Model-based optimization approaches for precision medicine: A
case study in presynaptic dopamine overactivity
Kai-Cheng Hsu1 and Feng-Sheng Wang2*
1
Department of Neurology, National Taiwan University Hospital Yunlin Branch,
Yunlin 64041, Taiwan
2
Department of Chemical Engineering, National Chung Cheng University, Chiayi
62102, Taiwan
Email: Kai-Cheng Hsu - [email protected]; Feng-Sheng Wang* [email protected]
Supplementary file 2:
Definition of the symbols in the pathogenesis problem and fuzzy multiobjective target
discovery problem and its solving strategy.
1
Pathogenesis problem:
min
ˆ ,z
x,α,u
 x  x
i
i DS

disease 2
i

 1  ˆ
ˆ basal

j
2
j
j Enz
 1  u
j EX
j
u basal

j
2
subjec to
Material balance equations:
m
 r
N
v
x,α

Bij u j  0, i   SP



 ij j
j 1
 j 1
n
v

Rxn
 w j  ˆ j   g jk yk , j  
k 1

v  exp w
 j
 j
 x  exp  y 
i
 i
Metabolite constraints:
 LB
UB
SP
 xi  xi  xi ; i  
Target constraints:

basal
UB
up
up-regulation: ˆ z  ˆ z  ˆ z , z   z
 basal
UB

u z  u z  u z


ˆ zLB  ˆ z  ˆ zbasal , z   down

z
down-regulation: u LB  u  uUB
 z
z
z

basal
up
down
ˆ  ˆ , z      
 z
z
z
z
z
(S1)
2
Fuzzy multiobjective target discovery (FMTD) problem:
Therapeutic effect: Fuzzy equal xid  xibasal ; i  TH , d   DS
ˆ ,z
x,α,u

 Adverse effect: Fuzzy min x ; j   AE , d   DS
jd

ˆ ,z
x,α,u

basal
TE
Rxn
DS
 Variation effect: Fuzzy equal ˆ zk  ˆ zk ; zk     \ 
ˆ ,z
x ,α,u


Fuzzy equal u zk  u zbasal
; zk   EX
k

ˆ ,z
x,α,u

min  zk ; zk  TE
 Number of targets: x,α,u
ˆ ,z
k



 Material balance equations:
m
 r
SP
DS
  N ij v jd  x, α    Bij u j  0; i   , d  
j 1
  j 1
 Metabolite constraints:

  xiLB  xid  xiUB

 Target constraints:
  LB
UB
TE
 ˆ zk  ˆ zk  ˆ zk ; zk  
 
basal
TE
 ˆ zk  ˆ zk ; zk  
  LB
UB
EX
 u zk  u zk  u zk ; zk  
 u  u basal ; z   EX
zk
k
  zk
(S2)
The ith reaction rate, vid, depends on each disease state and is expressed in the power
law function as following
n
v jd   jd  xk jk
g
k 1
 jd
(S3)
 j , j  TE and j   DS


  basal
, j   DS and j  d
j
 DS
DS

 j , j   and j  d
3
Notation:
Symbol
Definition
Bij
The connectivity matrix describing the corresponding control ij
Nij
The stoichiometric matrix describing the interconnecting fluxes ij
gjk
The kinetic order jk
uj
The external control j
basal
uj
uz
u zbasal
u zLB
uUB
z
u zk
The external control j at the basal (healthy) level
The external control z
The external control z at the basal (healthy) level
The external control z at the lower bound
The external control z at the upper bound
The external control zk
uzbasal
k
The external control zk at the basal (healthy) level
uzLBk
The external control zk at the lower bound
uUB
zk
The external control zk at the upper bound
vj
The rate constant of the reaction rate j
wj
The logarithmic reaction rate j
xi
The concentration of the metabolite i in the metabolic network
xibasal
xidisease
xiLB
xiUB
xid
x jd
yk
zk
The concentration of the metabolite i at the basal (healthy) level
The concentration of the metabolite i at the disease state in the
metabolic network
The concentration of the metabolite i at the lower bound
The concentration of the metabolite i at the upper bound
The concentration of the metabolite i in the disease state
The concentration of the metabolite j in the disease state
The logarithmic concentration k
The number of targets
4
ˆ j
The logarithmic rate constant of the reaction rate j
ˆ basal
j
The logarithmic rate constant of the reaction rate j at the basal
(healthy) level
ˆ zLBk
The logarithmic rate constant of the reaction rate zk at the lower
bound
ˆ UB
zk
The logarithmic rate constant of the reaction rate zk at the upper
bound
ˆ z
The logarithmic rate constant of the reaction rate z
ˆ zbasal
The logarithmic rate constant of the reaction rate z at the basal
ˆ zLB
The logarithmic rate constant of the reaction rate z at the lower
bound
ˆ UB
z
The logarithmic rate constant of the reaction rate z at the upper
ˆ z
The logarithmic rate constant of the reaction rate zk
(healthy) level
bound
k
ˆ zbasal
The logarithmic rate constant of the reaction rate zk at the basal
(healthy) level
DS
The set of disease metabolites

The set of enzyme targets
k
Enz
EX
The set of external controls
SP
The set of species

The set of reactions
Rxn
 up
z
The set of up-regulation reactions
 down
z
The set of down-regulation reactions
TH
The set of fuzzy equal objectives for therapeutic effects
 AE
The set of fuzzy minimization objectives for adverse effects

The set of manipulated variables
 Rxn
The set of nv -dimentional reaction rates
 EX
The set of nu -dimentional external control variable
TE
5
Strategy for solving fuzzy multiobjective target discovery
problem
To solve the FMTD framework (S1), we defined a membership function for each
fuzzy equal objective and fuzzy minimizing objective for quantifying each
corresponding satisfaction grade. The generalized membership function for each
fuzzy equal objective is described in Eq. (S4) as follows:
0, xi  xiLB

d ixL  xiLB  xi  xibasal , LB 

i ( xi )  1, xibasal , LB  xi  xibasal ,UB
 xR
basal ,UB
 xi  xiUB
d i , xi

UB
0, xi  xi
(S4)
The left-hand side membership function is a strictly monotonically increasing
function,
d ixL ,
whereas the right-hand side is a strictly monotonically decreasing
function,
dixR .
A membership function is similar to assess the effects of inaccuracies
in external control variables and reaction rate constants. Monte Carlo simulation is
generally applied to assess such an experimental imprecision. However, a designer
can define a membership function for fuzzy optimization problems in advance.
Sakawa [1] proposed five types of membership functions, namely linear, exponential,
hyperbolic, inverse, and piecewise linear functions, for quantifying the behavior of
fuzzy objectives or the constraints. Here,
xiLB
and
xiUB
are the lower and upper
bounds of the ith metabolite concentration or enzyme activity provided by the designer.
6
The satisfaction grade is zero when the metabolite concentration or enzyme activity is
beyond its lower or upper bounds. The satisfaction grade or membership function
value is equal to one when the corresponding metabolite concentration or enzyme
activity is between the lower and upper bounds of the basal value, represented as
xibasal ,LB , xibasal ,UB ,
and
xibasal ,
respectively. The satisfaction grade is between zero and
one when the metabolite concentration or enzyme activity is within its range on the
left- or right-hand side membership function. For each fuzzy minimizing objective,
the membership function is defined as a strictly monotonically decreasing function on
the right-hand side.
1, x j  xibasal


 j ( x j )  d ixR , xibasal  xi  xiUB

UB
0, xi  xi
(S5)
According to the membership functions expressed in Eqs. (S4) and (S5), we
conclude that the intersection for these membership functions is zero when either the
fuzzy equal objective functions are outside the corresponding lower and upper bounds
or the fuzzy minimizing objectives are greater than the corresponding upper bounds.
By contrast, when all objectives are within their corresponding bounds, the
intersection for all membership functions should show a certain degree of satisfaction.
For each membership function being introduced by the designer, the FMTD problem
is designed for determining a maximum intersection for all membership functions
7
between the desired bounds. The FMTD framework is then transferred to the
maximizing decision framework, which is a discontinuous function. The detailed
procedures have been discussed previously [2, 3]. Thus, the maximizing decision
problem can be rewritten as an equivalent optimization problem on the solving
domain to avoid a discontinuous computation, as follows:
 Z UB   z j 
max 
 
UB
 ,( x , αˆ , u , z ) 
Z

1


TH
i  xi    , i   , d   DS
 j  x j    , j   AE
k ˆ k    , k  TE
(S6)
l  ul    , l   EX
where the crisp feasible domain  includes the kinetic model and inequality
constraints described in Eqs. (S2) and (S3), respectively. The total number of the
identified enzyme targets is the crisp objective so that it is transformed to a
normalized value as expressed in Eq. (S6), where ZUB is the upper bound. The
advantage of this method is that the optimal membership grade corresponds to the
satisfaction level for each objective, and the optimal decision  represents the overall
satisfaction grade (equivalent to the lower bound) of the problem. The maximizing
decision problem can be then solved by the NHDE algorithm.
8
References
1.
Sakawa M: Fuzzy sets and interactive multiobjective optimization. New York:
Plenum; 1993. pp. 308.
2.
Hsu KC, Wang FS: Fuzzy optimization for detecting enzyme targets of
human uric acid metabolism. Bioinformatics 2013, 29, 3191-3198.
3.
Hsu KC, Wang FS: Fuzzy decision making approach to identify optimum
enzyme targets and drug dosage for remedying presynaptic dopamine
deficiency. PLOS ONE, 2016, Oct 13, 1-18
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