- Proceedings
- Open Access
Computational synchronization of microarray data with application to Plasmodium falciparum
- Wei Zhao^{1, 2},
- Justin Dauwels^{1, 2}Email author,
- Jacquin C Niles^{1, 3} and
- Jianshu Cao^{1, 4}Email author
https://doi.org/10.1186/1477-5956-10-S1-S10
© Zhao et al.; licensee BioMed Central Ltd. 2012
- Published: 21 June 2012
Abstract
Background
Microarrays are widely used to investigate the blood stage of Plasmodium falciparum infection. Starting with synchronized cells, gene expression levels are continually measured over the 48-hour intra-erythrocytic cycle (IDC). However, the cell population gradually loses synchrony during the experiment. As a result, the microarray measurements are blurred. In this paper, we propose a generalized deconvolution approach to reconstruct the intrinsic expression pattern, and apply it to P. falciparum IDC microarray data.
Methods
We develop a statistical model for the decay of synchrony among cells, and reconstruct the expression pattern through statistical inference. The proposed method can handle microarray measurements with noise and missing data. The original gene expression patterns become more apparent in the reconstructed profiles, making it easier to analyze and interpret the data. We hypothesize that reconstructed gene expression patterns represent better temporally resolved expression profiles that can be probabilistically modeled to match changes in expression level to IDC transitions. In particular, we identify transcriptionally regulated protein kinases putatively involved in regulating the P. falciparum IDC.
Results
By analyzing publicly available microarray data sets for the P. falciparum IDC, protein kinases are ranked in terms of their likelihood to be involved in regulating transitions between the ring, trophozoite and schizont developmental stages of the P. falciparum IDC. In our theoretical framework, a few protein kinases have high probability rankings, and could potentially be involved in regulating these developmental transitions.
Conclusions
This study proposes a new methodology for extracting intrinsic expression patterns from microarray data. By applying this method to P. falciparum microarray data, several protein kinases are predicted to play a significant role in the P. falciparum IDC. Earlier experiments have indeed confirmed that several of these kinases are involved in this process. Overall, these results indicate that further functional analysis of these additional putative protein kinases may reveal new insights into how the P. falciparum IDC is regulated.
Keywords
- Microarray Data
- Microarray Experiment
- Stage Parasite
- Normalize Life Span
- Infection Period
Introduction
Approximately 40% of the global population is at risk for contracting malaria, and an estimated 780,000 people die annually from this disease [1]. Human malaria is caused by five Plasmodium species, of which P. falciparum is responsible for the majority of human fatalities. The disease is transmitted when an infected mosquito bites a person, and injects sporozoites that migrate to and develop in the liver before merozoites are released into the bloodstream and invade red blood cells (RBCs) [2]. Within the RBC, P. falciparum undergoes a well-defined developmental cycle (IDC) during a 48-hour period that is characterized by three main stages, namely: rings, trophozoites and schizonts [2]. Schizont-infected RBCs rupture at the end of this cycle to release merozoites that can invade RBCs and reestablish a new IDC. In addition to the morphologic changes that characterize parasite development during the IDC, changes in gene expression accompany [3, 4] and most likely drive this developmental program.
Gene expression during the 48-hour IDC has been densely profiled at 1-hour intervals using microarray technology in an effort to understand how overall gene expression patterns help shape blood stage parasite biology [3, 4]. These studies revealed that the levels of many transcripts are reproducibly high or low at characteristic times within the IDC. Genes involved in key biological processes most relevant to a given IDC stage are generally coordinately up- and down- regulated. In fact, a 'just-in-time' model to describe transcriptionally regulated gene expression in P. falciparum has been proposed [4, 5]. Here, a transcript's level is proposed to peak just prior to when its encoded protein product is most critically needed. Regulation of general biological processes such as metabolism, DNA synthesis, protein turnover and red blood cell invasion, for example, is well-described by this model [3, 4]. While the full complement of parasite proteins controlling IDC progression has not been identified, the just-in-time principle could be useful for identifying key, transcriptionally regulated proteins that play important roles in regulating this process.
Protein kinases represent one such protein class, and have previously been implicated in regulating various aspects of the cell cycle and development in P. falciparum [6–8]. The genome encodes a predicted 85 [9] or 99 [10] protein kinases, which include an expanded and divergent FIKK family, of which there are 20 members [9, 11]. In Plasmodium spp., several protein kinases have been shown to be important in regulating blood stage biology, including parasite egress from infected RBCs [6–8]. Recently, a large-scale knockout screening effort identified 36 out of the 65 protein kinases evaluated (no FIKKs included) as likely essential to the development of P. falciparum blood stage parasites [12]. Overall, these studies highlight the important contribution of the protein kinases to regulating critical aspects of P. falciparum biology during the IDC.
Therefore, we have been interested in addressing whether computationally applying the just-in-time principle to publicly available microarray data for P. falciparum is a reasonably efficient strategy for identifying protein kinases that are important to transition through the IDC. Such an approach, if successful, could prioritize protein candidates for more exhaustive experimental analysis. Two fundamental assumptions underlie our analytical framework, namely: (1) an important subset of protein kinases regulating transition through the IDC is transcriptionally regulated, and this is captured in the publicly available microarray data; and (2) increases in protein kinase transcript levels predictably precede peak protein synthesis, the latter defining the time at which a given protein kinase plays its critical role in IDC progression. Our approach requires explicitly addressing the confounding factor of decaying synchrony of parasite cultures in order to improve the reliability of predictions. For microarray experiments, parasite cultures are initially synchronized. However, these cultures gradually lose synchrony over the experimental course [3]. Consequently, gene expression levels at discrete time points for individual parasites are not directly inferred from the observed microarray data, which reflect an ensemble of increasingly asynchronous parasite transcription profiles. To address this issue, we introduce a deconvolution approach for reconstructing the intrinsic gene expression pattern of individual parasites from microarray data. This work was partly presented in our previous paper [8]. We identify and account for three main factors driving asynchrony in gene expression profiles during a microarray experiment, namely: (1) diversity of infection time; (2) diversity of growth rate; and (3) the emergence of early stage parasites intermixed with later stage parasites. By including these considerations into our computational framework, we are able to more accurately reconstruct single cell, global gene expression profiles, from which the temporal expression profile for individual protein kinases are determined.
Methods
A publicly available microarray data set on three P. falciparum strains (HB3, 3D7 and Dd2) is used in this work [4, 13]. In the HB3 data set, the expression profiles of 4345 oligonucleotide sequences have been measured at 48 time points with 1 hour interval. The 23^{rd} and 29^{th} data points are missing for all oligonucleotide sequences. The oligonucleotide sequences associated with protein kinases involved in the P. falciparum life cycle are retrieved from the PlasmoDB database. Some protein kinases have several unique oligonucleotide sequences. For those protein kinases, an average trace is calculated from the curves associated with each oligonucleotide sequence. In this fashion, the gene expression profiles of 65 protein kinases are collected from the data set of HB3. Along the same lines, 52 protein kinases and 51 protein kinases are collected from the data set 3D7 and Dd2 respectively. In the rest of this section, we present a computational method to extract the intrinsic gene expression pattern from the microarray data.
Explanations of symbols
Symbols | Explanations |
---|---|
M | the total amount of cells in the media, it consists of RBCs and iRBCs |
S(t) | the number of schizonts which infect RBCs at time t |
R(t) | the number of fresh RBCs infected by schizont at time t |
a _{in} | the average number of RBCs infected by one schizont during the infection period |
a _{af} | the average number of RBCs infected by one schizont after the infection period |
$\stackrel{\u0303}{L}$ | the normalized life span of individual iRBCs |
${p}_{\stackrel{\u0303}{L}}\left(l\right)$ | the probability density function of normalized life span $\stackrel{\u0303}{L}$ |
L | the average life span of iRBCs |
L' | the individual life span of iRBCs |
ℓ | the cell age of iRBC in hours |
{f _{ i }(ℓ), ℓ ∈ [0, L]} | the intrinsic gene expression pattern of protein i |
ℓ_{re} | the rescaled cell age according to its normalized life span $\stackrel{\u0303}{L}$ |
{f _{ i }(ℓ_{re}), ${\ell}_{\mathsf{\text{re}}}=\stackrel{\u0303}{L}\ell $, ℓ ∈ [0, L']} | the gene expression profile of individual iRBC on protein i |
S _{ f } (t) | the number of fast-growing iRBCs which infect RBCs at time t |
R _{ f } (t) | the number of RBCs infected by fast-growing iRBCs at time t |
N(t) | the total number of iRBCs that have been infected at time t |
N(t, ℓ_{re}) | the number of iRBCs which reach rescaled cell age ℓ_{re} at time t |
e _{ i }(t) | the observed expression level of protein i at time t |
${\stackrel{\u0303}{f}}_{i}\left(\ell \right)$ | the normalized gene expression pattern on protein i |
s | the transitions between three stages of iRBC: ring, trophozoite, and schizont |
L _{ i } ( s ) | the likelihood that protein kinases i is involved in regulating the stage transition s |
T _{s} | the time point when stage transition s occur |
n | the average number of merozoites released by one schizont |
p | the probability that one merozoites does not infect any RBC |
V | the volume of merozoites can travel after it is released from schizont |
V _{total} | the volume of whole media |
m | the total number of RBCs in the media |
Analysis of decaying synchrony
We assume that three main factors drive the iRBCs out of synchronization in the microarray experiment, namely: (A) diversity of infection time, (B) diversity of growth rate, and (C) the emergence of early stage parasites intermixed with later stage parasites. These are discussed below.
Diversity of infection time
Let R(t) denote the number of fresh RBCs infected by schizont at time t (hours). In the perfectly synchronized case, R(t) should be a Dirac delta function, which means all iRBCs are simultaneously infected at the same time. In microarray experiment, however, R(t) has a high value during the invasion period, and it maintains positive as long as schizonts remain.
At the end of this section, we explain how to estimate the parameters a _{in}, a _{af} and function S(t).
Diversity of growth rate
where the value of σ is adjusted to fit the experimental observations. The details will be discussed later in this paper.
Let {f _{ i }(ℓ), ℓ ∈ [0, L]} be the intrinsic gene expression pattern of protein i on one complete life span, where ℓ denotes the cell age of iRBC (hours). Since f _{ i }(ℓ) represents the common pattern shared by individual RBC, the expression profile of individual iRBC is assumed to be $\left\{{f}_{i}\left({\ell}_{\mathsf{\text{re}}}\right),{\ell}_{\mathsf{\text{re}}}=\ell /\stackrel{\u0303}{L},\ell \in \left[0,{L}^{\prime}\right]\right\}$, where ℓ_{re} denotes the rescaled cell age according to its normalized life span $\stackrel{\u0303}{L}$ For instance, one iRBC has been infected for ℓ hours, its current expression level on the protein i is written as ${f}_{i}\left(\ell /\stackrel{\u0303}{L}\right),$ where $\stackrel{\u0303}{L}$ is the normalized life span of the corresponding RBC.
The emergence of early stage parasites intermixed with later stage parasites
The invasion factor a _{af} stands for the average number of fresh RBCs that will be infected by one schizont after the invasion period.
Simulation of iRBCs population distribution
where S(t) stands for the number of late-schizonts bursts at time t, R(t) denotes the number of RBCs infected by late-schizonts at time t, and R _{ f } (t) is the number of RBCs which are infected by fast-growing iRBCs at time t. The expressions of R(t) and R _{ f } (t) are given by (1) and (7) respectively. The expression of S(t) will be derived in (25).
Modeling of gene expression level
The observation matrix A can be calculated by means of the equation (12). The element of matrix A at row t and column ℓ_{re} denotes the number of iRBCs that reach the rescaled cell age l _{re} at time point t. The constant vector b stands for the gene expression levels observed in microarray experiment. The unknown variable vector x is the intrinsic gene expression pattern of individual cell. We can find x by solving the discrete linear inverse problem (15).
Reconstruction of synchronized expression profile
We solve (16) numerically by means of the function f mincon in the Optimization Toolbox of Matlab (MATLAB 7.9, The MathWorks Inc.)
Prediction of protein kinases
As we discussed earlier, an increased gene expression level before a cell stage transition is regarded as a sign that the corresponding protein kinase is involved in that stage transition. We denote the likelihood that protein kinases i is involved in regulating the stage transition s by L _{ i }(s). Let T _{ s } stand for the time point when stage transition s occur. The expression of likelihood L _{ i }(s) is derived in following.
Consequently, protein kinase i can be prioritized by its likelihood L _{ i }(s) of being involved in a stage transition.
Parameter estimation
The proposed method for reconstructing expression patterns consists of two main steps. First, the linear system described in (15) is built by calculating the cell age distribution N(t, ℓ_{re}) (12) for each element of the observation matrix A. Second, the expression pattern is reconstructed by solving the discrete linear inverse problem (16). Therefore, the performance of our method mainly depends on the calculation of N(t, ℓ_{re}). As shown in equation (12), N(t, ℓ_{re}) is dominated by three sets of parameters: the infection factors a _{in} and a _{af}, the burst rate of schizonts S(t), and the standard deviation σ of normalized life span $\stackrel{\u0303}{L}$.
Infection factors
After the invasion period, the cell concentration (both fresh and infected) is reduced from 14% to 3.3%. To estimate the value of a _{af}, we propose a simple model to describe how infection factor is influenced by the cell concentration.
At the end of the life span, the cell membrane of schizont bursts, and merozoites are released to infect other RBCs [14]. We denote by p the probability that a merozoite does not infect any RBC, and let n represent the average number of merozoites released by one schizont. The value of n is estimated to be 14, because one schizont usually contains 12 to 16 merozoites [14]. Therefore, the average number of RBCs infected by one bursted schizont is given by n(1 - p).
The derivation of V is related to the concept of mean free paths in physics, roughly the diffusion length for the first binding event or the lifetime [16].
Burst rate of schizonts
Standard deviation of normalized life span
Given a _{in}, a _{af}, and S(t), the iRBCs population distributions N(t, ℓ_{re}) (12) depend on the probability density function ${p}_{\stackrel{\u0303}{L}}$(l) of normalized life span $\stackrel{\u0303}{L}$ (3). The normalized life span $\stackrel{\u0303}{L}$ has been assumed to follow a normal distribution. By definition, the mean of $\stackrel{\u0303}{L}$ is equal to 1. The standard deviation σ is estimated such that the resulting iRBCs population distributions N(t, ℓ_{re}) are in agreement with the experimental data.
The percent of trophozoite and schizont can be calculated similarly.
Results on synthetic data
In this section, the proposed method is evaluated on synthetic microarray data.
Generate synthetic microarray data
The microarray experiments measure the superposition across cells. Due to the decay of synchrony among cells, the intrinsic expression pattern is blurred in microarray data. By comparing the intrinsic expression patterns and synthetic microarray data shown in Figure 10, we demonstrate how decaying synchrony can blur the expression pattern.
Tolerance to signal noise and missing data points
Due to the assay complexity in microarray experiments, signal noise [17] and missing time points [18] are common phenomena observed in real microarray data. To evaluate the performance of our method under similar conditions, expression patterns are reconstructed from synthetic microarray data contaminated by Gaussian white noise and/or missing data points. The resulting expression patterns are compared to the original expression patterns (A, B, C and D).
Sensitivity to model parameters
The performance of our method mainly depends on the calculation of cell age distribution N(t, ℓ_{re}). As shown in equation (12), N(t, ℓ_{re}) depends on three sets of parameters: the infection factors a _{in} and a _{af}, the bufirst rate of schizonts S(t), and the standard deviation σ of normalized life span $\stackrel{\u0303}{L}$. In the experiments of the previous section, the same parameters are used to generate synthetic data and to reconstruct expression patterns. However, in real microarray data, these parameters are unknown, and need to be estimated from experimental observations. The infection factors a _{in}, a _{af}, and burst rate S(t) of schizonts can be accurately calculated from parasitemia and percent representation of iRBC. It is difficult to precisely estimate the standard deviation σ of normalized life span $\stackrel{\u0303}{L}$. In this section, we investigate how results vary with regards to the choice of σ.
Results on real data
Expression patterns are reconstructed for 68 protein kinases collected from the microarray data of P. falciparum (HB3, 3D7 and Dd2) [4, 13]. Reconstructed expression patterns are substituted into the equation (18) to estimate the likelihood of each protein kinase being associated with a specific IDC transition, and hence, contributing to effecting either the stage transition itself or a process(es) needed in the subsequent stage. Data for the three broad transitions analyzed, namely ring-to-trophozoite, trophozoite-to-schizont and schizont-to-ring, are summarized in Additional Files 1, 2 and 3, respectively.
A primary motivation for developing this computational framework is to prioritize gene candidates with potentially important stage-dependent functions for detailed downstream experimental analysis of gene function. In the ring-to-trophozoite analysis, several members of the largely unstudied FIKK protein kinases emerge with relatively high probabilities of mediating important biology during this developmental transition. Several of these protein kinases are targeted to the infected RBC cytosol/membrane [19]. Two [MAL7P1.144, PFL0040c] have been previously studied using gene knockout approaches [20]. While non-essential to blood stage parasite growth and survival, these proteins help mediate the increased rigidity of infected RBCs observed in trophozoite stage parasites. Presumably, this requires modulation of the RBC cytoskeleton through a combination of RBC cytoskeletal and/or exported parasite protein phosphorylation and increased interactions between these [20]. The analysis here suggests that the other highly ranked family members (see Additional File 1) could also be mediating important yet unknown biology at this ring-trophozoite transition.
Interestingly, in the trophozoite-schizont and schizont-ring analyses, a number of protein kinases previously established to be essential and in some cases implicated in directly impacting the transition emerge with the highest probability rankings (Additional Files 2 and 3) [6–8, 12]. In the schizont-ring analysis, for example, PFB0815 has previously been implicated in parasite motility/invasion/egress [6], and PF13_{-}0211 in parasite egress from the RBC [7]. Furthermore, MAL13P1.278 is an essential Aurora kinase (Pfark1) that associates with spindle pole bodies during parasite schizogeny and is implicated in cell cycle regulation [8]. Overall, the top ranked genes in the trophozoite-schizont and schizont-ring analyses are highly enriched for protein kinases previously established to be essential to parasite survival. Therefore, it will be intriguing to experimentally examine the other highly ranked genes in both transition categories [PFI1415w, PF11_{-}0079, PF10_{-}0380, PF11_{-}0510, PFI0095c, PFE0045c and PFC0945w] as these may also play critical roles in regulating parasite development during these stages and could provide new opportunities for antimalarial drug development.
Conclusions
This study proposes a new methodology to reconstruct intrinsic expression patterns from microarray data. We derive a linear system that relates the microarray data to the expression patterns. By solving the corresponding linear inverse problem, the expression patterns are reconstructed. The experiments conducted on synthetic data suggest that the proposed method can reliably reconstruct the expression pattern, even though both signal noise and missing data points contaminate the microarray data. By applying this method to P. falciparum microarray data, protein kinases are prioritized in terms of their likelihood of being involved in regulating some aspect(s) of the IDC. Indeed, the results of our analyses are supported by previously published experimental data confirming the involvement of several protein kinases in regulating parasite biology at or between developmental stage transitions in the IDC. Our results indicate that experimentally investigating the function of other putative protein kinases not previously studied but ranked highly in our analysis may provide new insights into P. falciparum biology.
Declarations
Acknowledgements
This work was supported by the Singapore-MIT Alliance (SMA3) Graduate Fellowship granted to Z.W., and DOD ARO (Grant Number W911NF-09-1-0480).
This article has been published as part of Proteome Science Volume 10 Supplement 1, 2012: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2011: Proteome Science. The full contents of the supplement are available online at http://www.proteomesci.com/supplements/10/S1.
Authors’ Affiliations
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