使用生化分子作運算是合成生物學這門領域主要目標之一，如果將生化系統以化學反應層級做分析，便可知組合適當化學反應擁有實作出計算的潛力。在此我們將討論的是從合成的角度做算術運算，並特別想討論算術運算中基本的多項式運算。本篇論文提供兩種實作出多項式的方法，第一個是整數取值之多項式，此方法將一個成法器重複使用，並在時間軸上做分工，來達成多次乘法和加法運算組成之多項式。在這個架構下的化學反應，可類比於電腦中擁有確切順序和執行時間的指令。第二個方法是實數取值多項式，此多項式取值是利用控制化學反應平衡態濃度來達成，控制方法便是組合適當的生成反應和降解反應。兩種方法都經過電腦模擬一些範例來驗證正確性，在論文中將第二種方法的多項式運用到型態形成，有相當理想的成果。未來希望可以在我們的模型中套用上更多生物因子，並於活體細胞中實做出來。

Computation with biochemical elements is one of the major goals of synthetic biology. Engineering biochemical reactions has the potential to implement computations.
We discuss about synthetic approaches to biochemical arithmetic operation. In particular, computation of polynomial is fundamental and important because
we can approximate many non-linear functions with polynomials. In this thesis, we provide two bottom-up design strategies for polynomial evaluation. One is the
integer valued polynomial evaluation, where polynomials are computed by single multiplication module using a time multiplexing strategy. In the infrastructure, reactions
are regarded as atomic instruction marked with definite start time and finish time. The other is real valued polynomial evaluation, where the value is determined
by precise control of molecular concentrations at their biochemical equilibrium. To produce output, reactions are used as configurable controller for species generation and degradation. For both methods, we run deterministic computer simulation and verify their output correctness through case studies. Our biochemical polynomials
are applied to model pattern formations and provide a possible mechanism of reaction-diffusion system. In the future, we hope to impose more biological factors
to our model and realize it in living cells.

Acknowledgements i
Chinese Abstract ii
Abstract iii
List of Figures viii
List of Tables x
1 Introduction 1
1.1 Our Contributions . .. . . . . . . . . 3
1.2 Thesis Organization .. . . . . . . . . 4
2 Backgrounds 5
2.1 Terminologies and Definitions . . . . . .. . . . . . . . . . 5
2.2 Classical Chemical Kinetic Model . . . 7
2.3 Biochemical Program Control Flow . . . 8
2.4 Biochemical Linear Systems . . . . . . 10
2.5 Configurable Biochemical Linear Systems and Auxiliary Species . . . 12
3 Module Reusing and Integer Valued Polyomials 14
3.1 Integer Valued Multiplication . . . . . 15
3.2 Module Reusing and Scheduling . . . . . 17
3.2.1 Reduction of Species and Reaction Number . . 19
3.3 Integer Valued Polynomials . . . . . . . . . 20
4 Biochemical Construction of Real Valued Polynomials 21
4.1 Biochemical Module of Enzyme Genration and Degradation . . . . . 21
4.2 Construction of Real Valued Polynomials . . . . 23
4.2.1 Addition .. . . . . . . . . . 24
4.2.2 Subtraction . . . . . . . . . 24
4.2.3 Multiplication .. . . . . . . 25
4.2.4 Exponentiation . . . . . . . 25
4.2.5 Enable Signal . . . . . . . . 26
4.3 Feedback and the nth Root . . . 26
4.4 Features of our Framework . . . 28
4.4.1 Module Composition and Nested Polynomials . . . 28
4.5 Implementation Issues . . . . . 29
4.5.1 Model Compared with Michaelis–Menten Enzyme Kinetics . . 29
4.5.2 Effect of Reverse Reaction . . . . 31
4.5.3 Response Time . . . . . . . . . . 32
4.6 DNA Strand Displacement Implementation . . . . 33
5 Applications 36
5.1 Parameterized Polynomials Design .. . 36
5.2 Pattern Formation . .. . . . . . . 38
5.2.1 Turing’s Diffusion-reaction Model . .. . 38
5.2.2 Feasibility Evaluation of Pattern Formation by DNA Strand
Displacement Experiment . .. 42
6 Conclusion and Future Work 44
6.1 Summary . . . . . . . . . 44
6.2 Conclusion . . . . . . . . 45
6.3 Future Work . . .. . . . . 46

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