ r s ✾ ✁r s

✌

✌ r s ✏ r s ✾ ✁r s

ñ ✁ ♣

q Ñ ♣ q

Chemical Analog Computing with Negative Numbers Hendrik Happe 21/ 30

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SLIDE 60

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation

Idea ✌ One substance (YP) for the positive part

✌ rYPs ✏ rX1s ✾ ✁rX2s

✌

✌ r s ✏ r s ✾ ✁r s

ñ ✁ ♣

q Ñ ♣ q

Chemical Analog Computing with Negative Numbers Hendrik Happe 21/ 30

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SLIDE 61

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation

Idea ✌ One substance (YP) for the positive part

✌ rYPs ✏ rX1s ✾ ✁rX2s

✌ One substance (YN) for the negative part

✌ r s ✏ r s ✾ ✁r s

ñ ✁ ♣

q Ñ ♣ q

Chemical Analog Computing with Negative Numbers Hendrik Happe 21/ 30

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SLIDE 62

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation

Idea ✌ One substance (YP) for the positive part

✌ rYPs ✏ rX1s ✾ ✁rX2s

✌ One substance (YN) for the negative part

✌ rYNs ✏ rX2s ✾ ✁rX1s

ñ ✁ ♣

q Ñ ♣ q

Chemical Analog Computing with Negative Numbers Hendrik Happe 21/ 30

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SLIDE 63

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation

Idea ✌ One substance (YP) for the positive part

✌ rYPs ✏ rX1s ✾ ✁rX2s

✌ One substance (YN) for the negative part

✌ rYNs ✏ rX2s ✾ ✁rX1s

ñ ✁b : ♣Rq2 Ñ ♣Rq2

Chemical Analog Computing with Negative Numbers Hendrik Happe 21/ 30

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SLIDE 64

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation: Reaction Network

✁b:

X1 X2 YP Z1 ❍ YN Z2 ❍ ❍ ❍

Chemical Analog Computing with Negative Numbers Hendrik Happe 22/ 30

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SLIDE 65

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation: Example

rX1s ✏ 5 rX2s ✏ 3

Chemical Analog Computing with Negative Numbers Hendrik Happe 23/ 30

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SLIDE 66

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation: Example

rX1s ✏ 3 rX2s ✏ 5

Chemical Analog Computing with Negative Numbers Hendrik Happe 23/ 30

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SLIDE 67

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation: Addition

Input : X1 ✏ ♣rX1Ps, rX1Nsq, X2 ✏ ♣rX2Ps, rX2Nsq Output: Y ✏ ♣rYPs, rYNsq ✏ X1 X2 rYPs, rYNs ✏ ♣rX1Ps rX2Psq ✁b ♣rX1Ns rX2Nsq return rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 24/ 30

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SLIDE 68

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation: Subtraction

Input : X1 ✏ ♣rX1Ps, rX1Nsq, X2 ✏ ♣rX2Ps, rX2Nsq Output: Y ✏ ♣rYPs, rYNsq ✏ X1 ✁ X2 rYPs, rYNs ✏ ♣rX1Ps rX2Nsq ✁b ♣rX1Ns rX2Psq return rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 25/ 30

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SLIDE 69

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation: Multiplication

Input : X1 ✏ ♣rX1Ps, rX1Nsq, X2 ✏ ♣rX2Ps, rX2Nsq Output: Y ✏ ♣rYPs, rYNsq ✏ X1 ☎ X2 rYPs ✏ rX1Ps ☎ rX2Ps rX1Ns ☎ rX2Ns rYNs ✏ rX1Ps ☎ rX2Ns rX1Ns ☎ rX2Ps return rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 26/ 30

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SLIDE 70

Introduction Mass-action Kinetics Negative Numbers Conclusion

Bi-Concentration Representation: Division

Input : X1 ✏ ♣rX1Ps, rX1Nsq, X2 ✏ ♣rX2Ps, rX2Nsq Output: Y ✏ ♣rYPs, rYNsq ✏ X1④X2 rZs ✏ ♣rX1Ps rX1Nsq④♣rX2Ps rX2Nsq if ♣rX1Ps ☎ rX2Ps rX1Ns ☎ rX2Nsq → 0 then rYPs ✏ rZs else rYNs ✏ rZs end return rYPs, rYNs

Chemical Analog Computing with Negative Numbers Hendrik Happe 27/ 30

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SLIDE 71

Introduction Mass-action Kinetics Negative Numbers Conclusion

Comparison

Signed Number Bi-Concentration r s ✏✏ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 28/ 30

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SLIDE 72

Introduction Mass-action Kinetics Negative Numbers Conclusion

Comparison

Signed Number Bi-Concentration Speed Depends on concentration 2-3 times faster r s ✏✏ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 28/ 30

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SLIDE 73

Introduction Mass-action Kinetics Negative Numbers Conclusion

Comparison

Signed Number Bi-Concentration Speed Depends on concentration 2-3 times faster Result near zero Very slow Very slow r s ✏✏ r s

Chemical Analog Computing with Negative Numbers Hendrik Happe 28/ 30

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SLIDE 74

Introduction Mass-action Kinetics Negative Numbers Conclusion

Comparison

Signed Number Bi-Concentration Speed Depends on concentration 2-3 times faster Result near zero Very slow Very slow Result exact zero Very bad, sign is indicator Very bad (rVPs ✏✏ rVNs is indicator)

Chemical Analog Computing with Negative Numbers Hendrik Happe 28/ 30

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SLIDE 75

Introduction Mass-action Kinetics Negative Numbers Conclusion

Comparison

Signed Number Bi-Concentration Speed Depends on concentration 2-3 times faster Result near zero Very slow Very slow Result exact zero Very bad, sign is indicator Very bad (rVPs ✏✏ rVNs is indicator) Elementary arithmetic Addition and subtraction, complex, multiplication and division simple Addition, subtraction and multiplication simple, division complex

Chemical Analog Computing with Negative Numbers Hendrik Happe 28/ 30

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SLIDE 76

Introduction Mass-action Kinetics Negative Numbers Conclusion

Conclusion

✌ Both models can represent negative numbers ✌ ✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 29/ 30

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SLIDE 77

Introduction Mass-action Kinetics Negative Numbers Conclusion

Conclusion

✌ Both models can represent negative numbers ✌ all elementary arithmetic can be calculated ✌ ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 29/ 30

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SLIDE 78

Introduction Mass-action Kinetics Negative Numbers Conclusion

Conclusion

✌ Both models can represent negative numbers ✌ all elementary arithmetic can be calculated ✌ Problem with calculations near zero (further researches) ✌

Chemical Analog Computing with Negative Numbers Hendrik Happe 29/ 30

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SLIDE 79

Introduction Mass-action Kinetics Negative Numbers Conclusion

Conclusion

✌ Both models can represent negative numbers ✌ all elementary arithmetic can be calculated ✌ Problem with calculations near zero (further researches) ✌ Bi-Concentration is better than Signed Number

Chemical Analog Computing with Negative Numbers Hendrik Happe 29/ 30

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SLIDE 80

Introduction Mass-action Kinetics Negative Numbers Conclusion

Conclusion

✌ Both models can represent negative numbers ✌ all elementary arithmetic can be calculated ✌ Problem with calculations near zero (further researches) ✌ Bi-Concentration is better than Signed Number

Chemical Analog Computing with Negative Numbers Hendrik Happe 29/ 30

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SLIDE 81

Introduction Mass-action Kinetics Negative Numbers Conclusion

Conclusion

✌ Both models can represent negative numbers ✌ all elementary arithmetic can be calculated ✌ Problem with calculations near zero (further researches) ✌ Bi-Concentration is better than Signed Number Thank you for your attention

Chemical Analog Computing with Negative Numbers Hendrik Happe 29/ 30

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Introduction Mass-action Kinetics Negative Numbers Conclusion

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Chemical Analog Computers for Clock Frequency Control Based on P Modules. Lecture Notes in Computer Science Vol. 7184, pp. 182-202, Springer Verlag, 2012

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Environmental, Social, and Economic Implications of Global Reuse and Recycling of Personal Computers. Environ. Sci.

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Chemical Analog Computing with Negative Numbers Hendrik Happe 30/ 30